For a graph G=V,E, a subset F of E is called an edge dominating set of G if every edge not in F is adjacent to some edge in F. The edge domination number γ′G of G is the minimum cardinality taken over all edge dominating sets of G. Here, we determine the edge domination number for shadow graphs, middle graphs, and total graphs of paths and cycles.
1. Introduction
The domination in graphs is one of the concepts in graph theory which has attracted many researchers to work on it because of its many and varied applications in such fields as linear algebra and optimization, design and analysis of communication networks, and social sciences and military surveillance. Many variants of dominating models are available in the existing literature. For a comprehensive bibliography of papers on the concept of domination, readers are referred to Hedetniemi and Laskar [1]. The present paper is focused on edge domination in graphs.
We begin with simple, finite, connected, and undirected graph G=(V,E) of order n. The set S⊆V of vertices in a graph G is called a dominating set if every vertex v∈V is either an element of S or is adjacent to an element of S. A dominating set S is a minimal dominating set (or MDS) if no proper subset S′⊂S is a dominating set.
The minimum cardinality of a dominating set of G is called the domination number of G which is denoted by γ(G), and the corresponding dominating set is called a γ-set of G.
The open neighborhood N(v) of v∈V is the set of vertices adjacent to v, and the set N[v]=N(v)∪{v} is the closed neighborhood of v.
For any real number n, ⌈n⌉ denotes the smallest integer not less than n and ⌊n⌋ denotes the greatest integer not greater than n.
An edge e of a graph G is said to be incident with the vertex v if v is an end vertex of e. In this case, we also say that v is incident with e. Two edges e and f which are incident with a common vertex v are said to be adjacent.
In a graph G, a vertex of degree one is called a pendant vertex, and an edge incident with a pendant vertex is called a pendant edge.
A subset F⊆E is an edge dominating set if each edge in E is either in F or is adjacent to an edge in F. An edge dominating set F is called a minimal edge dominating set (or MEDS) if no proper subset F′ of F is an edge dominating set. The edge domination number γ′(G) is the minimum cardinality among all minimal edge dominating sets. The concept of edge domination was introduced by Mitchell and Hedetniemi [2] and it is explored by many researchers. Arumugam and Velammal [3] have discussed the edge domination in graphs while the fractional edge domination in graphs is discussed in Arumugam and Jerry [4]. The complementary edge domination in graphs is studied by Kulli and Soner [5] while Jayaram [6] has studied the line dominating sets and obtained bounds for the line domination number. The bipartite graphs with equal edge domination number and maximum matching cardinality are characterized by Dutton and Klostermeyer [7] while Yannakakis and Gavril [8] have shown that edge dominating set problem is NP-complete even when restricted to planar or bipartite graphs of maximum degree 3. The independent edge dominating sets of certain graphs are discussed in Mojdeh and Sadeghi [9] while a constructive characterization for trees with equal edge domination and end edge domination numbers is investigated by Muddebihal and Sedamkar [10]. The edge domination in graphs of cubes is studied by Zelinka [11].
Throughout the paper, Pn and Cn will denote the path and the cycle with n vertices, respectively.
We will give brief summary of definitions which are useful for the present investigations.
Definition 1.
The open neighborhood of an edge e∈E is denoted as N(e) and it is the set of all edges adjacent to e in G. Further, N[e]=N(e)∪{e} is the closed neighborhood of e in G.
Definition 2.
The degree of an edge e=uv of G is defined by deg(e)=deg(u)+deg(v)-2 and it is equal to the number of edges adjacent to it. The maximum degree of an edge in G is denoted by Δ′(G).
Definition 3 (see [12]).
The line graph of G, written L(G), is the simple graph whose vertices are the edges of G, with ef∈E(L(G)) when e and f have a common end vertex in G.
Definition 4 (see [13]).
The shadow graph of a connected graph G is constructed by taking two copies of G, say, G′ and G′′. Join each vertex u′ of G′ to the neighbors of the corresponding vertex u′′ of G′′. The shadow graph of G is denoted by D2(G).
Definition 5 (see [13]).
The middle graph of a connected graph G denoted by M(G) is the graph whose vertex set is V(G)∪E(G) where two vertices are adjacent if
they are adjacent edges of G, or
one is a vertex of G and the other is an edge incident with it.
Definition 6 (see [14]).
The total graph of G denoted by T(G) is the graph whose vertex set is V(G)∪E(G) and two vertices are adjacent in T(G) if
they are adjacent edges of G, or
one is a vertex of G and the other is an edge incident with it, or
they are adjacent vertices of G.
It is easy to see that T(G) always contains both G and the line graph L(G) as its induced subgraph. The total graph is the largest graph that is formed by the adjacency relations of elements of a graph.
For the various graph theoretic notations and terminology, we follow West [12] while the terms related to the concept of domination are used in the sense of Haynes et al. [15].
Generally, the following types of problems are considered in the field of domination in graphs:
to introduce new types of dominating models,
to determine bounds in terms of various graph parameters,
to obtain the exact domination number for some graphs or graph families,
to study the algorithmic and complexity results for particular dominating parameters, and
to characterize the graphs with certain dominating parameters.
The present work is intended to discuss the problem of the third kind in the context of edge domination in graphs. In this paper, we investigate the edge domination number of middle graphs, total graphs, and shadow graphs of Pn and Cn.
2. Main ResultsTheorem 7.
γ′(D2(Pn))=2⌈(n-1)/3⌉.
Proof.
Consider two copies of Pn. Let v1,v2,…,vn be the vertices of the first copy of Pn and let u1,u2,…,un be the vertices of the second copy of Pn. Let e1,e2,…,en-1 be the edges of the first copy of Pn and let e1′,e2′,…,en-1′ be the edges of the second copy of Pn where ei connects vi and vi+1 and ei′ connects ui and ui+1. Then |V(D2(Pn))|=2n and |E(D2(Pn))|=4(n-1).
For n≤4, F1={en-1,en-1′} is obviously an MEDS with minimum cardinality among all minimal edge dominating sets of D2(Pn). Hence, γ′(D2(Pn))=2=2⌈(n-1)/3⌉.
For n>4, we construct an edge set of D2(Pn) as follows:
(1)F={{e2,e2′,e5,e5′,…,e3i+2,e3i+2′}ifn≡0or1(mod3){e2,e2′,e5,e5′,…,e3i+2,e3i+2′}∪{en-1,en-1′}ifn≡2(mod3),
where 0≤i≤⌈(n-5)/3⌉ with |F|=2⌈(n-1)/3⌉.
Since each edge in E(D2(Pn)) is either in F or is adjacent to an edge in F, it follows that the above set F is an edge dominating set of D2(Pn).
Moreover, the above set F is an MEDS of D2(Pn) because for any edge e∈F, the set F-{e} does not dominate the edges in N(e) of D2(Pn). Now, deg(e3i+2)=6=deg(e3i+2′)=Δ′(D2(Pn)) for 0≤i≤⌈(n-5)/3⌉ and deg(en-1)=4=deg(en-1′)=Δ′(D2(Pn))-2 and pairs of edges {e3i+2,e3i+2′} for 0≤i≤⌈(n-5)/3⌉ will dominate maximum number of distinct edges of D2(Pn). Therefore, any set containing the edges less than that of F cannot be an edge dominating set of D2(Pn). This implies that the above edge dominating set F is of minimum cardinality.
Hence, the above set F is an MEDS with minimum cardinality among all minimal edge dominating sets of D2(Pn).
Thus, γ′(D2(Pn))=2⌈(n-1)/3⌉.
Theorem 8.
γ′(M(Pn))=⌊n/2⌋.
Proof.
Let v1,v2,…,vn be the vertices of path Pn and let u1,u2,…,un-1 be the added vertices corresponding to the edges f1,f2,…,fn-1 of Pn to obtain M(Pn). Thus, V(M(Pn))={v1,v2,…,vn,f1,f2,…,fn-1}.
Then |V(M(Pn))|=2n-1 and |E(M(Pn))|=3n-4. Let the edges e1,e2,…,en-2∈E(M(Pn)) where ek=ukuk+1.
Now, the edge sets F={v1u1} and F={u1u2} are clearly the minimal edge dominating sets of M(P2) and M(P3), respectively, with minimum cardinality among all minimal edge dominating sets of M(Pn). Hence, γ′(M(Pn))=1=⌊n/2⌋ for n=2,3.
For n≥4, we construct an edge set of M(Pn) as follows:
(2)F={e1,e3,…,e2i+1}∪{en-2}hhfor0≤i≤⌊n-42⌋hhhhhhhhhhhhhhhhhhhhhhhhlhhlwith|F|=⌊n2⌋.
Since each edge in E(M(Pn)) is either in F or is adjacent to an edge in F, it follows that the above set F is an edge dominating set of M(Pn). Moreover, the above set F is an MEDS of M(Pn) because for any edge e∈F, the set F-{e} does not dominate the edges in N(e) of M(Pn).
Now, each graph M(Pn), for n≥4, has two nonadjacent pendant edges and there is no edge which is adjacent to both pendant edges. Hence, at least two distinct edges are required to dominate these pendant edges. Moreover, deg(ei)=6=Δ′(M(Pn)) for 1<i<n-2 and for n≥5; each edge of at most ⌊(n-2)/3⌋ distinct edges out of total 3n-4 edges of M(Pn) can dominate seven distinct edges of M(Pn) including itself and each of the remaining edges can dominate less than six distinct edges of M(Pn) at a time. Therefore, any set containing edges less than that of F cannot be an edge dominating set of M(Pn). This implies that the above edge dominating set F is of minimum cardinality.
Hence, the above set F is an MEDS with minimum cardinality among all minimal edge dominating sets of M(Pn).
Thus, γ′(M(Pn))=⌊n/2⌋.
Theorem 9.
For path Pn,
(3)γ′(T(Pn))={⌈2n-13⌉ifn≡0or2(mod3)⌊2n-13⌋otherwise.
Proof.
Let v1,v2,…,vn be the vertices of path Pn and let u1,u2,…,un-1 be the added vertices corresponding to the edges e1,e2,…,en-1 of Pn to obtain T(Pn). Thus, V(T(Pn))={v1,v2,…,vn,u1,u2,…,un-1}. The graph T(Pn) will have (2n-1) vertices and (4n-5) edges. Let the edges f1,f2,…,f2(n-1)∈E(T(Pn)) where fi=u⌈i/2⌉v⌈i/2⌉ for odd i and fi=ui/2v(i/2)+1 for even i.
Now, we construct the edge sets of T(Pn) as follows:
(4)F={{f2,f5,…,f3j+2}ifn≡2(mod3){f2,f5,…,f3j+2}∪{f2(n-1)}ifn≡0(mod3),
for 0≤j≤2⌈(n-3)/3⌉ with |F|=⌈(2n-1)/3⌉, and F={f2,f5,…,f3k+2} if n≡1(mod3), for 0≤k≤⌊(2n-3)/3⌋ with |F|=⌊(2n-1)/3⌋.
The above set F is an edge dominating set of T(Pn) because each edge in E(T(Pn)) is either in F or is adjacent to an edge in F. Also, since for any edge e∈F, the set F-{e} does not dominate the edges in N(e) of T(Pn), it follows that the above set F is an MEDS of T(Pn).
Now, the sets F={f2} and F={f2,f4} are clearly minimal edge dominating sets of T(P2) and T(P3) respectively with minimum cardinality. Hence, γ′(T(P2))=1 and γ′(T(P3))=2. For n>3, Δ′(T(Pn))=6 implying that an edge of T(Pn) can dominate at most seven distinct edges of T(Pn) including itself. But, from the nature of graph, we can observe that each of at most ⌊(n-2)/2⌋ distinct edges of T(Pn) can dominate seven distinct edges including itself and each of the remaining edges can dominate less than six edges of T(Pn). Therefore, any set containing the edges less than that of F cannot be an edge dominating set of T(Pn). This implies that the above edge dominating set F is of minimum cardinality.
Hence, the above set F is an MEDS with minimum cardinality among all minimal edge dominating sets of T(Pn).
This implies that
(5)γ′(T(Pn))={⌈2n-13⌉ifn≡0or2(mod3)⌊2n-13⌋otherwise.
Theorem 10.
For cycle Cn,
(6)γ′(D2(Cn))={2⌈n-13⌉ifn≡0or2(mod3)2⌈n+13⌉otherwise.
Proof.
Consider two copies of Cn. Let v1,v2,…,vn be the vertices of the first copy of Cn and let u1,u2,…,un be the vertices of the second copy of Cn. Let e1,e2,…,en be the edges of the first copy of Cn and e1′,e2′,…,en′ be the edges of the second copy of Cn. Then |V(D2(Cn))|=2n and |E(D2(Cn))|=4n-1.
First, we construct an edge set of D2(Cn) as follows:
(7)F={{e2,e5,…,e3i+2,e2′,e5′,…,e3i+2′}ifn≡0or2(mod3){e2,e5,…,e3i+2,e2′,e5′,…,e3i+2′}∪{en,en′}otherwise,hhhhhhhhhhhhhhhhhhhlwhere0≤i≤⌊n-23⌋
with |F|=2⌈(n-1)/3⌉ for n≡0 or 2(mod3) and |F|=2⌈(n+1)/3⌉ for n≡1(mod3).
Since each edge in E(D2(Cn)) is either in F or is adjacent to an edge in F, it follows that the above set F is an edge dominating set of D2(Cn).
Now, the above set is an MEDS of D2(Cn) because for any edge e∈F, the set F-{e} does not dominate the edges in N(e) of D2(Cn). Moreover, deg(ei)=6=deg(ei′)=Δ′(D2(Cn)) for 1≤i≤n and each edge of D2(Cn) can dominate at most seven distinct edges of D2(Cn) including itself. But, at a time, each of at most ⌈(n-2)/3⌉ distinct edges of D2(Cn) can dominate seven distinct edges of D2(Cn) including itself and each of the remaining edges can dominate less than six distinct edges of D2(Cn). Therefore, any set containing the edges less than that of F cannot be an edge dominating set of D2(Cn). This implies that the above edge dominating set F is of minimum cardinality.
Hence, the above set F is an MEDS with minimum cardinality among all minimal edge dominating sets of D2(Cn).
Let v1,v2,…,vn be the vertices of cycle Cn and let u1,u2,…,un be the added vertices corresponding to the edges f1,f2,…,fn of Cn to obtain M(Cn). Then |V(M(Cn))|=2n and |E(M(Cn))|=3n. Let the edges e1,e2,…,en∈E(M(Cn)), where ek=ukuk+1.
Now, we construct an edge set of M(Cn) as follows:
(9)F={e2,e4,…,e2i}∪{en}for1≤i≤⌊n-12⌋hhhhhhhhhhhhhhhhhhlhhlwith|F|=⌊n+12⌋.
Since each edge in E(M(Cn)) is either in F or is adjacent to an edge in F, it follows that the above set F is an edge dominating set of M(Cn). Moreover, the above set F is an MEDS of M(Cn) because for any edge e∈F, the set F-{e} does not dominate the edges in N(e) of M(Cn).
Now, deg(ei)=6=Δ′(M(Cn)) for 1≤i≤n and each edge of M(Cn) can dominate at most seven distinct edges of M(Cn) including itself. But, at a time, each of at most ⌊(n-2)/3⌋ edges of M(Cn) can dominate seven distinct edges of M(Cn) including itself and each of the remaining edges can dominate less than six distinct edges of M(Cn). Therefore, any set containing the edges less than that of F cannot be an edge dominating set of M(Cn). This implies that the above edge dominating set F is of minimum cardinality.
Hence, the above set F is an MEDS with minimum cardinality among all minimal edge dominating sets of M(Cn).
Thus, γ′(M(Cn))=⌊(n+1)/2⌋.
Theorem 12.
For cycle Cn(10)γ′(T(Cn))={⌈2n-13⌉ifn≡0or1(mod3)⌈2n+13⌉otherwise.
Proof.
Let v1,v2,…,vn be the vertices of cycle Cn and let u1,u2,…,un be the added vertices corresponding to the edges e1,e2,…,en of Cn to obtain T(Cn). Then |V(T(Cn))|=2n and |E(T(Cn))|=4n. Let the edges f1,f2,…,f2n∈E(T(Cn)) where fk=u⌈k/2⌉v⌈k/2⌉ for odd k and fk=u(k/2)v(k/2)+1 for even k.
First, we construct an edge set of T(Cn) as follows:
(11)F={{f2,f5,…,f3i+2}ifn≡0or1(mod3){f2,f5,…,f3i+2}∪{f2n}otherwise,
for 0≤i≤⌈(2n-4)/3⌉, with |F|=⌈(2n-1)/3⌉ if n≡0or1(mod3) and |F|=⌈(2n+1)/3⌉ if n≡2(mod3).
The above set F is an edge dominating set of T(Cn) because each edge in E(T(Cn)) is either in F or is adjacent to an edge in F. Since for any edge e∈F, the set F-{e} does not dominate the edges in N(e) of T(Cn), it follows that the above set F is an MEDS of T(Cn).
Now, deg(fi)=6=Δ′(T(Cn)) for 1≤i≤2n and each edge of T(Cn) can dominate at most seven distinct edges of T(Cn) including itself. But, at a time, each of at most ⌊n/2⌋ distinct edges of T(Cn) can dominate seven distinct edges of T(Cn) including itself and each of the remaining edges can dominate less than six distinct edges of T(Cn). Therefore, any set containing the edges less than that of F cannot be an edge dominating set of T(Cn). This implies that the above edge dominating set F is of minimum cardinality.
Hence, the above set F is an MEDS with minimum cardinality among all minimal edge dominating sets of T(Cn).
Here, we have taken up a problem to determine the edge domination number for the larger graphs obtained by means of three graph operations on paths and cycles. To derive similar results in the context of other variants of domination is an open area of research.
Conflict of Interests
The authors declare that they have no conflict of interests regarding the publication of this paper.
Acknowledgment
The authors are highly thankful to the anonymous referees for their kind comments and fruitful suggestions on the first draft of this paper.
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