On Bondage Numbers of Graphs -- a survey with some comments

The bondage number of a nonempty graph $G$ is the cardinality of a smallest edge set whose removal from $G$ results in a graph with domination number greater than the domination number of $G$. This lecture gives a survey on the bondage number, including the known results, problems and conjectures. We also summarize other types of bondage numbers.


Introduction
For terminology and notation on graph theory not given here, the reader is referred to [117]. Let G = (V, E) be a finite, undirected and simple graph. We call |V | and |E| the order and size of G, and denote them by υ = υ(G) and ε = ε(G), respectively. A dominating set S is called a γ-set of G if |S| = γ(G).
The domination is so an important and classic conception that it has become one of the most widely studied topics in graph theory, and also is frequently studied property of networks. The domination, with its many variations, is now well studied in graph and networks theory. A thorough study of domination appears in [46,47]. However, the problem determining domination number for general graphs was early proved to be NP-complete (see GT2 in Appendix in Garey and Johnson [37], 1979).
Among various problems related with the domination number, some focus on graph alterations and their effects on the domination number. Here we are concerned with a particular graph alternation, the removal of edges from a graph.
Graphs with domination numbers changed upon removal of an edge were first investigated by Walikar and Acharya [114] in 1979. A graph is called edge domination-critical graph if γ(G − e) > γ(G) for every edge e ∈ E(G). The edge domination-critical graph was were characterized by Bauer et al. [7] in 1983, that is, a graph is edge dominationcritical if and only if it is the union of stars. The proof is simple. The sufficiency is clear. Suppose that S is a γ-set of G. Then every vertex of degree at least two must be in S, and no two vertices in S can be adjacent. Hence G is a union of stars.
However, for lots of graphs, the domination number is out of the range of one-edge removal. It is immediate that γ(H) γ(G) for any spanning subgraph H of G. Every graph G has a spanning forest T with γ(G) = γ(T ) and so, in general, a graph will have a nonempty set of edges F ⊆ E(G) for which γ(G − F ) = γ(G).
Then it is natural for the alternation to be generalized to the removal of several edges, which is just enough to enlarge the domination number. That is the idea of the bondage number.
A measure of the efficiency of a domination in graphs was first given by Bauer et al. [7] in 1983, who called this measure as domination line-stability, defined as the minimum number of lines (i.e. edges) which when removed from G increases γ.
In 1990, Fink et al. [31] formally introduced the bondage number as a parameter for measuring the vulnerability of the interconnection network under link failure. The minimum dominating set of sites plays an important role in the network for it dominates the whole network with the minimum cost. So we must consider whether its function remains good under the with attack. Suppose that someone such as a saboteur does not know which sites in the network take part in the dominating role, but does know that the set of these special sites corresponds to a minimum dominating set in the related graph. Then how many links does he have to attack so that the cost can not remains the same in order to dominate the whole network? That minimum number of links is just the bondage number.
The bondage number b(G) of a nonempty undirected graph G is the minimum number of edges whose removal from G results in a graph with larger domination number. The precise definition of the bondage number is defined as follows.
Since the domination number of every spanning subgraph of a nonempty graph G is at least as great as γ(G), the bondage number of a nonempty graph is well defined.
We call such an edge set B that γ(G−B) > γ(G) the bondage set and the minimum one the minimum bondage set. In fact, if B is a minimum bondage set, then γ(G−B) = γ(G) + 1, because the removal of one single edge can not increase the domination number by more than one. If b(G) does not exist, for example empty graphs, we define b(G) = ∞.
It is quite difficult to compute the exact value of the bondage number for general graphs since it strongly depends on the domination number of the graphs. Much work focused on the bounds of the bondage number as well as the restraints on particular classes of graphs. The purpose of this lecture is to give a survey of results and research methods related to these topics for graphs and digraphs. For some results, we will give detailed proofs. For some results and research methods, we will make some comments to develop our study further.
The rest of the lecture is organized as follows. Section 2 gives some preliminary results and complexity. Section 3 and Section 4 survey the study on the upper bounds and lower bounds, respectively. The results for some special classes of graphs and planar graphs are stated in Section 5 and Section 6, respectively. In Section 7, we introduce some results on crossing number restraints. In Section 8 and Section 9, we are concerned about other and generalized types of bondage numbers, respectively. In Section 10, we introduce some results for digraphs. In the last section we introduce some results for vertex-transitive graphs by applying efficient dominating sets.

Simplicity and Complexity
As we have known from Introduction, the bondage number is an important parameter for measuring the stability or the vulnerability of a domination in a graph or a network. Our aim is to compute the bondage number for any given graphs or networks. One has determined the exact value of the bondage number for some graphs with simple structure. For arbitrarily given graph, however, it has been proved that determining its bondage number is NP-hard.

Exact Values for Ordinary Graphs
We begin our investigation of the bondage number by computing its value for several well-known classes of graphs with simple structure. In 1990, Fink et al. [31] proposed the concept of the bondage number, and completely determined the exact values of bondage numbers of some ordinary graphs, such as complete graphs, paths, cycles and complete multipartite graphs.
By definition, to compute the exact value of bondage number for a graph strongly depends upon its domination number. It is just that the domination numbers for these graphs can be easily determined, Fink et al. [31] determined the exact values of bondage number for these graphs when they proposed the concept of the bondage number.
Theorem 2.1.1 (Fink et al. [31], 1990) (a) For a complete graph K n of order n 2, b(K n ) = n 2 ; (b) For a path P n of order n 2, b(P n ) = 2 if n ≡ 1(mod 3), 1 otherwise; (c) For a cycle C n of order n, b(C n ) = 3 if n ≡ 1(mod 3), 2 otherwise; (d) For a complete t-partite graph G = K n 1 ,n 2 ,...,nt with n 1 n 2 · · · n t and n t > 1, if n j = 1 and n j+1 2, for some j, 1 j < t, 2t − 1 if n 1 = n 2 = · · · = n t = 2, Proof. We give the proof of the assertion (a). Clearly, γ(K n ) = 1. Let H is a spanning subgraph of K n obtained by removing fewer than ⌈ n 2 ⌉ edges from K n . Then H contains a vertex of degree n − 1, which can dominate all other vertices, and hence γ(H) = 1. Thus, b(K n ) ⌈ n 2 ⌉. If n is even, the removal of a prefect matching from K n reduces the degree of each vertex to n − 2 and therefore yields a graph H with γ(H) = 2. If n is odd, the removal of a prefect matching from K n leaves a graph having exactly one vertex of degree n − 1; by removing one edge incident with this vertex, we obtain a graph H with γ(H) = 2. In both cases, we can a spanning subgraph H by removal of ⌈ n 2 ⌉ edges from K n with γ(H) = 2. This implies b(K n ) ⌈ n 2 ⌉. Thus, b(K n ) = ⌈ n 2 ⌉. Now, we show the assertion (b). Since γ(C n ) = γ(P n ) = n 3 for n 3, we see that b(C n ) 2.
As an immediate corollary to the assertion (b), we have the assertion (c). The proof of the assertion (d) is left to the reader as an exercise. Theorem 2.1.1 shows b(K n ) = n 2 for an (n − 1)-regular graph K n of order n 2, b(G) = n − 1 for an (n − 2)-regular graph G of order n 2, where G is a t-partite graph K n 1 ,n 2 ,...,nt with n 1 = · · · = n t = 2 and t = n 2 for an even integer n ≥ 4. For an (n − 3)-regular graph G of order n ≥ 4, Hu and Xu [57] obtained the following result.
The exact value of bondage number for a general graph, there is a result as follows.
Proof. Let S be the γ-set of G, and let x / ∈ S. Furthermore, let y ∈ N G (x) ∩ S. If N G (x) ∩ S| 2 for each vertex x / ∈ S, then S ′ = (S − {y}) ∪ {x} dominates G and |S ′ | = |S|, so that S is a γ-set of G as well, which is a contradiction to the uniqueness of S. Thus, |N G (x) ∩ S| = 1 for a vertex x / ∈ S. Then γ(G − xy) > γ(G), which implies that b(G) = 1.
The following result is easy to verify. Theorem 2.1.4 (Bauer et al. [7], 1983) If any vertex of a graph G is adjacent with two or more pendant vertices, then b(G) = 1.
Bauer et al. [7] observed that the star is the unique graph with the property that the bondage number is 1 and the deletion of any edge results in the domination number increasing. Hartnell and Rall [45] concluded by determining when this very special property holds for higher bondage number. A graph is called to be uniformly bonded if it has bondage number b and the deletion of any b edges results in a graph with increased domination number. Theorem 2.1.5 (Hartnell and Rall [45], 1999) The only uniformly bonded graphs with bondage number 2 are C 3 and P 4 . The unique graph with bondage number 3 that is uniformly bonded is the graph C 4 . There are no such graphs for bondage number greater than 3.
Comments As we mentioned above, to compute the exact value of bondage number for a graph strongly depends upon its domination number. In this sense, studying the bondage number can greatly inspire one's research interesting to dominations. However, determining the exact value of domination number for a given graph is quite difficulty. In fact, even if the exact value of the domination number for some graph is determined, it is still very difficulty to compute the value of the bondage number for that graph. For example, for the hypercube Q n , we have γ(Q n ) = 2 n−1 , but we have not yet determined b(Q n ) for any n 2.
Perhaps Theorem 2.1.3 and Theorem 2.1.4 provide an approach to compute the exact value of bondage number for some graphs by establishing some sufficient conditions for b(G) = b. In fact, we will see later that Theorem 2.1.3 plays an important role in determining the exact values of the bondage numbers for some graphs. Thus, to study the bondage number, it is importance to present various characterizations of graphs with a unique minimum dominating set.

Characterizations of Trees
For trees, Bauer et al. [7] in 1983 from the point of view of the domination line-stability, independently, Fink et al. [31] in 1990 from the point of view of the domination edgevulnerability, obtained the following result. Proof. If υ(T ) = 2 then b(T ) = 1. Assume υ(T ) 3 and let (x 0 , x 1 , . . . , x k ) be a longest path in T . Clearly, d T (x 0 ) = d T (x k ) = 1 and k 2. If d T (x 1 ) = 2 then B = {x 0 x 1 , x 1 x 2 } is a bondage set, and so b(T ) |B| = 2. If d T (x 1 ) > 2, then x 1 is adjacent to another vertex y of degree one, the single edge x 1 y is bondage set, and so b(T ) 1.
It is natural to classify all trees according to their bondage numbers. Fink et al. [31] proved that a forbidden subgraph characterization to classify trees with different bondage numbers is impossible, since they proved that if F is a forest, then F is an induced subgraph of a tree T with b(T ) = 1 and a tree T ′ with b(T ′ ) = 2. However, they pointed out that the complexity of calculating the bondage number of a tree is at most O(n 2 ) by methodically removing each pair of edges.
Even so, some characterizations, whether a tree has bondage number 1 or 2, have been found by several authors, see example [43,105,111].
First we describe the method due to Hartnell and Rall [43], by which all trees with bondage number 2 can be constructed inductively. An important tree F t in the construction is shown in Figure 2. To characterize this construction, we need some terminologies.
1. Attach a path P n to a vertex x of a tree means to link x and one end-vertex of the P n by an edge.

2.
Attach F t to a vertex x means to link x and a vertex y of F t by an edge.
The following are four operations on a tree T : x belongs to at least one γ-set of T (such a vertex exists, say, one end-vertex of P 5 ).
, where x belongs to at least one γ-set of T . Type 4: Attach F t , t 2, to x ∈ V (T ), where x can be any vertex of T .
Let C = {T : T is a tree and T = K 1 , T = P 4 , T = F t for some t 2, or T can be obtained from P 4 or F t (t 2) by a finite sequence of operations of Type 1, 2, 3, 4}.  [105], 1997) A tree T has bondage number 1 if and only if T has a universal vertex or an edge xy satisfying 1) x and y are neither universal nor idle; and 2) all neighbors of x and y ( except for x and y) are idle.
For a positive integer k, a subset I ⊆ V (G) is called a k-independent set (also called a k-packing) if d G (x, y) > k for any two distinct vertices x and y in I. When k = 1, 1-set is the normal independent set. The maximum cardinality among all k-independent sets is called the k-independence number (or k-packing number) of G, denoted by α k (G).
for every edge e of G. There are two important results on k-independent sets.

Complexity for General Graphs
As mentioned above, the bondage number of a tree can be determined within polynomial time. Indeed, in 1998, Hartnell et at. [41] designed a linear time algorithm to compute the bondage number of a tree. According to this algorithm, we can determine within polynomial time the domination number of any tree by removing each edge and verifying whether the domination number is enlarged according to the known linear time algorithm for domination numbers of trees.
However, it is impossible to find a polynomial time algorithm for bondage numbers of general graphs. If such an algorithm A exists, then the domination number of any nonempty undirected graph G can be determined within polynomial time by repeatedly using A.
we can always find the minimum B i whose removal from G i enlarges the domination number, until As known to all, if NP = P , the minimum dominating set problem is NP-complete, and so polynomial time algorithms for the bondage number do not exist unless NP = P .
In fact, Hu and Xu [64] have recently shown that the problem determining the bondage number of general graphs is NP-hard. We first state the decision problem.
Hu and Xu [64] showed that the bondage problem is NP-hard. The basic way of the proof is to follow Garey and Johnson's techniques for proving NP-hardness [37] by describing a polynomial transformation from the known NP-complete problem: 3satisfiability problem. To state the 3-satisfiability problem, we recall some terms.
Let U be a set of Boolean variables. A truth assignment for U is a mapping t : U → {T, F }. If t(u) = T , then u is said to be " true" under t; if If t(u) = F , then u is said to be" false" under t. If u is a variable in U, then u andū are literals over U. The literal u is true under t if and only if the variable u is true under t; the literalū is true if and only if the variable u is false.
A clause over U is a set of literals over U. It represents the disjunction of these literals and is satisfied by a truth assignment if and only if at least one of its members is true under that assignment. A collection C of clauses over U is satisfiable if and only if there exists some truth assignment for U that simultaneously satisfies all the clauses in C . Such a truth assignment is called a satisfying truth assignment for C . The 3-satisfiability problem is specified as follows.
Question: Is there a truth assignment for U that satisfies all the clauses in C ? Lemma 2.3.2 (Theorem 3.1 in [37]) The 3-satisfiability problem is NP-complete. Theorem 2.3.3 (Hu and Xu [64], 2012) The bondage problem is NP-hard.
Proof. We show the NP-hardness of the bondage problem by transforming the 3satisfiability problem to it in polynomial time.
Let U = {u 1 , u 2 , . . . , u n } and C = {C 1 , C 2 , . . . , C m } be an arbitrary instance of the 3-satisfiability problem. We will construct a graph G and take a positive integer k such that C is satisfiable if and only if b(G) ≤ k. Such a graph G can be constructed as follows.
For each i = 1, 2, . . . , n, corresponding to the variable u i ∈ U, associate a triangle T i with vertex-set {u i ,ū i , v i }. For each j = 1, 2, . . . , m, corresponding to the clause C j = {x j , y j , z j } ∈ C , associate a single vertex c j and add an edge-set E j = {c j x j , c j y j , c j z j }. Finally, add a path P = s 1 s 2 s 3 , join s 1 and s 3 to each vertex c j with 1 j m and set k = 1. Figure 2 shows an example of the graph obtained when U = {u 1 , u 2 , u 3 , u 4 } and To prove that this is indeed a transformation, we must show that b(G) = 1 if and only if there is a truth assignment for U that satisfies all the clauses in C . This aim can be obtained by proving the following four claims. Proof. Let D be a γ-set of G. By the construction of G, the vertex s 2 can be dominated only by vertices in P , which implies |D ∩ V (P )| ≥ 1; for each i = 1, 2, . . . , n, the vertex v i can be dominated only by vertices in T i , which implies |D ∩ V (T i )| ≥ 1. It follows that γ(G) = |D| ≥ n + 1.
Suppose that γ(G) = n + 1. Then |D ∩ V (P )| = 1 and |D ∩ V (T i )| = 1 for each i = 1, 2, . . . , n. Consequently, c j / ∈ D for each j = 1, 2, . . . , m. If s 1 ∈ D, then |D ∩ V (P )| = 1 implies that D ∩ V (P ) = {s 1 }, and so s 3 could not be dominated by D, a contradiction. Hence s 1 / ∈ D. Similarly s 3 / ∈ D and, thus, Proof. Suppose that γ(G) = n + 1 and let D be a γ-set of G. By Claim 3.1, We will show that t is a satisfying truth assignment for C . It is sufficient to show that every clause in C is satisfied by t. To this end, we arbitrarily choose a clause C j ∈ C with 1 j m. Since the corresponding vertex c j in G is adjacent to neither s 2 nor v i for any i with 1 i n, there exists some i with 1 i n such that c j is dominated by u i ∈ D orū i ∈ D. Suppose that c j is dominated by u i ∈ D. Since u i is adjacent to c j in G, the literal u i is in the clause C j by the construction of G. Since u i ∈ D, it follows that t(u i ) = T by (2.3.1), which implies that the clause C j is satisfied by t. Suppose that c j is dominated by ū i ∈ D. Sinceū i is adjacent to c j in G, the literalū i is in the clause C j . Sincē u i ∈ D, it follows that t(u i ) = F by (2.3.1). Thus, t assignsū i the truth value T , that is, t satisfies the clause C j . By the arbitrariness of j with 1 j m, we show that t satisfies all the clauses in C , that is, C is satisfiable.
Conversely, suppose that C is satisfiable, and let t : U → {T, F } be a satisfying truth assignment for C . Construct a subset D ′ ⊆ V (G) as follows. If t(u i ) = T , then put the vertex u i in D ′ ; if t(u i ) = F , then put the vertexū i in D ′ . Clearly, |D ′ | = n. Since t is a satisfying truth assignment for C , for each j = 1, 2, . . . , m, at least one of literals in C j is true under the assignment t. It follows that the corresponding vertex c j in G is adjacent to at least one vertex in D ′ since c j is adjacent to each literal in C j by the construction of G. Thus D ′ ∪ {s 2 } is a dominating set of G, and so γ(G) ≤ |D ′ ∪ {s 2 }| = n + 1. By Claim 3.1, γ(G) ≥ n + 1, and so γ(G) = n + 1. Proof. Let E 1 = {s 2 s 3 , s 1 c j , u iūi , u i v i , : i = 1, 2, . . . , n; j = 1, 2, . . . , m} (induced by heavy edges in Figure 2) and let If e is either s 2 s 3 or incident with the vertex s 1 , then D ′′ is a dominating set of G − e, clearly. If e is either u iūi or u i v i for some i (1 i n), then we use the vertex either v i orū i instead of u i in D ′′ to obtain D ′′′ ; and hence D ′′′ is a dominating set of G − e. These facts imply that γ(G − e) n + 2. Proof. Assume γ(G) = n + 1 and consider the edge e = s 1 s 2 . Suppose γ(G) = γ(G − e). Let D ′ be a γ-set in G − e. It is clear that D ′ is also a γ-set of G. By Claim 3.1 we have c j / ∈ D ′ for each j = 1, 2, . . . , m and D ′ ∩ V (P ) = {s 2 }. But then s 1 is not dominated by D ′ , a contradiction. Hence, γ(G) < γ(G − e), and so b(G) = 1. Now, assume b(G) = 1. By Claim 3.1, we have that γ(G) ≥ n + 1. Let e ′ be an edge such that γ(G) < γ(G − e ′ ). By Claim 3.3, we have that γ(G − e ′ ) n + 2. Thus, n + 1 γ(G) < γ(G − e ′ ) n + 2, which yields γ(G) = n + 1.
By Claim 3.2 and Claim 3.4, we prove that b(G) = 1 if and only if there is a truth assignment for U that satisfies all the clauses in C . Since the construction of the bondage instance is straightforward from a 3-satisfiability instance, the size of the bondage instance is bounded above by a polynomial function of the size of 3satisfiability instance. It follows that this is a polynomial transformation.
The theorem follows.
Comments Theorem 2.3.3 shows that we are unable to find a polynomial time algorithm to determine bondage numbers of general graphs unless NP = P . At the same time, this result also shows that the following study is of important significance.
• Find approximation polynomial algorithms with performance ratio as small as possible.
• Find the lower and upper bounds with difference as small as possible.
• Determine exact values for some graphs, specially well-known networks.
Unfortunately, we can not proved whether or not determining the bondage is NPproblem, since for any subset B ⊂ E(G), it is not clear that there is a polynomial algorithm to verify γ(G−B) > γ(G). Since the problem of determining the domination number is NP-complete, we conjecture that it is not in NP . This is a worthwhile task to study further.
However, Hartnell et at. [41] designed a linear time algorithm to compute the bondage number of a tree. Motivated by this fact, we can made an attempt to consider whether there is a polynomial time algorithm to compute the bondage number for some special classes of graphs such as planar graphs, Cayley graphs, or graphs with some restrictions of graph-theoretical parameters such as degree, diameter, connectivity, domination number and so on.

Upper Bounds
By Theorem 2.3.3, since we can not find a polynomial time algorithm for determining the exact values of bondage numbers of general graphs, it is weightily significative to establish some sharp bounds of the bondage number of a graph. In this section, we survey several known upper bounds of the bondage number in terms of some other graph-theoretical parameters.

Most Basic Upper Bounds
Along with the exact values of bondage numbers for some ordinary graphs computed, several general upper bounds were also derived. In this subsection, we will survey some simple and important upper bounds in terms of the sum of degrees of two vertices with distance 1 or 2. To show the simpleness of these upper bounds, we give their proofs. We start this subsection with an easy observation.  Proof. Let x be a vertex in G such that γ(G − x) γ(G). Then x is not in any γ-set of G, and so it is dominated by some y ∈ N G (x). Let H = G − E x + xy. Then H is a spanning subgraph obtained by removing k edges from G, where k = d(x) − 1. It is clear that the removal of the edge xy from H results in increase of the domination number of H, and so b(H) = 1. The result follows from Lemma 3.1.1 immediately.
The following early result obtained by Bauer et al. [7] and Fink et al. [31], respectively, can be derived from Lemma 3.1.1. This theorem gives a natural corollary obtained by several authors. In 1999, Hartnell and Rall [45] extended Theorem 3.1.3 to the following more general case, which can be also derived from Lemma 3.1.1 by adding "H = G−E x −E y +(x, z, y) if d G (x, y) = 2, where (x, z, y) is a path of length 2 in G " in the above proof of Theorem 3.1.3.
Theorem 3.1.5 (Hartnell and Rall [45], 1999) b(G) d(x) + d(y) − 1 for any distinct two vertices x and y in a graph G with d G (x, y) 2, that is, Corollary 3.1.6 (Fink et al. [31], 1990) If a vertex of a graph G is adjacent with two or more vertices of degree one, then b(G) = 1.
We remark that the bounds stated in Corollary 3.1.4 and Theorem 3.1.5 are sharp. As indicated by Theorem 2.1.1, one class of graphs in which the bondage number achieves these bounds is the class of cycles whose orders are congruent to 1 modulo 3.
On the other hand, Hartnell and Rall [44] sharpened the upper bound in Theorem 3.1.3 as follows, which can be also derived from Lemma 3.1.1.
Then H is a spanning subgraph obtained by removing k edges from G, It is clear that at least one of x and y is in any γ-set of H, and so the removal of e results in increase of the domination number of H, and so b(H) = 1. The result follows from Lemma 3.1.1 immediately.
These results give simple but important upper bounds on the bondage number of a graph, and is also the foundation of almost all results on bondage numbers upper bounds obtained till now.
By careful consideration of the nature of the edges from the neighbors of x and y, Wang [115] further refined the bound in Theorem 3.1.7. For any edge xy ∈ E(G), N G (y) contains the following four subsets. ).
x y T 1 (x, y) The illustrations of T 1 (x, y), T 2 (x, y), T 3 (x, y) and T 4 (x, y) are shown in Figure 3   Proof. Let H = G − E x ∪ T 4 (x, y) + xy. Then H is a spanning subgraph obtained by removing k edges from G, where k = |E x ∪F y |+1 = d(x)+|T 4 (x, y)|−1. Without loss of generality, assume d y (G) d G (x). If y is not in any γ-set of H, then x must be in every γ-set of G, and so y is not in any γ-set of G. By Theorem 3.
Assume that y is in some γ-set of H below. Then x can be dominated by y in H.
Thus, the edge {xy} is a bondage set of H, and so b(H) = 1. By Lemma 3.
The graph shown in Figure 3 shows that the upper bound given in Theorem 3.1.8 is better than that in Theorem 3.1.5 and Theorem 3.1.7, for the upper bounds obtained from these two theorems are d G (x) + d G (y) − 1 = 11 and d G (x) + d G (y) − |N G (x) ∩ N G (y)| = 9, respectively, while the upper bound given by Theorem 3.1.8 is d G (x) + |T 4 (x, y)| = 6.
The following result is also an improvement of Theorem 3.1.3, in which t = 2.
Proof. Let xy ∈ E(K t ), F y be the set of edges incident with y in G but in K t , and let H = G − F y + xy.
Following Fricke et al. [34], a vertex x of a graph G is γ-good if x belongs to some γ-set of G and γ-bad if x belongs to no γ-set of G. Let A(G) be the set of γ-good vertices, and let B(G) be the set of γ-bad vertices in G. Clearly, {A(G), B(G)} is a partition of V (G). Note there exists x ∈ A such that γ(G − x) = γ(G), say, one endvertex of P 5 . Samodivkin [93] presented some sharp upper bounds for b(G) in terms of γ-good and γ-bad vertices of G.
Proof. Notice that if x is an isolated vertex in G then x ∈ A(G), and if γ(G−x) > γ(G) then x is not an isolated vertex and is in every γ-set of G.
(i) Let x ∈ C(G) and let γ(G − x) = γ(G) + p. Then p 0. If p = 0, then b(G) d G (x) by Theorem 3.1.2. Now assume p 1. Then γ(G − x) > γ(G). By the above explanation, it follows that x is in every γ-set of G. Let S be a γ-set of G.
To prove (ii), we only need to prove that γ(G − E G (x, A)) > γ(G). Assume to the contrary that there is some But this is clearly impossible.
Proof. Let x ∈ A + (G) and S be a γ-set of G − x. Then clearly no neighbor of x is in S, which implies Proposition 3.1.12 (Samodivkin [93], 2008) Under the notation of Theorem 3.1.8, if Hence Theorem 3.1.8 can be seen to follow from Theorem 3.1.11. Any graph G with b(G) achieving the upper bound of some of Theorem 3.1.8 can be used to show that the bound of Theorem 3.1.11 is sharp. Let t 2 be an integer. Samodivkin [93] constructed a very interesting graph G t to show that the upper bound in Theorem 3.1.11 is better than the known bounds. Let H 0 , H 1 , H 2 , . . . , H t+1 be mutually vertex-disjoint graphs such that H 0 ∼ = K 2 , H t+1 ∼ = K t+3 and H i ∼ = K t+3 − e for each i = 1, 2, . . . , t. Figure 4 when t = 2.
Observe that γ( Hence each of the bounds stated in theorems 3.1.2 -3.1.9 is greater than or equals t + 2. Consider the graph G t − xy. Clearly γ(G t − xy) = γ(G t ) and Therefore, N Gt (x) ∩ G(G t − y) = {x t+1 } which implies that the upper bound stated in Theorem 3.1.11 is equals to d Gt (y) + |{x t+1 }| = 2. Clearly b(G t ) = 2 and hence this bound is sharp for G t .
From the graph G t , we obtain the following statement immediately. Comments Although Theorem 3.1.11 supplies us with the upper bound that is closer to b(G) for some graph G than what any one of theorems 3.1.2 -3.1.9 provides, it is not easy to determine the sets A + (G) and B(G) mentioned in Theorem 3.1.11 for an arbitrary graph G. Thus the upper bound given in Theorem 3.1.11 is of theoretical importance, but not applied since, until now, we have not found a new class of graphs whose bondage numbers are determined by Theorem 3.1.11.
The above-mentioned upper bounds on the bondage number are involved in only degrees of two vertices. Hartnell and Rall [45] established an upper bound of b(G) in terms of the numbers of vertices and edges of G. For any connected graph G, letδ(G) represent the average degree of vertices in G. Hartnell and Rall first discovered the following proposition.
Proposition 3.1.14 For any connected graph G, there exist two vertices x and y with distance at most two and, with the property that d G (x) + d G (y) 2δ(G).
Using Proposition 3.1.14 and Theorem 3.1.5, Hartnell and Rall gave the following bound.
Proof. Let G be a graph satisfying the hypothesis. By Proposition 3.1.14, there are two vertices x and y with distance at most two and, with the property that d G (x)+d G (y) 2δ(G). By Theorem 3.1.5, we have that from which, we have that 4m(G) = 2 nδ(G) n (b(G) + 1).
That is, b(G) 4m n − 1.  Hartnell and Rall [45] gave examples to show that for each value of b(G), the lower bound given in the Corollary 3.1.17 is sharp for some values of n.
If b(G) = 1, simply take n = 2 (necessary for G to be connected) and G isomorphic to K 2 . If b(G) = 2, consider n = 4 and G isomorphic to P 4 .
For b(G) = k with 2 < k < n 2 , let G be the graph on n = 4m vertices constructed as follows. Start with a k-graph H with order 2m. In fact, if k is even,

Bounds Implied by Connectivity
Use κ(G) and λ(G) to denote the vertex-connectivity and the edge-connectivity of a connected graph G, respectively, which are the minimum numbers of vertices and edges whose removal result in G disconnected. The famous Whitney's inequality can be stated as κ(G) λ(G) δ(G) for any graph or digraph G. Corollary 3.1.4 was improved by several authors as follows.
Proof. Let G be a connected graph with edge-connectivity λ(G) and F be λ-cut of Assume now that any γ-set of H contains all vertices incident with edges in F . Arbitrarily choose an edge xy ∈ F . Then there exists a vertex The upper bound given in Theorem 3.2.1 can be attained. For example, a cycle C 3k+1 of order 3k + 1 with k 1, b(C 3k+1 ) = 3 by Theorem 2.1.
Motivated by Corollary 3.1.4, Theorems 3.2.1 and the Whitney's inequality, Dunbar et al. [27] naturally proposed the following conjecture.
However, Liu and Sun [79] presented a counterexample to this conjecture. They first constructed a graph H showed in Figure 5 with γ(H) = 3 and b(H) = 5. Then let G be the disjoint union of two copies of H by identifying two vertices of degree two. They proved b(G) 5. Clearly, G is a 4-regular graph with κ(G) = 1 and λ(G) = 2, and so b(G) 5 by Theorem 3.2.1. Thus, b(G) = 5 > 4 = ∆(G) + κ(G) − 1. With a suspicion of the relationship between the bondage number and the vertexconnectivity of a graph, the following conjecture is proposed. To the knowledge of the author, until now no results have been known about this conjecture.
Comments We conclude this subsection with following comments.
From Theorem 3.2.1, if Conjecture 3.2.3 holds for some connected graph G, then λ(G) > κ(G) + r, which implies that G is of large edge-connectivity and small vertexconnectivity.
Use ξ(G) to denote the minimum edge-degree of G, that is,  Use λ ′ (G) to denote the restricted edge-connectivity of a connected graph G, which is the minimum number of edges whose removal result in G disconnected and no isolated vertices. Esfahanian and Hakimi [A. H. Esfahanian and S. L. Hakimi, On computing a conditional edge-connectivity of a graph. Information Processing Letters, 27 (1988), 195-199] showed the following result.
Combining Proposition 3.2.4 and Proposition 3.2.5, we propose a conjecture as follows.
For the graph H shown in Figure 5, λ ′ (H) = 4 and δ(H) = 2, and so b( For the 4-regulae graph G constructed by Liu and Sun [79] obtained from the disjoint union of two copies of the graph H showed in Figure 5 by identifying two vertices of degree two, For the 4-regulae graph G t constructed by Samodivkin [93], see Figure 4 for These examples show that if Conjecture 3.2.6 is true then the given upper bound is tight.

Bounds Implied by Degree Sequence
Now let us return to Theorem 3.1.5, from which Teschner [105] obtained some other bounds in terms of the degree sequences. The degree sequence π(G) of a graph G with vertex-set The following result is essentially a corollary of Theorem 3.1.5. Theorem 3.3.1 (Teschner [105], 1997) Let G be a nonempty graph with degree sequence π(G).
Proof. Let I = {x 1 , x 2 , . . . , x t , x t+1 }, the set of vertices corresponding the first t + 1 elements in the degree sequence π(G). If there are two vertices x and y in I with d G (x, y) 2, then the lemma follows by Theorem 3.1.5 immediately. Otherwise, I is a 2-independent set, and so α 2 (G) |I| = t + 1, a contradiction.
In [105], Teschner showed that these two bounds are sharp for arbitrarily many graphs. Hartnell and Rall [45] established an upper bound of the bondage number b(G) in terms of order and the sum of all degrees. For a connected graph G with order υ, let [45], 1999) For a connected graph G, there exists a pair of vertices x and y such that d G (x, y) 2 and d G (x) + d G (y) 2 µ(G).

Proposition 3.3.3 (Hartnell and Rall
Proof. Assume that there is a connected graph G such that the proposition is false.
By our assumption, X is an independent set in G. Hence, each x ∈ X has only vertices in Y as its neighbors. Also each y ∈ Y has at most one vertex of X as its neighbor otherwise, if there were two, they would contradict our assumption. These facts imply that G has a matching M that saturate every vertex in X and |X| = |Y |.
By our assumption, a contradiction. The proposition follows.
Combining Proposition 3.3.3 with Theorem 3.1.5, they obtained the following result. Proof. Let G be a connected graph with order υ and average degree µ. By Proposition 3.3.3, there is such two vertices, say x and y, that The theorem follows. Note that the number of edges 2 ε(G) = υ µ(G). Theorem 3.3.4 implies the following bound in terms of vertex-number υ(G) and edge-number ε(G).
Hartnell and Rall [45] also observed that for each value of b(G), the upper bound given in Eq. (3.3.1) is sharp for some values of υ.
If b(G) = 1 or 2, simply take G = K 2 or P 4 , respectively. For b(G) = k > 2, let H be a circulant undirected graph with order 2m and degree k − 1, and let G be a graph obtained from H by attaching a leaf to each of the 2m vertices. It is easy to see that ε(G) = m(k + 1) and b(G) = k.
Comments Although various of upper bounds have been establish as the above, we find that the appearance of these bounds is essentially based upon the local structures of a graph, precisely speaking, the structures of the neighborhoods of two vertices within distance 2. Even if these bounds can be achieved by some special graphs, it is more often not the case. The reason lies essentially in the definition of the bondage number, which is the minimum value among all bondage sets, an integral property of a graph. While it easy to find upper bounds just by choosing some bondage set, the gap between the exact value of the bondage number and such a bound obtained only from local structures of a graph is often large. For example, a star K 1,∆ , however large ∆ is, b(K 1,∆ ) = 1. Therefore one has been longing for better bounds upon some integral parameters. However, as what we will see below, it is difficult to establish such upper bounds.

Bounds in γ-critical Graphs
A graph G is called a vertex domination-critical graph ( vc-graph or γ-critical for short) if γ(G − x) < γ(G) for any vertex x ∈ V (G), proposed by Brigham, Chinn and Dutton [9] in 1988.
Several families of graphs are known to be γ-critical. From definition, it is clear that if G is a γ-critical graph, then γ(G) 2. The class of γ-critical graphs with γ = 2 is characterized as follows.
Proposition 3.4.1 (Brigham, Chinn and Dutton [9], 1988) A graph G with γ(G) = 2 is a γ-critical graph if and only if G is a complete graph K 2t (t 2) with a perfect matching removed.
The reason why the γ-critical graphs are of special interest in this context is easy to see that they play an important role in the study of the bondage number. For instance, it immediately follows from Theorem 3.
The γ-critical graphs are defined exactly in this way. In order to find graphs G with a high bondage number (i.e. higher than ∆(G)) and beyond its general upper bounds for the bondage number we therefore have to look at γ-critical graphs.
The bondage numbers of some γ-critical graphs have been examined by several authors, see for example [93,94,103,105]. From Theorem 3.1.2 we know that the bondage number of a graph G is bounded from above by ∆(G) if G is not a γ-critical graph. For γ-critical graphs it is more difficult to find an upper bound. We will see that the bondage numbers of γ-critical graphs in general are not even bounded from above by ∆ + c for any fixed natural number c.
In this subsection we introduce some upper bounds for the bondage number of a γ-critical graph. By Proposition 3.4.1, we can easily prove the following result.
In Section 4, by Theorem 4.0.9, we will see the equality in Theorem 3.4.2 holds, Theorem 3.4.3 (Teschner [105], 1997) Let G be a γ-critical graph with degree sequence π(G). Then b(G) max{∆(G) + 1, Let x be a vertex of maximum degree ∆(G). It is easy to see that G−x has a unique minimum dominating set. Let H = G−E x . Then H must also have a unique minimum dominating set. Then by Theorem 2.1.3 we have b(H) = 1, and so b(G) ∆(G) + 1.
As we mention above, if G is a γ-critical graph with γ(G) = 2 then b(G) = ∆ + 1, which shows that the bound given in Theorem 3.4.3 can be attained for γ = 2. However, we have not known whether this bound is tight for general γ 3. Theorem 3.4.3 gives the following corollary immediately.
The following result shows that this bound is not tight.
Until now, we have not known whether the bound given in Theorem 3.4.5 is tight or not. We state two conjectures on γ-critical graphs proposed by Samodivkin [93]. The first of them was motivated by Theorem 3.1.11 and Theorem 3.1.8.
Conjecture 3.4.6 (Samodivkin [93], 2008) For every connected nontrivial γ-critical graph G, min To state the second conjecture, we need the following result on γ-critical graphs.
moreover, if the equality holds, then G is regular.
The bound of Proposition 3.4.7 is the best possible in the sense that equality holds for the infinite class of γ-critical graphs G m,n defined in the beginning of this subsection. In Proposition 3.4.7, the first result is due to Brigham, Chinn and Dutton [9] in 1988; the second is due to Fulman, Hanson and MacGillivray [35] in 1995.
We have not yet known whether the equality in Theorem 3.4.8 holds or not. However, Samodivkin proposed the following conjecture.
In general, based on Theorem 3.4.5, Teschner proposed the following conjecture. Comments We conclude this subsection with some comments.
Graphs which are minimal or critical with respect to a given property or parameter frequently play an important role in the investigation of that property or parameter. Not only are such graphs of considerable interest in their own right, but also a knowledge of their structure often aids in the development of the general theory. In particular, when investigating any finite structure, a great number of results are proven by induction. Consequently it is desirable to learn as much as possible about those graphs that are critical with respect to a given property or parameter so as to aid and abet such investigations.
In this subsection we survey some results on the bondage number for γ-critical graphs. Although these results are not very perfect, it provides a feasible method to approach the bondage number from different angles. In particular, the methods given in Teschner [103] worthily further explore and develop.
The following proposition is maybe useful for us to further investigate the bondage number of a γ-critical graph.
Proposition 3.4.11 (Brigham, Chinn and Dutton [9] Proof. Let y ∈ N G (x). Any minimum dominating set of G − y includes a vertex of N G [v] and hence must also be a minimum dominating set of G.
This simple fact shows that any γ-critical graph contains no vertices of degree one.

Bounds Implied by Domination
In preceding subsection, we introduce some upper bounds of the bondage numbers for γ-critical graphs by consideration of dominations. In this subsection, we introduce related results for general graphs.
Proof. Let x and y be adjacent vertices with υ − 1 − |D| and y / ∈ D. Now, if F y denotes the set of edges from y to a vertex in D, then since y / ∈ D we must have This proves the first assertion. The proofs of other two assertions are omitted here, and left the reader as an exercise.
While the bound b(G) υ − 1 is not particularly good for many classes of graphs (e.g. trees and most cycles), it is an attainable bound. For example, if G is a complete t-partite graph G = K 2,2,...,2 , then the three bounds on b(G) in Theorem 3.5.1 are sharp.
Teschner [103] consider γ(G) = 1 and γ(G) = 2. The next result is almost trivial but useful, the proof is similar to the proof of Theorem 2.1.1 (a).
Lemma 3.5.2 Let G be a graph with order n and γ(G) = 1, t be the number of vertices of degree n − 1. Then b(G) = ⌈t/2⌉.
Since t ∆(G) clearly, Lemma 3.5.2 yields the following result immediately.
Remarks For a complete graph K 2k+1 , b(K 2k+1 ) = k + 1 = 1 2 ∆ + 1, which shows that the upper bound given in Theorem 3.5.3 can be attained. For a graph G with γ(G) = 2, by Theorem 4.0.9 later, the upper bound given in Theorem 3.5.4 can be also attained by a 2-critical graph (see Proposition 3.4.1).

Two Conjectures
In 1990, when Fink et al. [31] introduced the concept of the bondage number, they proposed the following conjecture. Although these results partially support Conjecture 3.6.1, Teschner [102] in 1993 found a counterexample to this conjecture, the cartesian product K 3 × K 3 , as shown in Figure 6, which shows b(G) = ∆(G) + 2. If a graph G is a counterexample to Conjecture 3.6.1, it must be a γ-critical graph by Theorem 3.1.2. It is why the vertex domination-critical graphs are of special interest in the literature. Now we return to Conjecture 3.6.1. Hartnell and Rall [44] and Teschner [104], independently, proved that b(G) can be much greater than ∆(G) by showing the following result.
Theorem 3.6.2 (Hartnell and Rall [44], 1994; Teschner [104], 1996) For an integer n 3, let G n be the cartesian product This theorem shows that there exist no upper bounds of the form b(G) ∆(G) + c for any integer c. Teschner [103] proved that b(G) 3 2 ∆(G) for any graph G with γ(G) 2 (see Theorem 3.5.3 and Theorem 3.5.4) and for some class of graphs with γ(G) = 3, and proposed the following conjecture.
We believe that this conjecture is valid, but so far there are no much work about it.

Lower Bounds
Since the bondage number is defined as the smallest number of edges whose removal results in increase of domination number, each constructive method that creates a concrete bondage set leads to an upper bound on the bondage number. For that reason it is hard to find lower bounds. Nevertheless, there are still a few lower bounds obtained by Teschner [105], the first one of them can be got in terms of its spanning subgraph.    The vertex covering number β(G) of G is the minimum number of vertices that are incident with all edges in G. If G has no isolated vertices, then γ(G) β(G) clearly. In [113], Volkmann gave a lot of graphs with β = γ.
2) It is clear from the proof of 1) that for any bondage set B, E x ⊆ B for some vertex x, and so x is The graph shown in Figure 7 shows that the bound given in  Proposition 4.0.8 (Sanchis [95], 1991) Let G be a graph G of order υ ( 6). If G has no isolated vertices and 3 γ(G) υ/2, then ε(G) Using the idea in the proof of Theorem 4.0.7, every upper bound for γ(G) can lead to a lower bound for b(G). In this way Teschner [105] obtained another lower bound from Proposition 4.0.8. Theorem 4.0.9 (Teschner [105], 1997) Let G be a graph G of order υ ( 6), and 2 γ(G) υ/2 − 1.
Proof. Let G be a graph with 2 γ(G) υ/2 − 1 and B be a minimum bondage set.
If G − B has no isolated vertices, then Proposition 4.0.8 yields that The lower bound in Theorem 4.0.9 is sharp for a class of γ-critical graphs with domination number 2. By Proposition 3.4.1, G is a complete graph K 2t (t 2) with a perfect matching removed. Then b(G) δ(G) + 1 = ∆(G) + 1 by Theorem 4.0.9 and b(G) ∆(G) + 1 by Theorem 3.4.2, and so b(G) = ∆(G) + 1 = 2t − 1.
So far as we know, there are no more lower bounds. In view of applications of the bondage number, a network is vulnerable if its bondage number is small while it is stable if its bondage number is large. Therefore better lower bounds let us learn better stability of networks from this point of view. In our opinion, it is of great significance to seek more lower bounds for various classes of graphs.

Results on Graphs-operations
Generally speaking, it is quite difficult to determine the exact value of the bondage number for a given graph since it strongly depends on the dominating number of the graph. Thus, determining bondage numbers for some special graphs is interesting if the dominating numbers of those graphs are known or can be easily determined. In this section, we will introduce results on bondage numbers for some special classes of graphs.

Cartesian Product Graphs
, or x 2 = y 2 and x 1 y 1 ∈ E(G 1 ). The cartesian product is a very effective method for constructing a larger graph from several specified small graphs. For the Cartesian product C n × C m of two cycles C n and C m , where n 4 and m 3, Klavzar and Seifter [72] determined γ(C n × C m ) for some n' and m's. For example, γ(C n × C 4 ) = n for n 3 and γ( For a general n 4, the exact values of the bondage numbers of C n × C 3 and C n × C 4 were determined as follows.  For larger m and n, Huang and Xu [63] obtained the following result, see Theorem 11.2.2 for more details. Theorem 5.1.4 (Huang and Xu [63], 2008) b(C 5i × C 5j ) = 3 for any positive integers i and j.
Theorem 5.1.5 (Hu, Cao and Xu [54], 2009) For n ≥ 4, From the proof of Theorem 5.1.5, we find that if P n = {0, 1, . . . , n − 1} and P m = {0, 1, . . . , m − 1}, then the removal of the vertex (0, 0) in P n × P m does not change the domination number. If m increase, the effect of (0, 0) for the domination number will be smaller and smaller from the probability. Therefore we expect it is possible that γ(P n × P m − (0, 0)) = γ(P n × P m ) for m ≥ 5 and give the following conjecture.

Block Graphs and Cactus Graphs
In this subsection, we introduce some results for corona graphs, block graphs and cactus graphs.
The corona G 1 • G 2 , proposed by Frucht and Harary [R. Frucht, F. Harary, On the corona of two graphs, Aequationes Math. 4 (1970), 322-324], is the graph formed from a copy of G 1 and υ(G 1 ) copies of G 2 by joining the i-th vertex of G 1 to the i-th copy of G 2 . In particular, we are concerned with the corona H • K 1 , the graph formed by adding a new vertex v i and a new edge u i v i for every vertex u i in H. Carlson and Develin [14] determined the bondage number of H • K 1 . . Similarly, u j ∈ S for j > k, so the only thing we need to check is that u j is adjacent to an element of S for j k. Since d H (u j ) δ(H) k, u j is adjacent to at least one element of {u k+1 , . . . , u n }, completing the proof of this direction.
To show b(G) δ(H) + 1, take any vertex of minimum degree in H, and delete its pendant edge and the pendant edges incident on its δ(H) neighbors. It is easy to see that deletion of these δ(H) + 1 pendant edges of G increases the domination number of the graph.
A block graph is a graph whose blocks are complete graphs. Each block in a cactus graph is either a cycle or a K 2 . If each block of a graph is either a complete graph or a cycle, then we call this graph a block-cactus graph. Teschner [106] first studied the bondage numbers for these graphs.
Theorem 5.2.2 (Teschner [106], 1997) b(G) ≤ ∆(G) for any block graph G Teschner [106] characterized all block graphs with b(G) = ∆(G). At the same paper, Teschner found that γ-critical graphs were instrumental in determining bounds for the bondage number of cactus and block graphs and obtained the following result.  Some upper bounds for block-cactus graphs were also obtained.

Results on Planar Graphs
From Section 2, we have seen that the bondage number for a tree has been completely solved. Moreover, a linear time algorithm to compute the bondage number of a tree was designed by Hartnell et at. [41]. It is quite to consider the bondage number for a planar graph. In this section, we will state some results and problems on the bondage number for a planar graph.
Recall some results on planar graphs used in this section. For any planar graph G, δ(G) 5 and ε(G) 3υ(G) − 6, where υ(G) is the number of vertices and ε(G) is the number of edges in G.
The well-known Euler's formula is stated as follows. For a connected planar graph G, where φ(G) is the number of faces in any embedding of G in the plane or the sphere.  Here, we note that it is sufficient to prove this conjecture for connected planar graphs, since the bondage number of a disconnected graph is simply the minimum of the bondage numbers of its components.

Bounds Implied by Maximum Degree
The first paper attacking this conjecture is due to Kang and Yuan [69], which confirmed the conjecture for every connected planar graph G with ∆ 7. The proofs mainly base on Theorem 3.1.5, Theorem 3.1.7 and the following lemma, which is a simple observation.  Obviously, in view of Theorem 6.2.2, Conjecture 6.1.1 is true for any connected planar graph with ∆ 7.

Bounds Implied by Degree-conditions
As we have seen from Theorem 6.2.2, to attack Conjecture 6.1.1, we only need to consider connected planar graphs with maximum ∆ 6. Thus, studying the bondage number of planar graphs by degree-conditions is of significance. The first result on bounds implied by degree-conditions was obtained by Kang and Yuan [69]. As further applications of Lemma 6.2.1, Fischermann, Rautenbach and Volkmann [33] generalized Theorem 6.3.1 as follows.

Bounds Implied by Girth-conditions
The girth g(G) of a graph G is the length of the shortest cycle in G. If G has no cycles we define g(G) = ∞.
Combining Theorem 6.2.2 with Theorem 6.3.4, we find that if a planar graph contains no triangles and has maximum degree ∆ 5 then Conjecture 6.1.1 holds. This fact motivated Fischermann et al. [33] attempting to attack Conjecture 6.1.1 by grith restraints. They showed that the conjecture is valid for all connected planar graphs of girth g(G) 4 and maximum degree ∆(G) 5 as well as for all not 3-regular graphs of girth g(G) 5. In this subsection, we will introduce their results and some opened problems.
First, they improved the inequality ε 3υ − 6 for a planar graph as follows.
Lemma 6.4.1 If G is a planar graph with finite girth g(G) 3, then where c(G) is the number of cut-edges in G.
Proof. Since every cut-edge is on the boundary of exactly one face and every noncutedge is on the boundary of two faces, we reduce that g(G)φ(G) 2 ε(G) − c(G). Applying Euler's Formula, the result follows.
Then, by Lemma 6.4.1 and considering the relations among υ i 's, the following result for planar graphs with girth restraints is obtained.
Substituting k 4, 5, 6 into (6.4.3), respectively, yields the desired conclusions. The first result in Theorem 6.4.2 shows that Conjecture 6.1.1 is valid for all connected planar graphs with g(G) 4 and ∆(G) 5. It is easy to verify that the second result in Theorem 6.4.2 implies that Conjecture 6.1.1 is valid for all not 3-regular graphs of girth g(G) 5, which is stated the following corollary.  We conclude this subsection with a question on bondage numbers of planar graphs. In 2006, Carlson and Develin [14] showed that the corona G = H • K 1 for a planar graph H with δ(H) = 5 has the bondage number b(G) = δ(H) + 1 = 6 (see Theorem 5.2.1). Since the minimum degree of planar graphs is at most 5, then b(G) can attach 6. If we take H as the graph of the icosahedron (see Figure 8 (a)), then G = H • K 1 is such an example.
The question for the existence of planar graphs with bondage number 7 or 8 remains open.

Comments on the Conjectures
Conjecture 6.1.1 is true for all connected planar graphs with minimum degree δ 2 by Theorem 3.1.5, or maximum degree ∆ 7 by Theorem 6.2.2, or not γ-critical planar graphs by Theorem 3.1.2. Thus, to attack Conjecture 6.1.1, we only need to consider connected critical planar graphs with degree-restriction 3 δ ∆ 6.
Recalling and anatomizing the proofs of all results mentioned in the preceding subsections on the bondage number for connected planar graphs, we find that they strongly depend upon Theorem 3.1.5 or Theorem 3.1.7. In other words, a basic way used in the proofs is to find two vertices x and y with distance at most two in a considered planar graph G such that which bounds b(G), is as small as possible. Let .
Thus, using Theorem 3.1.5 or Theorem 3.1.7, if we can prove Conjecture 6.1.1 and Conjecture 6.4.4, then we can prove the following statement. To construct these counterexamples, we recall the operation of subdividing an edge, i.e, replacing the edge xy by a 2-path xvy through a new vertex v, called the subdividing vertex or s-vertex for short. We say subdividing the edge xy twice if xy is replaced by a 3-path xv 1 v 2 y.   Proof. The existence of H is guaranteed by the icosahedron (see Figure 8 (a)). The construction is showed as Figure 9. Assume V (H) = {u 1 , . . . , u n }. Let S = {v 1 , . . . , v m } be the set of subdividing vertices. Then V (G) = V (H) ∪ S. It is easy to observe that, d G (u i ) = 5 for i = 1, . . . , n for each i = 1, . . . , n, and d G (v j ) = 6 for each j = 1, . . . , m. Furthermore, d G (u i , u j ) = 2 (i = j) and |N G (x) ∩ N G (y)| 2 for every edge xy of G. Thus B(G) = 5 + 6 − 1 − 2 = 8 = ∆ + 2.
Example 6.5.5 For any 3-regular planar graph H with g(H) 4, we can subdivide each edge twice and then link the subdividing vertices properly such that the resulting graph G is a planar graph with ∆(G) = 4, g(G) 4 and B(G) = 6, which disproves the first and the third conclusions in Statement 6.5.3. Proof. The construction is showed as Figure 10 where H is the cube. The resulting graph G is a planar graph with g(G) 4. Note that all subdividing vertices have degree 4 and d G (u, v) 3 for any two vertices u, v ∈ V (H). Thus B(G) = 4 + 3 − 1 = 6. It is easy to verify that the result holds for any 3-regular planar graph. Example 6.5.6 The dodecahedron (see Figure 8 (b)) G is a 3-regular planar graph with g(G) = 5 and B(G) = 5, which disproves the fist and the late conclusions in Statement 6.5.3. Examples 6.5.4 ∼ 6.5.6 disprove Statement 6.5.3. As a result, we can state the following conclusion. Therefore, a new method is need to prove these conjectures, if they are right.

Minimum Counterexamples to the Conjectures
As mentioned in the preceding subsection, now that we can not prove these conjectures, then we may consider to disprove them. If one of these conjectures is invalid, then there exists a minimum counterexample G with respect to υ(G) + ε(G). Huang and Xu [64] investigated the property of the minimum counterexample. Let G 1 be the possible existing minimum counterexamples to Conjecture 6.1.1, G 2 , G 3 and G 4 be the possible existing minimum counterexamples to three conjectures in Conjecture 6.4.4, respectively. From the above discussions, G i is a connected planar graph with 3 ∆ 6 for each i = 1, 2, 3, 4. In particular, • G 1 is γ-critical and b(G 1 ) = ∆(G 1 ) + 2 (by Theorem 3.1.2).
In order to obtain further properties of these minimum counterexamples, we consider how the bondage number changes under some operation of a graph G which decreases υ(G) + ε(G) and preserves planarity. A simplest operation satisfying this requirement is the edge deletion. Lemma 6.6.1 Let G be any graph and e ∈ E(G). Proof. By Lemma 6.6.1, b(G i − e) b(G i ) − 1 for any edge e ∈ E(G i ). Note that G i − e is a planar graph with g(G i − e) g(G i ) and ∆(G i − e) ∆(G i ). Thus, if b(G i −e) b(G i ), then G i −e is also a counterexample, a contradiction to the minimum of G i . Hence b(G i − e) = b(G i ) − 1 for any edge e ∈ E(G).
From Lemma 6.6.2 we obtain the following conclusion immediately. Theorem 6.6.3 It is unable to construct minimum counterexamples to Conjecture 6.1.1 and Conjecture 6.4.4 by the operation of an edge deletion.
Next we consider the effect of the edge contraction on the bondage number. Given a graph G, the contraction of G by the edge e = xy, denoted by G/xy, is the graph obtained from G by contracting two vertices x and y to a new vertex n xy and then deleting all multi-edges. It is easy to observe that υ(G/xy) + ε(G/xy) < υ(G) + ε(G) and G/xy is also planar if G is planar.
First, we investigate the influence of the edge contraction on the domination and the bondage numbers for general graphs. Lemma 6.6.4 Let G be any graph. Then γ(G) − 1 γ(G/xy) γ(G) for any edge xy of G.
Proof. Let S be a γ-set of G. If neither x nor y belongs to S, then S is a dominating set in G/xy. If S ∩ {x, y} = ∅, then (S \ {x, y}) ∪ {n xy } is a dominating set in G/xy, since n xy dominates all neighbors of x and y. It follows that γ(G/xy) |S| = γ(G).
On the other hand, let S ′ be a γ-set in G/xy. If n xy ∈ S ′ , then S = S ′ \{n xy }∪{x, y} is a dominating set of G. If n xy / ∈ S ′ , then S ′ contains a vertex z such that zn xy ∈ E(G/xy). By the definition of edge contraction, zx ∈ E(G) or yz ∈ E(G), which implies that S ′ ∪ {y} or S ′ ∪ {x} is a dominating set of G. Thus γ(G) |S ′ | + 1 = γ(G/xy) + 1.
Theorem 6.6.5 Let G be any graph and xy be any edge in G. If N G (x) ∩ N G (y) = ∅ and γ(G/xy) = γ(G), then b(G/xy) b(G). (6.6.6) Since N G (x) ∩ N G (y) = ∅, the set of edges incident with x and y except xy in G is the set of edges incident with n xy in G/xy. It is easy to verify that Then by Lemma 6.6.4, (6.6.7) and (6.6.6), we have The above examples show that the conditions of Theorem 6.6.5 are necessary. Clearly γ(C n ) = ⌈n/3⌉ and γ(K n ) = 1; for any edge xy, C n /xy = C n−1 and K n /xy = K n−1 . By Example 6.6.6, if n ≡ 1 (mod 3), then γ(C n /xy) < γ(G) and b(C n /xy) = 2 < 3 = b(C n ). Thus the result in Theorem 6.6.5 is generally invalid without the hypothesis γ(G/xy) = γ(G). Furthermore, the condition N G (x) ∩ N G (y) = ∅ can not be omitted even if γ(G/xy) = γ(G), since for odd n, b(K n /xy) = ⌈ n−1 2 ⌉ < ⌈ n 2 ⌉ = b(K n ), by Example 6.6.7.
On the other hand, the above examples also show that the equality in b(G/xy) b(G) may hold (b(C n /xy) = b(C n ) = 2 if n ≡ 0 (mod 3), b(K n /xy) = b(K n ) if n is even). Thus the bound in Theorem 6.6.5 is tight. However, provided all the conditions, b(G/xy) can be arbitrarily larger than b(G). Given a graph H, let G be the graph formed from Now we apply Theorem 6.6.5 to G 2 . Corollary 6.6.8 If γ(G 2 /xy) = γ(G 2 ) for some edge xy, then N G 2 (x) ∩ N G 2 (y) = ∅.
By Theorem 6.4.2, G 2 must contain triangles, and so there is an edge xy with N G 2 (x) ∩ N G 2 (y) = ∅. From Theorem 6.6.5 and Corollary 6.6.8 we obtain the following conclusion immediately. Theorem 6.6.9 It is unable to construct a minimum counterexample to the first conjecture in Conjecture 6.4.4, that is G 2 , by the operation of an edge contraction.
Finally we consider G 1 . Lemma 6.6.10 Let G be any graph and xy be any edge in G. Then γ(G/xy) γ(G − x).
Proof. Let S be a γ-set of G − x. If y / ∈ S, then there exists a vertex z ∈ S such that yz ∈ E(G − x). Thus zn xy ∈ E(G/xy), i.e., n xy is dominated by z ∈ S. Therefore S is also a dominating set of G/xy and γ(G/xy) |S| = γ(G). Now assume y ∈ S and let S ′ = (S \ {y}) ∪ {n xy }. If yz ∈ E(G − x), then zn xy ∈ E(G/xy), which means that the vertices dominated by y in G − x are all dominated by n xy in G/xy. Thus S ′ is a dominating set of G/xy and γ(G/xy) |S ′ | = γ(G). Lemma 6.6.11 γ(G 1 /xy) = γ(G 1 ) − 1 for every edge xy in G 1 .

Results on Crossing Number Restraints
It is quite natural to generalize the known results on the bondage number for planar graphs to for more general graphs in terms of graph-theoretical parameters. In this section, we consider graphs with crossing number restraints.
The crossing number cr(G) of G is the smallest number of pairwise intersections of its edges when G is drawn in the plane. If cr(G) = 0, then G is a planar graph. A spanning subgraph H of G is called a maximum planar subgraph of G if H is planar and contains as many edges of G as possible. we can easily observe the following property of maximum planar subgraphs.

General Methods
Use υ i (G) to denote the number of vertices of degree i in G for i = 1, 2, . . . , ∆(G). By using Lemma 7.0.12 with some more effort of computations, Huang and Xu obtained the following results.   In order to generalized other known results for planar graphs, we make the following construction. For a connected graph G, let G 0 be a subgraph of G without isolated vertices, H 0 be a maximum planar subgraph of G 0 and let After these constructions we obtained a planar graph H k . Then ε(H k ) 3υ(H k ) − 6 = 3υ(G k ) − 6. Applying this inequality, Huang and Xu [62] obtained the following results for graphs with crossing number restraints, which generalize some known results for planar graphs.   Theorem 7.1.8 (Huang and Xu [62], 2007) Let G be a connected graph. If G satisfies 1) 5cr(G) + υ 5 < 2υ 2 + 3υ 3 + 2υ 4 + 12; or 2) 7cr(G) + 2υ 5 < 3υ 2 + 4υ 4 + 24, then b(G) 7.

Carlson and Develin's Methods
In this subsection, we introduce an elegant method presented by Carlson and Develin [14] to obtain some upper bounds for the bondage number of a graph.
Suppose that G is a connected graph. We say that G has genus ρ if G can be embedded in a surface S with ρ handles such that edges are pairwise disjoint except possibly for end-vertices. Let D be an embedding of G in surface S, and let φ(G) denote the number of regions in D. The boundary of every region contains at least three edges and every edge is on the boundary of at most two regions (the two regions are identical when e is a cut-edge). For any edge e of G, let r 1 G (e) and r 2 G (e) be the numbers of edges comprising the regions in D which the edge e borders. It is clear that (7.2.1) Following Carlson and Develin [14], for any edge e = xy of G, we define By the well-known Euler's Formula it is easy to see that If G is a connected planar graph, that is, ρ(G) = 0, then Combining these formulas with Theorem 3.1.7, Carlson and Develin [14] gave a simple and intuitive proof of Theorem 6.2.2 and obtained the following result. Proof. Suppose that G is a graph which can be embedded on a torus, for which ρ(G) = 2. By Corollary 3.1.4, if δ(G) 4 then theorem holds, so we can assume δ(G) 5. For the sake of contradiction, assume b(G) ∆(G) + 4. Let e = xy be an arbitrary edge of G and let, without loss of generality, d G (x) d G (y). By Theorem 3.1.7 we should have, By the way, Cao, Xu and Xu [13] generalized the result in (6.4.1) to a connected graph G that can be embedded on a torus, that is, Several authors used this method to obtain some results on the bondage number. For example, Fischermann et al. [33] used this method to prove the second conclusion in Theorem 6.3.4. Recently, Cao, Huang and Xu [12] have used this method to deal with more general graphs with small crossing numbers. First, they found the following property by Lemma 7.0.12.  By using Carlson and Develin's method, Samodivkin [94] obtained some results on the bondage number for graphs with some given properties.
Kim [71] showed that b(G) ∆(G) + 2 for a connected graph G with genus 1, ∆(G) 5 and having a torodial embedding of which at least one region is not 4-sided. Recently, Gagarin and Zverovich [36] further have extended Carlson and Develin's ideas to establish a nice upper bound for arbitrary graphs that cab be embedded on orientable or nonorientable topological surface. Theorem 7.2.7 (Gagarin and Zverovich [36], 2010) Let G be a graph embeddable on an orientable surface of genus h and a non-orientable surface of genus k. Then b(G) min{∆(G) + h + 2, ∆(G) + k + 1}.
This result generalizes the corresponding upper bounds of Theorems 6.2.2 and Theorems 7.2.1 for any orientable or nonorientable topological surface.
By investigating the proof of Theorem 7.2.7, Huang [59] found that the issue of the orientability can be avoided by using the Euler characteristic χ(= υ(G) − ε(G) + φ(G)) instead of the genera h and k, the relations are χ = 2 − 2h and χ = 2 − k. To have the best result from Theorem 7.2.7, one wants h and k as small as possible, this is equivalent to having χ as large as possible.
According to Theorem 7.2.7, if G is planar (h = 0, χ = 2) or can be embedded on the real projective plane (k = 1, χ = 1), then b(G) ∆(G) + 2. In all other cases, Huang [59] had the following improvement for Theorem 7.2.7, the proof is based on the technique developed by Carlson-Develin and Gagarin-Zverovich, and includes some elementary calculus as a new ingredient, mainly the intermediate value theorem and the mean value theorem.
Theorem 7.2.8 (Huang [59], 2011) Let G be a graph embeddable on a surface whose Euler characteristic χ is as large as possible. If χ 0 then b(G) ∆(G) + ⌊r⌋, where r is the largest real root of the following cubic equation in z: In addition, if χ decreases then r increases.
The following result is asymptotically equivalent to Theorem 7.2.8.

Other Types of Bondage Numbers
Since the concept of the bondage is based upon the domination, all sorts of dominations, which are generalizations of the normal domination or adding restricted conditions to the normal dominating set, can develop a new "bondage number" as long as a variation of domination number is given. In this section, we will survey some results on the bondage number under some restricted conditions. The concept of restrained domination was introduced by Telle and Proskurowski [101] in 1997, albeit indirectly, as a vertex partitioning problem. Here conditions are imposed on a set S, the complementary setS and on edges between the sets S andS. For example, if we require that every vertex inS should be adjacent to some other vertex of S (the condition on the setS) and to some vertex in S (the condition on edges between the sets S andS), then S is a restrained dominating set. Concerning the study of the restrained domination numbers, the reader is referred to [22,23,24,50,101].

Restrained Bondage Numbers
In 2008, Hattingh and Plummer [40] defined the restrained bondage number b r (G) of a nonempty graph G to be the minimum cardinality among all subsets B ⊆ E(G) for which γ r (G − B) > γ r (G).
For some simple graphs, their restrained domination numbers can be easily determined, and so restrained bondage numbers have been also determined. For example, it is clear that γ r (K n ) = 1. Domke et al. [24] showed that γ r (C n ) = n − 2⌊n/3⌋ for n 3, and γ r (P n ) = n−2⌊(n−1)/3⌋ for n 1. Using this results, Hattingh and Plummer [40] obtained the restricted domination numbers for K n , C n , P n and G = K n 1 ,n 2 ,···,nt . For a path P n of order n ( 4), b r (P n ) = 1. For a complete t-partite graph G = K n 1 ,n 2 ,···,nt with n 1 n 2 · · · n t (t 2), if n m = 1 and n m+1 2 (1 m < t); 2t − 2 if n 1 = n 2 = · · · = n t = 2 (t 2); 2 if n 1 = 2 and n 2 3 (t = 2); Proof. We first show the conclusion for a complete graph K n . If n = 3 then γ r (K 3 ) = 1 clearly. Now, removing any edge from K 3 yields P 3 . Since γ r (P 3 ) = 3, it follows that b r (K 3 ) = 1. Let n 4 and let H be a spanning subgraph of K n that is obtained by removing fewer than ⌈n/2⌉ edges from K n . Then H contains a vertex of degree n − 1. Moreover, for every x ∈ V (H), d H (x) 2. Hence, γ r (H) = 1. It follows that b r (K n ) ⌈n/2⌉.
Let H be the graph obtained by removing a perfect matching M from K n . If n is even, then |M| = n/2 = ⌈n/2⌉. Thus, for every x ∈ V (H), d H (x) = n − 2, whence γ r (H) = 2. If n is odd then, then |M| = (n − 1)/2 = ⌈n/2⌉ − 1. there is exactly one vertex x ∈ V (H) such that d H (x) = n − 1. Let H ′ be the graph obtained by removing from H one edge incident with x. It follows that γ r (H ′ ) = 2.
In either case, we can obtain a graph, by removing ⌈n/2⌉ edges from K n , whose restrained bondage number is lager than γ r (K n ). Thus b r (K n ) ⌈n/2⌉, whence b r (K n ) = ⌈n/2⌉.
Thus, b r (P n ) 1. Hence, b r (P n ) = 1. The proof of the last conclusion is little complex and omitted here.  Consequently, it is significative to establish some sharp bounds of the restrained bondage number of a graph in terms of some other graphic parameters.
Suppose to the contrary that b r (G) > b r . Let E xy denote the set of edges that are incident with at least one of x and y, but not both. Then |E xy | = b r and γ r (G − E xy ) = γ r (G) since b r (G) > b r . Since x and y are vertices of degree one in G − E xy , it follows that Since δ(G) 2, it follows that N xy = ∅. If N xy ⊆ R, then R is a restrained dominating set of G of cardinality γ r (G − x − y) = γ r (G) − 2, a contradiction. Hence, N xy R and there is a vertex z ∈ N xy such that z / ∈ R. Without loss of generality, assume z is adjacent to x. Then R ∪ {y} is a restrained dominating set of G of cardinality γ r (G − x − y) + 1 = γ r (G) − 1, a contradiction.
Notice that the bounds stated in Theorem 8.1.5 and Corollary 8.1.6 are sharp. Indeed the class of cycles whose orders are congruent to 1, 2 (mod 3) have a restrained bondage number achieving these bounds (see Theorem 8.1.1). Proof. We proceed by induction on γ r (G). Let γ r (G) = 2, and suppose b r (G)

Remarks
Let e be any edge incident with y, and let H = (V (G − x), E(G − x − e)). Since b r (G − x) 2, it follows that γ r (H) = 1. Hence, there is a vertex z ∈ V (G − x) such that z = y and z is adjacent to every vertex in V (G − x). Since y is the only vertex not in N G (x), we have z ∈ N G (x). Hence, d G (z) = |V (G)| − 1, a contradiction. Thus, b r (G) ∆(G) + 1, for γ r (G) = 2. Now, assume that, for any graph G ′ such that γ r (G ′ ) = k 2, b r (G ′ ) (k − 1)∆(G ′ ) + 1. Let G be a graph such that γ r (G) = k + 1. Suppose to the contrary that b r (G) > k∆(G) + 1. Let x ∈ V (G) and notice that γ r (G − x) = γ r (G) − 1 = k.
We now consider the relation between b r (G) and b(G). Proof. Indeed, assume γ r (G) = γ(G). Let B be a set of edges such that γ(G − B) > γ(G), where |B| = b(G). Then However, we do not have b r (G) = b(G) for any graph G even if γ r (G) = γ(G). Observe that γ r (K 3 ) = γ(K 3 ), yet b r (K 3 ) = 1 and b(K 3 ) = 2. We still may not claim that b r (G) = b(G) even in the case that every γ(G)-set is a γ r (G)-set. The example K 3 again demonstrates this. In fact, there exists a graph G ∈ B such that b r (G) can be much larger than b(G), which is stated as the following theorem.  Comments Corollary 8.1.6 and Theorem 8.1.8 provides a possibility to attack Conjecture 6.1.1 by characterizing planar graphs with γ r = γ and b r = b when 3 ∆ 6.

Total Bondage Numbers
A dominating set S of a graph G without isolated vertices is called to be total if the induced subgraph G[S] contains no isolated vertices. The minimum cardinality of a total dominating set is called the total domination number of G and denoted by γ t (G). It is clear that γ(G) γ t (G) 2γ(G) for any graph G without isolated vertices.
The total domination in graphs was introduced by Cockayne et al. [18] in 1980. Pfaff et at. [87,76] in 1983 showed that the problem determining total domination number for general graphs is NP-complete, even for bipartite graphs, and chordal graphs. Even now, total domination in graphs has been extensively studied in the literature. In 2009, Henning [51] gave a survey of selected recent results on total domination in graphs.
The total bondage number of G, denoted by b t (G), is the smallest cardinality of a subset B ⊆ E(G) with the property that G − B contains no isolated vertices and γ t (G − B) > γ t (G).
From definition, b t (G) may not exist for some graphs, for example, G = K 1,n . We put b t (G) = ∞ if b t (G) does not exist. In fact, b t (G) is finite for any connected graph G other than K 1 , K 2 , K 3 and K 1,n . Since there is a path of length 3 in G, we can find In 1991, Kulli and Patwari [74] first studied the total bondage number of a graph and calculated the exact values of b t (G) for some standard graphs. Recently, Hu, Lu and Xu [55] have obtained some results on the total bondage number of the Cartesian product P m × P n of two paths P m and P n . Theorem 8.2.2 (Hu, Lu and Xu [55], 2009) For the Cartesian product P m × P n of two paths P m and P n , Generalized Petersen graphs are an important class of commonly used interconnection networks and have been studied recently. By constructing a family of minimum total dominating sets, Cao et al. [11] determined the total bondage number of the Generalized Petersen graphs.
From Theorem 2.2.1, we know that b(T ) 2 for a nontrivial tree T . But given any position integer k, Sridharan et al. [99] constructed a tree T for which b t (T ) = k. Let H k be the tree obtained from the star K 1,k+1 by subdividing k edges twice. The tree H 7 is shown in Figure 11. It can be easily verified that b t (H k ) = k. We state this fact as the following theorem.  However, Sridharan et al. [99] gave an upper of the total bondage number of a tree in terms of its maximum degree: For any tree T of order n, if T == K 1,n−1 , then b t (T ) = min{∆(T ), 1 3 (n − 1)}. Rad and Raczek [91] improved this upper and gave a constructive characterization of a certain class of trees attaching the upper bound. In general, the decision problem for b t (G) is NP-complete for any graph G. We state the decision problem for the total bondage as follows. Consequently, it is significative to establish some sharp bounds of the total bondage number of a graph in terms of some other graphic parameters.
Theorem 8.2.9 Let G be a graph with order υ 5. Then

Rad and
Raczek [91] also established some upper bounds of b t (G) for a general graph G. In particular, they gave an upper bound of b t (G) in terms of the girth of a graph.

Paired Bondage Numbers
A dominating set S of G is called to be paired if the induced subgraph G[S] contains a perfect matching. The paired domination number of G, denoted by γ p (G), is the minimum cardinality of a paired dominating set of G. Clearly, γ t (G) γ p (G) for every connected graph G with order at least two, where γ t (G) is the total domination number of G, and γ p (G) 2γ(G) for any graph G without isolated vertices. Paired domination was introduced by Haynes and Slater [48,49], and further studied in [53,88,89,96].
The paired bondage number of G with δ(G) 1, denoted by b p (G), is the minimum cardinality among all sets of edges B ⊆ E such that δ(G − B) 1 and γ p (G − B) > γ p (G).
The concept of the paired bondage number was first proposed by Raczek [90] in 2008. The following observations follow immediately from the definition of the paired bondage number. (c) If xy ∈ E(G) such that d G (x), d G (y) > 1, and xy belongs to each perfect matching of each minimum paired dominating set of G, then b p (G) = 1.
Based on these observations, b p (P n ) 2. In fact, the paired bondage number of a path has been determined. Foe a cycle C n of order n 3, since γ p (C n ) = γ p (P n ), from Theorem 8.3.2 we obtain the following result. A wheel W n , where n 4, is a graph with n vertices, formed by connecting a single vertex to all vertices of a cycle C n−1 . Of course γ p (W n ) = 2. Theorem 3.1.5 shows that, if T is a non-trivial tree, then b(T ) 2. However, no similar result exists for paired bondage. For any non-negative integer k, let T k be a tree obtained by subdividing all but one edge of the star K 1,k+1 (as shown in Figure 12). It is easy to see that b p (T k ) = k. We state this fact as the following theorem.  A constructive characterization of trees with b(T ) = 2 is given by Hartnell and Rall in [43]. Raczek [90] provided a constructive characterization of trees with b p (T ) = 0. In order to state the characterization, we define a labeling and three simple operations on a tree T . Let y ∈ V (T ) and let ℓ(y) be the label assigned to y.
Let P 2 = (u, v) with ℓ(u) = A and ℓ(v) = B. LetT be the class of all trees obtained from the labeled P 2 by a finite sequence of Operations T 1 , T 2 , T 3 .
A tree T in Figure 13 belongs to the family T . Raczek [90] obtained the following characterization of all trees T with b p (T ) = 0. We state the decision problem for the paired bondage as follows.

Independence Bondage Numbers
A subset I ⊆ V (G) is called an independent set if no two vertices in I are adjacent in G. The maximum cardinality among all independent sets is called the independence number of G, denoted by α(G). A dominating set S of a graph G is called to be independent if S is an independent set of G. The minimum cardinality among all independent dominating set is called the independence domination number of G and denoted by γ i (G).
Since an independent dominating set is not only a dominating set but also an independent set, γ(G) γ i (G) α(G) for any graph G.
It is clear that a maximal independent set is certainly a dominating set. Thus, an independent set is maximal if and only if it is an independent dominating set, and so γ i (G) is the minimum cardinality among all maximal independent sets of G. This graph-theoretical invariant has been well studied in the literature, see for example Haynes, Hedetniemi and Slater [39].
In 2003, Zhang, Liu and Sun [119] defined the independence bondage number b i (G) of a nonempty graph G to be the minimum cardinality among all subsets B ⊆ E(G) for which γ i (G − B) > γ i (G). For some ordinary graphs, their independence domination numbers can be easily determined, and so independence bondage numbers have been also determined. Clearly, b i (K n ) = 1 if n 2. For a complete bipartite graph K m,n , b i (K m,n ) = max{m, n}.
Proof. We will give the proof of an assertion, say for P n to show a basic method. Let P n = (x 1 , x 2 , . . . , x n ) be a path. Clearly, γ i (P n ) = ⌈ n 2 ⌉. We compute b i (P n ) according as n is even or odd.
If n is odd, then γ i (P n ) = ⌈ n 2 ⌉ = n+1 2 . Let e be any edge in P n . Then the edge e partitions P n into two subpaths P k and P ℓ , where k + ℓ = n. Since n is odd, k and ℓ are of have different parity. Without loss of generality, let k be even and ℓ odd. Then On the other hand, Comments Apart from the above-mentioned results, as far as we know, there are no other results on the independence bondage number. We never so much as know any result on this parameter for a tree.

Generalized Types of Bondage Numbers
There are various generalizations of the classical domination, such as distance domination, fractional domination and so on. Every such a generalization can lead to a corresponding bondage. In this section, we introduce some of them.

p-Bondage Numbers
In 1985, Fink and Jacobson [30] introduced the concept of p-domination. Let p be a positive integer. A subset S of V (G) is a p-dominating set of G if |S ∩ N G (y)| p for every y ∈S. The p-domination number γ p (G) is the minimum cardinality among all p-dominating sets of G. Any p-dominating set of G with cardinality γ p (G) will be called a γ p -set of G. Note that the γ 1 -set is the classic minimum dominating set. Notice that every graph has a p-dominating set since the vertex set V (G) is such a set. We also note that the 1-dominating set is a dominating set, and so γ(G) = γ 1 (G). The p-domination number has received much research attention, see a state-of-the-art survey articles by Chellali et. al. [15].
It is clear from definition that every p-dominating set of a graph certainly contains all vertices of degree at most p − 1. By this simple observation, to avoid happening the trivial case, we always assume ∆(G) p. For p 2, Lu et al. [82] gave a constructive characterization of trees with unique minimum p-dominating sets.
Recently, Lu and Xu [83] have introduced the concept to the p-bondage number of G, denoted by b p (G), as the minimum cardinality among all sets of edges Lu and Xu [83] established a lower bound and an upper bound of γ p (T ) for any integer p 2 and any tree T with ∆(T ) p, and characterized all trees achieving the lower bound and the upper bound, respectively. Then, all trees achieving the lower bound 1 can be characterized as follows.
Theorem 9.1.2 (Lu and Xu [83], 2011) Let T be a tree with ∆(T ) p 2. Then b p (T ) = 1 if and only if for any γ p -set S of T there exists an edge xy ∈ (S, S) such that y ∈ N p (x, S, T ) The symbol S(a, b) denotes the double star obtained by adding an edge between the central vertices of two stars K 1,a−1 and K 1,b−1 . And the vertex with degree a (resp., b) in S(a, b) is called the L-central vertex (resp., R-central vertex) of S(a, b).
To characterize all trees attaining the upper bound given in Theorem 9.1.1, we define three types of operations on a tree T with ∆(T ) = ∆ p + 1.

Distance Bondage Numbers
A subset S of vertices of a graph G is said to be a distance k-dominating set for G if every vertex in G not in S is at distance at most k from some vertex of S. The minimum cardinality of all distance k-dominating sets is called the distance k-domination number of G and denoted by γ k (G) (does not confuse with above-mentioned γ p (G)!). When k = 1, a distance 1-dominating set is a normal dominating set, and so γ 1 (G) = γ(G) for any graph G. Thus, the distance k-domination is a generalization of the classical domination.
A subset I ⊆ V (G) is called a distance k-independent set if d G (x, y) > k for any two distinct vertices x and y in I. When k = 1, a distance 1-independent set is a classical independent set. The maximum cardinality among all distance k-independent sets is called the distance k-independence number of G, denoted by α k (G). The relation between γ k and α k for a tree obtained by Meir and Moon [86], who proved that γ k (T ) = α 2k (T ) for any tree T . The further research results can be found in Henning et al [42], Tian and Xu [107,108,109,110] and Liu et al. [80].
In 1998, Hartnell et al. [41] defined the distance k-bondage number of G, denoted by b k (G), to be the cardinality of a smallest subset B of edges of G with the property that γ k (G − B) > γ k (G). From Theorem 2.2.1 it is clear that if T is a nontrivial tree, then 1 b 1 (T ) 2. Hartnell et al. [41] generalized this result to any integer k 1. Hartnell et al. [41] and Topp and Vestergaard [111] also characterized the trees having distance k-bondage number 2. In particular, the class of trees for which b 1 (T ) = 2 are just those which have a unique maximum 2-independent set (see Theorem 2.2.6).
Since, when k = 1, the distance 1-bondage number b 1 (G) is the classical bondage number b(G), Theorem 2.3.3 gives the NP-completeness of deciding the distance kbondage number of general graphs. Comments For a vertex x ∈ V (G), the open k-neighborhood N k (x) of x is defined as

Roman Bondage Numbers
A Roman dominating function on a graph G is a labeling f : V → {0, 1, 2} such that every vertex with label 0 has at least one neighbor with label 2. The weight of a Roman dominating function is the value f (V (G)) = u∈V (G) f (u), denoted by f (G). The minimum weight of a Roman dominating function on a graph G is called the Roman domination number, denoted by γ R (G).
A Roman dominating function f : V → {0, 1, 2} can be represented by the ordered partition dominating set when f is a Roman dominating function, and since placing weight 2 at the vertices of a dominating set yields a Roman dominating function, in [19], it was observed that γ(G) γ R (G) 2γ(G).
The definition of the Roman dominating function was given implicitly by Stewart [100] and ReVelle and Rosing [92]. Roman dominating numbers have been studied. In particular, Bahremandpour et al. showed [5] that the problem determining the Roman domination number is NP-complete even for bipartite graphs.
Let G be a graph with maximum degree at least two. The Roman bondage number b R (G) of G is the minimum cardinality of all sets E ′ ⊆ E for which γ R (G−E ′ ) > γ R (G). Since in the study of Roman bondage number the assumption ∆(G) 2 is necessary, we always assume that when we discuss b R (G), all graphs involved satisfy ∆(G) 2. The Roman bondage number b R (G) was introduced by Jafari Rad and Volkmann in [66].
Recently, Bahremandpour et al. have showed [5] that the problem determining the Roman bondage number is NP-hard even for bipartite graphs.

Roman bondage number problem:
Instance: A nonempty bipartite graph G and a positive integer k.
Question: Is b R (G) k? Theorem 9.4.1 (Bahremandpour et al. [5], 2012) The Roman bondage number problem is NP-hard even for bipartite graphs.
The exact value of b R (G) is known only for a few family of graphs including the complete graphs, cycles and paths.
Lemma 9.4.3 (Cockayne et al. [19]) If G is a graph of order n and contains vertices of degree n − 1, then γ R (G) = 2.
Using Lemma 9.4.3, the third conclusion in Theorem 9.4.2 can be generalized to more general case, which is similar to Lemma 3.5.2.
Proposition 9.4.4 Let G be a graph with order n ≥ 3 and t be the number of vertices Proof. Let H be a spanning subgraph of G obtained by removing fewer than ⌈ t 2 ⌉ edges from G. Then H contains vertices of degree n − 1 and, hence, γ R (H) = 2 = γ R (G) by Lemma 9.4.3, which implies b R (G) ≥ ⌈ t 2 ⌉. Since G contains t vertices of degree n − 1, it contains a complete subgraph K t induced by these t vertices. We can remove ⌈ t 2 ⌉ edges such that no vertices have degree n − 1 and, hence, if m = n = 3; m otherwise.
For a tree T with order n 3, Ebadi and PushpaLatha [29], and Jafari Rad and Volkmann [66], independently, obtained an upper bound of b R (T ).
Theorem 9.4.7 b R (T ) 3 for any tree T with order n 3.
(a) If G is a graph and (x, y, z) a path of length 2 in G, Theorem 9.4.9 (a) implies b R (G) δ(G) + 2∆(G) − 3. Note that for a planar graph G, δ(G) 5, moreover, δ(G) 3 if the girth at least 4 and δ(G) 2 if the girth at least 6. These two facts show that b R (G) 2∆(G) + 2 for connected planar graphs G. Jafari Rad and Volkmann [67] improved this bound.
Theorem 9.4.10 (Jafari Rad and Volkmann [67], 2011) Let G be a connected planar graph of order n 3 with girth g(G). Then According to Theorem 9.4.5, b R (C n ) = 3 for a cycle C n of length n 8 with n ≡ 2 (mod 3), and therefore the last result in Theorem 9.4.10 is best possible, at least for ∆ = 2.
Combining the fact that every planar graph G with minimum degree 5 contains an edge xy with d G (x) = 5 and d G (y) ∈ {5, 6} with Theorem 9.4.9 (a), Akbari, Khatirinejad and Qajar [1] obtained the following result. It remains open to show whether the bound in Theorem 9.4.11 is sharp or not. Though finding a planar graph G with b R (G) = 15 seems to be difficult, Akbari, Khatirinejad and Qajar [1] constructed an infinite family of planar graphs with Roman bondage number equal to 7 by proving the following result.
Theorem 9.4.12 (Akbari, Khatirinejad and Qajar [1], 2012) Let G be a graph of order n and G is the graph of order 5n obtained from G by attaching the central vertex of a copy of a path P 5 to each vertex of G (see Figure 14). Then γ R ( G) = 4n and b R ( G) = δ(G) + 2.
By Theorem 9.4.12, infinitely many planar graphs with Roman bondage number 7 by considering any planar graph G with δ(G) = 5 (e.g. the icosahedron graph). For general bounds, the following observation is directly. G Figure 14: The graph G is constructed from G.
Observation 9.4.14 Let G be a graph of order n with maximum degree at least two. Assume that H is a spanning subgraph of G with γ R (H) = γ R (G). Proof. Let F be a minimum edge-set of G for which γ(G − F ) > γ(G), ie, |F | = b(G). By (9.4.1), γ(G) γ R (G). By assumption, The bound is sharp for cycles on n vertices where n ≡ 0 (mod 3).
A graph G is called to be vertex Roman domination-critical if γ R (G − x) < γ R (G) for every vertex x in G. If G has no isolated vertices, then γ R (G) 2γ(G) 2β(G). If γ R (G) = 2β(G), then γ R (G) = 2γ(G) and hence G is a Roman graph. In [113], Volkmann gave a lot of graphs with γ(G) = β(G).
The following result is similar to Theorem 4.0.7.
Dehgardi, Sheikholeslami and Volkmann [20] posed the following problem: If G is a connected graph of order n 4 with γ R (G) 3, then The following result shows that (9.4.2) holds for all graphs G of order n 4 with γ R (G) = 3, 4, which improves Theorem 9.4.9 (a).
with the first equality if and only if G ∼ = C 4 .
Dehgardi et al. [20] proved that for any connected graph G of order n 3, b R (G) n − 1 and posed the following problems.
Prove or disprove: For any connected graph G of order n ≥ 3, b R (G) = n − 1 if and only if G ∼ = K 3 .
Prove or disprove: If G is a connected graph of order n ≥ 3, then b R (G) n − γ R (G) + 1.
Since γ R (K 3,3,...,3 ) = 4, Theorem 9.4.2 shows that the above two problems are false. Recently Akbari and Qajar [2] proved the following result. In [29], Ebadi and PushpaLatha conjectured that b R (G) n − 1 for any graph G of order n 3. Akbari and Qajar [2] showed that this conjecture is true.

Remarks and Comments
There are many variants of domination except mentioned-above ones.
Generally speaking, the concept of k-domination has an analog for all dominations with various restrained conditions.  [69][70][71][72][73][74][75][76][77][78][79]. A dominating set S of G is called a total k-dominating set of G if every vertex in G is within distance k from some vertex of S other than itself.
It is quite natural to propose bondage numbers for k-dominations of these types. However, we have not yet seen any research results on these topics.

Results on Digraphs
Although domination has been extensively studied in undirected graphs, it is natural to think of a dominating set as a one-way relationship between vertices of the graph. Indeed, among the earliest literature on this subject, J. van [47] for an overview of the domination literature). Thus, there are few, if any, such results on domination for digraphs in the literature.
The bondage number and its related topics for undirected graph have become one of major areas both in theoretical and applied researches. However, until recently, Carlson and Develin [14], Shan and Kang [97], Huang and Xu [60,61] studied the bondage number for digraphs, independently. In this section, we will introduce their results for general digraphs. Results for some special digraphs such as vertex-transitive digraphs are introduced in the next section.

Results for Some Special Digraphs
The exact values and bounds of b(G) for some standard digraphs were determined. Like undirected graphs, we can define the total domination number and the total bondage number. On the total bondage numbers for some special digraphs, the known results are as follows.
Theorem 10.2.2 (Huang and Xu [61], 2007) For a directed cycle C n and a directed path P n , b t (P n ) and b t (C n ) all do not exist. For a complete digraph K n , The extended de Bruijn digraph EB(d, n; q 1 , . . . , q p ) and the extended Kautz digraph EK(d, n; q 1 , . . . , q p ) were introduced by Shibata and Gonda 4 . If p = 1, then they are the de Bruijn digraph B(d, n) and the Kautz digraph K(d, n), respectively. Huang and Xu [61] determined their total domination numbers. In particular, their total bondage numbers for general cases are determined as follows: Theorem 10.2.3 (Huang and Xu [61], 2007) If d 2 and q i 2 for each i = 1, 2, . . . , p, then b t (EB(d, n; q 1 , . . . , q p )) = d p − 1 and b t (EK(d, n; q 1 , . . . , q p )) = d p .
In particular, for the de Bruijn digraph B(d, n) and the Kautz digraph K(d, n), b t (B(d, n)) = d − 1 and b t (K(d, n)) = d.
Zhang et al. [120] determined the bondage number in complete t-partite digraphs.
Theorem 10.2.4 (Zhang et al. [120], 2009) For a complete t-partite digraph K n 1 ,n 2 ,...,nt , where n 1 n 2 · · · n t , b(K n 1 ,n 2 ,...,nt ) =        m if n m = 1 and n m+1 2 for some m (1 m < t); 4t − 3 if n 1 = n 2 = · · · = n t = 2; Comments Since an undirected graph can be thought of a symmetric digraph, any result for digraphs has an analogy for undirected graphs in general. In view of this point, studying the bondage number for digraphs is more significant than for undirected graphs. Thus, we should further study the bondage number of digraphs and try to generalized known results on the bondage number and related variants for undirected graphs to digraphs, prove or disprove Conjecture 10.1.4. In particular, determine the exact values of b(B(d, n)) for an even n, and b t (K n ) for n 4.

Efficient Dominating Sets
A dominating set S of a graph G is called to be efficient if for every vertex x in G, The efficient domination has important applications in many areas, such as errorcorrecting codes, and receives much attention in the late years.
The concept of efficient dominating sets is a measure of the efficiency of domination in graphs and proposed by Bange et al. [4] in 1988. Unfortunately, as shown in [4], not every graph has an efficient dominating set and, moreover, it is an NP-complete problem to determine whether a given graph has an efficient dominating set. In addition, it has been shown by Clark [17] in 1993 that for a wide range of p, almost every random undirected graph G ∈ G (υ, p) has no efficient dominating sets. This means that undirected graphs possessing an efficient dominating set are rare. However, it is easy to show that every undirected graph has an orientation with an efficient dominating set (see Bange et al. [3]).
In 1993, Barkauskas and Host [6] showed that determining whether an arbitrary oriented graph has an efficient dominating set is NP-complete. Even so, the existence of efficient dominating sets for some graphs has been examined, see, for example, Dejter and Serra [21] and Lee [77] for Cayley graph, Gu, Jia and Shen [38] for meshes and tori, Huang and Xu [63] for circulant graphs, Harary graphs and tori; Van Wieren, Livingston and Stout [112] for cube-connected cycles.
In this section, we introduce some results of the bondage number for some graphs with an efficient dominating set.

Results for General Graphs
In this subsection, we introduce some results on bondage numbers obtained by applying efficient dominating sets, due to Huang and Xu [63]. We first state the two following lemmas.
Lemma 11.1.1 Let G be a k-regular graph or digraph of order υ. Then γ(G) υ/(k + 1), with equality if and only if G has an efficient dominating set. In addition, if G has an efficient dominating set, then every efficient dominating set is certainly a γ-set, and vice versa.
Proof. Since G is k-regular, then |N + [x]| = k + 1 for each x ∈ V (G). Hence γ(G) ⌈υ(G)/(k + 1)⌉. It is easy to observe that the equality holds if and only if there exists a dominating set D such that N + [D] is a partition of V (G), equivalently, D is an efficient dominating set. Now suppose that G has an efficient dominating set, i.e., γ(G) = υ(G)/(k + 1). Then a dominating set D is a γ-set if and only if |D| = υ(G)/(k + 1). On the other hand, D is efficient if and only if |D| = υ(G)/(k + 1). The lemma follows.
Let e be an edge and S a dominating set in G. We say e supports S if e ∈ (S,S), where (S,S) = {(x, y) ∈ E(G) : x ∈ S, y ∈S}. Denote by s(G) the minimum number of edges which support all γ-sets in G.
Lemma 11.1.2 For any graph or digraph G, b(G) s(G), with equality if G is regular and has an efficient dominating set.
Proof. Assume E ′ ⊆ E(G) with |E ′ | < s(G). Then E ′ can not support all γ-sets in G. Let D be a γ-set not supported by E ′ . We prove by contradiction that D is still a dominating set in G − E ′ .
Suppose to the contrary that there exists some y ∈ V (G) \ D such that D can not dominate it in G − E ′ . Since D is a dominating set in G, there exists a vertex x ∈ D which dominates y in G. Hence (x, y) ∈ E(G) supports D, which implies that (x, y) / ∈ E ′ . It follows that x dominates y in G − E ′ , a contradiction. Thus, γ(G − E ′ ) = γ(G) for any set E ′ ⊆ E(G) with |E ′ | < s(G), and so b(G) s(G). Now let G be a regular graph with an efficient dominating set, and E ′ a set of s(G) edges which supports all γ-sets. We show that any γ-set D is not a dominating set in H = G − E ′ . Since E ′ supports D, there exists an edge (x, y) ∈ E ′ such that x ∈ D and y / ∈ D. Hence y is not dominated by x in H. By lemma 11.1.1, D is efficient, which implies that D dominate y only by x. Thus, D can not dominate y in H. It follows that γ(H) > γ(G), and b(G) |E ′ | = s(G). The lemma follows.
A graph G is called to be vertex-transitive if its automorphism group Aut(G) acts transitively on its vertex-set V (G). A vertex-transitive graph is regular. Applying Lemma 11.1.1 and Lemma 11.1.2, Huang and Xu obtained some results on bondage numbers for vertex-transitive graphs or digraphs.
Theorem 11.1.3 (Huang and Xu [63], 2008) Let G be a vertex-transitive graph or digraph. Then Proof. Assume V (G) = {x 1 , . . . , x υ }. Let D i be the family of all γ-sets that contain x i in G. We first show that |D i | = |D j | for any i and j. Since G is vertex-transitive, there exists an automorphism φ of G such that φ(x i ) = x j . Clearly φ(D i ) = φ(D ′ i ) for any distinct D i , D ′ i ∈ D i . On the other hand, for any D j ∈ D j , it holds that φ −1 (D j ) ∈ D i and φ(φ −1 (D j )) = D j . Thus, φ is a bijection from D i to D j , and so |D i | = |D j | = s for any i, j ∈ {1, 2, . . . , υ}.
Note that ∪ υ i=1 D i contains all γ-sets of G and every γ-set appears γ(G) times in it. Hence there are exactly υ(G)s/γ(G) γ-sets in G.
If G is undirected, then an edge x i x j may only support those γ-sets in D i and D j whose number is at most 2s. Hence it needs at least (υ(G)s/γ(G))/2s edges to support ∪ υ i=1 D i . It follows from Lemma 11.1.2 that b(G) s(G) ⌈υ(G)/2γ(G)⌉.
If G is directed, then an edge (x i , x j ) only supports those γ-sets in D i . Hence b(G) s(G) ⌈υ(G)s/(γ(G)s)⌉ = ⌈υ(G)/γ(G)⌉. The theorem follows. Next we will establish a better upper bound of b(G). To this aim, we introduce the following terminology, which generalizes the concept of the edge-covering of a graph G. For V ′ ⊆ V (G) and E ′ ⊆ E(G), we say E ′ covers V ′ and call E ′ an edge-covering for V ′ if there exists an edge (x, y) ∈ E ′ for any vertex x ∈ V ′ . For y ∈ V (G), let β ′ [y] be the minimum cardinality over all edge-coverings for N − G [y].
Proof. For any y ∈ V (G), let E ′ be the smallest set of edges that covers N The upper bound of b(G) given in Theorem 11.1.5 is tight in view of b(C n ) for a cycle or a directed cycle C n (see Theorem 2.1.1 and Theorem 10.2.1, respectively).
It is easy to see that for a k-regular graph G, ⌈(k + 1)/2⌉ β ′ [y] k when G is undirected and β ′ [y] = k + 1 when G is directed. By this fact and Lemma 11.1.1, the following theorem is merely a simple combination of Theorem 11.1.3 and Theorem 11.1.5.
Theorem 11.1.6 Let G be a vertex-transitive graph of degree k. If G has an efficient dominating set, then Theorem 11.1.7 If G is an undirected vertex-transitive cubic graph with order 4γ(G) and girth g(G) 5, then b(G) = 2.
Proof. Since G is a cubic graph of order υ(G) = 4γ(G), then by Lemma 11.1.1, any γ-set in G is efficient. By Theorem 11.1.6, 2 b(G) 3. Thus, we only need to show b(G) 2. Let D be an efficient dominating set in G. By the proof of Theorem 11.1.3, there are n(G)s/γ(G) = 4s distinct efficient dominating sets in G provided that a vertex of G belongs to s distinct efficient dominating sets.
If g = 4 or 5, there exists a cycle (u 1 , u 2 , u 3 , u 4 ) or (u 1 , u 2 , u 3 , u 4 , u 5 ). For any 1 i < j 4, it is easy to observe that d(u i , u j ) 2. Note that two distinct vertices u, v in D satisfy d(u, v) 3, since N[u] ∩ N[v] = ∅. Hence there exists no efficient dominating set containing both u i and u j . Suppose that D i is the family of efficient dominating sets containing u i for i = 1, 2, 3, 4. Then D i ∩ D j = ∅. It follows that E ′ = {(u 1 , u 2 ), (u 3 , u 4 )} supports exactly 4s efficient dominating sets, i.e., all sets in ∪ 4 i=1 D i . Since there are only 4s distinct efficient dominating sets in G, then by Lemma 11.1.2, b(G) = s(G) |E ′ | = 2.
Remarks The above proof leads to a byproduct. In the case of g = 5 we have D i ∩ D j = ∅ for i = 1, 2, . . . , 5. Then G has at least 5s efficient dominating sets. But there are only υ(G)s/γ(G) = 4s distinct efficient dominating sets in G. This contradiction implies that an undirected vertex-transitive cubic graph with girth five has no efficient dominating sets. But a similar argument for g(G) = 3, 4 or g(G) 6 could not give any contradiction. This is consistent with the result that CCC(n), a vertex-transitive cubic graph with girth n if 3 n 8, or girth 8 if n 9, has efficient dominating sets for all n 3 except n = 5 (see Theorem 11.2.4).

Results for Cayley Graphs
In this subsection, we will use Theorem 11.1.6 to determine the exact values or approximative values of bondage numbers for some special vertex-transitive graphs by characterizing the existence of efficient dominating sets in these graphs.
Let Γ be a non-trivial finite group, S be a non-empty subset of Γ without the identity element of Γ. A digraph G defined as follows V (G) = Γ; (x, y) ∈ E(G) ⇔ x −1 y ∈ S for any x, y ∈ Γ.
is called a Cayley digraph of the group Γ with respect to S, denoted by C Γ (S). If S −1 = {s −1 : s ∈ S} = S, then C Γ (S) is symmetric, and is called a Cayley undirected graph, a Cayley graph for short. Cayley graphs or digraphs are certainly vertextransitive.
A circulant graph G(n; S) of order n is a Cayley graph C Zn (S), where Z n = {0, 1, . . . , n − 1} is the addition group of order n and S is a nonempty subset of Z n without the identity element and, hence, is a vertex-transitive digraph of degree |S|. If S −1 = S, then G(n; S) is an undirected graph. If S = {1, k}, where 2 k n − 2, we write G(n; 1, k) for G(n; {1, k}) or G(n; {±1, ±k}), and call it a double loop circulant graph.
For directed G = G(n; 1, k), we showed that ⌈n/3⌉ γ(G) ⌈n/2⌉ and G has an efficient dominating set if and only if 3|n and k ≡ 2 (mod 3). For directed G = G(n; 1, k), k = n/2, we showed that ⌈n/5⌉ γ(G) ⌈n/3⌉ and G has an efficient dominating set if and only if 5|n and k ≡ ±2 (mod 5). By Theorem 11.1.6, we can obtain the bondage number of a double loop circulant graph if it has an efficient dominating set.
The m × n torus is the cartesian product C m × C n of two cycles, and is a Cayley graph C Zm×Zn (S), where S = {(0, 1), (1, 0)} for directed cycles and S = {(0, 1), (1, 0)} for undirected cycles and, hence, is vertex-transitive. Gu et al [38] showed that the undirected torus C m × C n has an efficient dominating set if and only if both m and n are multiples of 5. We showed that the directed torus C m × C n has an efficient dominating set if and only if both m and n are multiples of 3. Moreover, we found a necessary condition for a dominating set containing the vertex (0, 0) in C m × C n to be efficient, and obtained the following result.
Theorem 11.2.2 Let G = C m ×C n . If G is undirected and both m and n are multiples of 5, or if G is directed and both m and n are multiples of 3, then b(G) = 3.
The hypercube Q n is the Cayley graph C Γ (S), where Γ = Z 2 × . . . × Z 2 = (Z 2 ) n and S = {100 · · · 0, 010 · · · 0, . . . , 00 · · · 01}. Lee [77] showed that Q n has an efficient dominating set if and only if n = 2 m − 1 for a positive integer m. Then we obtain the following result by Theorem 11.1.6. The n-dimensional cube-connected cycle, denoted by CCC(n), is constructed from the n-dimensional hypercube Q n by replacing each vertex x in Q n with an undirected cycle C n of length n and linking the ith vertex of the C n to the ith neighbor of x. It has been proved that CCC(n) is a Cayley graph and, hence, is a vertex-transitive graph with degree 3. Van Wieren et al. [112] proved that CCC(n) has an efficient dominating set if and only if n = 5. Then we derive the following result from Theorem 11.1.6 and Theorem 11.1.7.
Remarks Whether we can determine the exact value of b(CCC(n)) for n 5.

Remarks
F Q n is a Cayley graph and hence vertex-transitive. The n-dimensional augmented cube AQ n is vertex-symmetric. The n-dimensional star graph S n is vertex-and edge-transitive The n-dimensional pancake graph P n is a Cayley graph and, hence, is vertex transitive.
The (n, k)-arrangement graph A n,k is a regular graph of degree k(n − k) with n ! (n−k) ! vertices and diameter ⌊ 3 2 k⌋. A n,1 is isomorphic to a complete graph K n and A n,n−1 is isomorphic to a star graph S n . Moreover, A n,k is vertex-transitive and edge-transitive.
An n-dimensional alternating group graph AG n is a Cayley graph and, hence, is vertex-transitive.