Graphs with Constant Sum of Domination and Inverse Domination Numbers

A subsetD of the vertex set of a graph G, is a dominating set if every vertex in V −D is adjacent to at least one vertex inD. The domination number γ G is the minimum cardinality of a dominating set of G. A subset of V −D, which is also a dominating set of G is called an inverse dominating set of G with respect toD. The inverse domination number γ ′ G is the minimum cardinality of the inverse dominating sets.Domke et al. 2004 characterized connected graphsGwith γ G γ ′ G n, where n is the number of vertices in G. It is the purpose of this paper to give a complete characterization of graphs Gwith minimum degree at least two and γ G γ ′ G n − 1.


Introduction
Let G V, E be a simple graph. For D ⊆ V , if every vertex in V − D is adjacent to at least one vertex in D, then D is said to be a dominating set of G 1 . A dominating set D is said to be a minimal dominating set if no proper subset of D is a dominating set of G. The minimum cardinality among all dominating sets of G is called domination number of G, and it is denoted by γ G . Any dominating set of G with cardinality γ G is noted as a γ set of G 1 . Let D be a γ-set of G. If V − D contains a dominating set D of G, then D is called an inverse dominating set with respect to D. The minimum cardinality of all inverse dominating sets is called the inverse domination number 2 and is denoted by γ G . An inverse dominating set D is called a γ -set if |D | γ . By virtue of the definition of the inverse domination number, γ G ≤ γ G . The concept of the inverse domination number was introduced by Kulli and Sigarkanti 2 . It is well known by Ore's Theorem 3 that if a graph G has no isolated vertices, then the complement V − D of every γ-set D contains a dominating set. Thus any graph with no isolated vertices contains an inverse dominating set. However, for graphs with isolated vertices, one cannot find an inverse dominating set. For this reason, hereafter, we restrict ourselves to graphs with no isolated vertices.
A Gallai-type theorem has the form α G β G n, where α G and β G are parameters defined on the graph G, and n is the number of vertices in G. Cockayne The total cardinality of a pair D 1 , D 2 is |D 1 | |D 2 |, and the minimum cardinality of a dominating pair is the disjoint domination number γγ G of G. As mentioned earlier, by Ore's observation, γγ G ≤ |V G | for every graph G without isolated vertices and Hedetniemi et al. characterized all extremal graphs for this bound. In this connection, the existence of two disjoint minimum dominating sets in trees was first studied by Bange et al. 10 . In a related paper, Haynes and Henning 11 studied the existence of two disjoint minimum independent dominating sets in a tree.
Another application of finding two disjoint γ sets is the one in respect of networks. In any network or graphs , dominating sets are central sets, and they play a vital role in routing problems in parallel computing 12 . Also finding efficient dominating sets is always concern in finding optimal central sets in networks 13 . Suppose that S is a γset in a graph or network G, when the network fails in some nodes in S, the inverse dominating set in V − S will take care of the role of S. In this aspect, it is worthwhile to concentrate on dominating and inverse dominating sets. Note that γ G ≥ γ G . From the point of networks, one may demand γ G γ G , where as many graphs do not enjoy such a property. For example consider the star graph K 1,n . Clearly γ K 1,n 1 where as γ K 1,n n. If we consider the graph G K 1,n K 2 with n ≥ 3, then γ G 2 and γ G n. In both the cases if n is large, then γ G is sufficiently large compare to γ G .
The purpose of this paper is to characterize all graphs G with δ G ≥ 2 for which γ G γ G n − 1. In this regards, it may be possible that γ G is larger than γ G and γ G γ G n − 1. But we prove that graphs G with γ G γ G n − 1 having exactly two disjoint minimum dominating sets. Hereafter G denotes a simple graph on n vertices with no isolated vertices. The minimum degree of a graph G is denoted by δ G . The set of neighbors of a vertex v in a graph G is denoted by N G v , and the set of neighbors of v in an induced subgraph of G induced by A ⊆ V G is denoted by N A v . Also P n and C n denote the path and cycle on n vertices, respectively. The Cartesian product of graphs G 1 and G 2 is the graph G 1 G 2 whose vertex set is V G 1 × V G 2 and whose edge set is the set of all pairs Let us first recall the following characterizations of graphs for which γ G γ G n. 1 V − S is an independent set and 2 for every vertex x ∈ V − S ∪ L , every stem in N x is adjacent to at least two leaves.

Graphs with
Tamizh Chelvam and Grace Prema 8 characterized graphs for which γ G γ G n − 1 /2. In this context, we attempt to characterize graphs G with δ G ≥ 2 for which γ G γ G n − 1. To attain this aim, we first present the theorem which is useful in the further discussion. To prove the following theorem, since no better proof technique is available, authors prefer case by case analysis.
Let S ⊆ D be those vertices that are adjacent to more than one vertex in D . Suppose that γ G < γ G . Then |D| < |D | and so S / ∅. Let S N S ∩ D . Claim 1. There is at most one vertex in S which is adjacent to a vertex in D − S.
Suppose not, there are at least two vertices t , r in S and t, r ∈ D − S such that t is adjacent to t and r is adjacent to r. Then either both t, r ∈ D − S are adjacent to w or at least one of t, r is not adjacent to w.
Suppose that both t, r ∈ D − S are adjacent to w. Since t , r are the only vertices in V G − {D ∪ {w}} which are adjacent to t, r, and t , r are dominated by some vertices in S, D 1 D ∪ {w} − {t, r} ⊂ D is a dominating set of G, which is a contradiction to the fact that D is a γ-set of G.
When at least one of them, say t, is not adjacent to w. Since t ∈ D − S and δ G ≥ 2, t is adjacent to a vertex u in D. Therefore D 1 D − {t} is a dominating set of G, which is a contradiction to the fact that D is a γ-set of G. Hence, at most one vertex t ∈ S which is adjacent to a vertex in D − S. By similar argument as given above, one can prove that t is adjacent to exactly one vertex in D − S. Let us take Note that each vertex in S has at least two neighbors in S and so S 1 / ∅.

Claim 2. S 1 is independent.
Suppose that there exists a vertex x ∈ S 1 which is adjacent to y ∈ S 1 . Suppose that w is not adjacent to both x and y . By the fact that each vertex in S has at least two neighbors in D and Claim 1, D − {x } is a γ -set of G, a contradiction. If w is adjacent to one of x or y , say x , then D − {y } is a γ -set of G, which is a contradiction. Hence S 1 is independent. Now we have the following three possibilities.
1 w is not adjacent to any of the vertices in S 1 .
2 w is adjacent to exactly one vertex in S 1 .
3 w is adjacent to more than one vertex in S 1 . If there is no vertex x ∈ S such that N S 1 x {u , v }, then, by Claim 2, D 1 D − {v} ∪ {v } is a γ-set of G, and so D 1 D − {u , v } ∪ {v} is a γ -set of G, which is a contradiction. Case 1.2. Suppose that, for each pair of vertices x, y ∈ S, there exist at least two vertices x , y ∈ S 1 such that {x , y } ⊆ N S 1 x ∩ N S 1 y .
Suppose that, for each pair of vertices x , y ∈ S 1 , there exist at least two vertices x, y ∈ S such that {x, y} ⊆ N S x ∩ N S y . Note that |S| ≥ 2 and |S 1 | ≥ 2. Assume that |S| k and S {u 1 , u 2 , . . . , u k }. If w is adjacent to some vertex in S, say u 1 , then by the assumption in Case 1.2, there exist u 2 ∈ S and u 1 , u 2 ∈ S 1 such that {u 1 , u 2 , u 1 , u 2 } K 2,2 as S, and S 1 are independent.
Assume that |S| k ≥ 3. Suppose that N s 1 u 3 {u 1 , u 2 }. Since u 2 dominates both u 2 , u 3 Hence u k is adjacent to at least one vertex in S − {u 1 , . . . , u k } ∅, which is not possible.
Let |S| k 2. If |S 1 | ≥ 3, then u 3 is adjacent to u 1 and u 2 only. Therefore If |S 1 | 2, then D D , which is a contradiction. Case 2. Suppose that w is adjacent to exactly one vertex x ∈ S 1 . If u ∈ S 1 − {x } is adjacent to a vertex in D , then D − {u } is a γ -set of G, a contradiction. Thus every vertex in S 1 − {x } has at least two neighbors in S.
If |S 1 | ≥ 3, then, as in Case 1, replacing S 1 by S 1 − {x }, we get contradiction in all the possibilities.
If |S 1 | 1, then since |D| < |D |, S ∅, a contradiction to S / ∅. Case 3. Suppose w is adjacent to more than one vertex in S 1 , say u , v ∈ S 1 . If no vertex in S is adjacent to only u , v , then Suppose that N D x {x}. If x is adjacent to w, then, D − {u , x } ∪ {w} is a γ -set of G. If x is not adjacent to w, then as δ G ≥ 2, x is adjacent to at least a vertex say y ∈ D . Since x ∈ N y , x, u ∈ N w and N D x {x}, we get D − {u , x } ∪ {w} is a γ -set of G, which is a contradiction. Hence, γ G γ G .
Bange et al. 10 characterized trees with two disjoint minimum dominating sets. In the following corollary, we give the necessary condition for graphs with minimum degree at least two having two disjoint minimum dominating sets.

Corollary 2.2.
Let G be a connected graph on n vertices with δ G ≥ 2. If γ G γ G n − 1, then G has two disjoint γ-sets.
The following example shows that in general Theorem 2.1 is not true whenever δ G 1.