Independent strong domination in complemen- tary prisms

Let G = (V,E) be a graph and u, v ∈ V . Then, u strongly dominates v if (i) uv ∈ E and (ii) deg(u) ≥ deg(v). A setD ⊂ V is a strong-dominating set ofG if every vertex in V −D is strongly dominated by at least one vertex in D. A set D ⊆ V is an independent set if no two vertices of D are adjacent. The independent strong domination number is(G) of a graph G is the minimum cardinality of a strong dominating set which is independent. Let Ḡ be the complement of a graph G. The complementary prism GḠ of G is the graph formed from the disjoint union of G and Ḡ by adding the edges of a perfect matching between the corresponding vertices of G and Ḡ. In this paper, we consider the independent strong domination in complementary prisms, characterize the complementary prisms with small independent strong domination numbers, and investigate the relationship between independent strong domination number and the distance-based parameters.


Introduction
Graph theoretic techniques provide a convenient tool for the investigation of networks. It is well-known that an interconnection network can be modeled by a graph with vertices representing sites of the network and edges representing links between sites of the network. Therefore various network problems can be studied by graph theoretical methods. The study of domination in graphs is an important research area, perhaps also the fastest-growing area within graph theory. The reason for the steady and rapid growth of this area may be the diversity of its applications to both theoretical and real-world problems. For instance, dominating sets in graphs are natural models for facility location problems in operations research. Research on domination in graphs has not only important theoretical signification, but also varied application in such fields as computer science, communication networks, ad hoc networks, biological and social networks, distributed computing, coding theory, and web graphs. Domination and its variations have been extensively studied [1,6,7,9,12,14,16,17]. In general, the concept of dominating sets in graph theory finds wide applications in different types of communication networks. A broadcast from a communication vertex is received by all its neighbors. This is captured by the notion of domination in a graph. The minimum dominating set of sites plays an important role in the network for it dominates the whole network with the minimum cost. A thorough study of domination appears in [16,17]. Independent sets play an important role in graph theory and other areas like discrete and combinatorial optimization. They appear in matching theory, coloring of graphs, and in trees. In a communication graph, an independent set consists of vertices that cannot communicate with one another directly. From an application point of view, independent and dominating set in a communication network are important structures, and many optimization approaches rely on these. In some sense, one could say that the domination and independence based parameters reveal an underlying efficient and stable communication network. Among the domination-type parameters that have been studied, the independent strong domination number is one of the fundamental ones. A set D ⊆ V is a dominating set if every vertex not in D is adjacent to at least one vertex in D.
. The domination number of G (strong domination number), denoted γ(G)(γ s (G)) is the minimum size of a dominating set (strong dominating set) of G. A set D ⊆ V is an independent set if no two vertices of D are adjacent. The independent domination number of G (independent strong domination number), denoted i(G)(i s (G)) is the minimum size of an independent dominating set (independent strong dominating set) of G. Then a minimal independent strong dominating set of cardinality i s (G) is called an i s (G) − set [2,5,11,13]. It follows from the definitions that for any graph G, γ(G) ≤ i(G) ≤ i s (G) and γ(G) ≤ γ s (G) ≤ i s (G) [4]. Every graph admits an independent strong dominating set [11]. For example, to find an independent strong dominating set, say D, in a graph G, the following algorithm can be applied: The concept of strong domination was introduced by Sampathkumar and Pushpa Latha in [5] by the following motivation. Consider a network of roads connecting a number of locations. In such a network, the degree of a vertex v is the number of roads meeting at v. Suppose deg(u) ≥ deg(v).
Naturally, the traffic at u is heavier than that at v. If we consider the traffic between u and v, preference should be given to the vehicles going from u to v. Thus, in some sense, u strongly dominates v. In [11], it is shown that the problems of computing i s is NP-hard, even for bipartite graphs. Since computing the independent strong domination of a graph is NP-hard in general, it becomes an interesting question to calculate the independent strong domination for some special classes of interesting or practically useful graphs. In the following two sections we will deal with this question.
In this paper, we consider finite undirected graphs without loops and multiple edges. The order of G is the number of vertices in A vertex of degree zero is an isolated vertex or an isolate. A leaf or an endvertex or a pendant vertex is a vertex of degree one and its neighbor is called a support vertex. The maximum degree of G is The distance d(u, v) between two vertices u and v in is the length of a shortest path between them. If u and v are not connected, then d(u, v) = ∞, and for Complementary prisms are a deeply intriguing family of graphs and the study in complementary prisms is just beginning. Complementary prisms were first introduced by Haynes, Henning, Slater and Van der Merwe in [15]. For a graph G, its complementary prism, denoted by GḠ, is formed from a copy of G and a copy ofḠ by adding a perfect matching between corresponding vertices. For each v ∈ V (G), letv denote the vertex v in the copy ofḠ. Formally GḠ is formed from G ∪Ḡ by adding the edge vv for every v ∈ V (G). For example, if G is a 5-cycle, then GḠ is the Petersen graph. Complementary prisms form an interesting family of graphs, generalizing the concept of graph products as well as graphs such as the Petersen graph. Hence parameters for these graphs are important to study. Independent and domination based parameters of complementary prims were also studied in [18,19].
Let GḠ be the complementary prism of a graph G = (V, E). For notational convenience, we letV = V (Ḡ) . Note that V (GḠ) = V ∪V , and to simplify our discussion of complementary prisms, we say simply G andḠ to refer to the subgraph copies of G andḠ , respectively, in GḠ. Also, for a vertex v of G , we letv be the corresponding vertex inḠ, and for a set X ⊆ V , we let X be the corresponding set of vertices inV .
The paper proceeds as follows. In the following section, existing literature on independent strong domination is reviewed. The independent strong domination numbers for specific types of graphs and complementary prism GḠ when G is a specified family of graphs are computed and exact formulae are derived. Independent strong domination numbers for the graphs and vertices with specific distance-based parameters are investigated. The graphs and vertices for which i s is small are characterized. Proof. If G has diameter one, then G is the complete graph on n vertices in which all vertices are adjacent to each other and all vertices having the same vertex degree n − 1. Therefore, the proof is immediate. Proof. The sufficiency is immediate since if G has a vertex v with eccentricity one, then this implies that vertex v is adjacent to all other vertices of G with deg G (v) = n − 1. Thus, the set including only the vertex v is the i s (G)-set and strongly dominates V . Now, suppose that i s (G) = 1. This implies that G has a support vertex v that dominates V . Since N G [v] = V , vertex v has eccentricity one. This establishes the necessity.

Specific families
We begin our investigation of independent strong domination for this subsection by computing its value for several well-known classes of graphs. We use x to denote the least integer not less than x.
Observation 2.1. The independent strong domination of (a) the complete graph K n is 1; (b) the null graphK n is n; (c) the star K 1,n is 1; (d) the path P n is n/3 ; (e) the cycle C n is n/3 ; (f) the wheel W n is 1; (g) any complete multipartite graph of order p and least partite set of order r is r; (h) the comet C s,t is 1 + (t − 2)/3 , where s and t are positive integers, the comet C s,t denotes the tree obtained by identifying the center of the star K 1,s with an end-vertex of P t , the path of order t. So C s,t ∼ = K 1,s and C 1,p−1 ∼ = P p .

Complementary prisms
We next determine the independent strong domination number of the complementary prism GḠ when G is a specified family of graphs. We shall make use of the following lemmata.
Lemma 2.1. (a) γ(P n ) = n/3 ; (b) γ(C n ) = n/3 ; (c) γ(C n ) = 2 for n > 3; Definition 2.1. Let G and H be two graphs of order n and m, respectively. The corona product G • H is defined as the graph obtained from G and H by taking one copy of G and n copies of H, and then joining the ith vertex of G to every vertex in the ith copy of H.
The following lemma gives a formula for the independent strong domination number of corona of two graphs in terms of the independent strong domination numbers of these two graphs.  To

Proof. In a graph
.., t] and t > 1, both verticesā k andb k are included in i s (GḠ)-set to strongly dominate V (Ḡ) and the vertices a k and b k of G. Then, any combination of the nondominated vertices with cardinality t − 1 in which including the vertices a r and b s where r = s, k,s = r, k (∀r, s ∈ [1, .., t]) can be in i s (GḠ)-set. Consequently, we have i s (GḠ) = t + 1, and taking any subset of V (GḠ) with cardinality less than t + 1 does not yield an i s (GḠ)-set.
To prove (c), let G = K t • K 1 . If t = 1, then G = K 2 and from (a), i s (GḠ) = 2. Assume that t ≥ 2, and label the vertices of G as follows: let A = {a i |1 ≤ i ≤ t} be the set of t vertices that induce the subgraph K t of G, and let B = {b i |1 ≤ i ≤ t} be the end-vertices in G adjacent to the vertices in A such that a i b i ∈ E(G). Let I be an i s (GḠ)-set. Since deg GḠ (b k ) = ∆(GḠ) for a vertexb k (k ∈ [1, .., t]) in GḠ,b k ∈ I to strongly dominate the vertices ofB,Ā \ {a i }, and the vertex b i ∈ B. Among the remaining non-dominated vertices, since I should be independent, being deg GḠ (ā i ) = t < deg GḠ (a i ) = t + 1(∀i), a k ∈ I to strongly dominateā k ∈Ā and A \ {a k }. To strongly dominate the nondominated vertex set B \ {b k }, all t − 1 vertices of B \ {b k } should be included in the independent set I. Then, we have I =b k ∪ a k ∪ B \ {b k } with cardinality |I| = t + 1, and taking any other subset of V (GḠ) with cardinality less than t + 1 does not yield an i s (GḠ)-set.
To prove (d), since G is a star, the support vertex t in G is an isolated vertext inḠ, and a leaf in GḠ. Denote the n leaves of G by {u i |1 ≤ i ≤ n}. The leaves in G form a complete graph on n vertices inḠ. The support vertex t of G has a vertex degree deg GḠ (t) = n + 1 in GḠ which is greater than or equal to of all its neighboring vertices, deg GḠ (t) > deg GḠ (t) = 1 and To prove (g), let G be a wheel of order n + 1 and consider GḠ. If I is an i s (GḠ)-set, then first, the center vertex c ∈ V (G) which is adjacent to every other vertex of G and which has the maximum vertex degree in GḠ belongs to I in order to strongly dominate N GḠ [c] = {c} ∪ V (G).
. Eventually, GḠ[S] =C n which is regular. Therefore, a minimum dominating set ofC n belongs to I. By Lemma 2.1, we have that i s (GḠ) = |I| = 1 + γ(C n ) = 3, and taking any other subset of V (GḠ) with cardinality less than |I| = 3 does not yield an i s (GḠ)set.
Theorem 2.7. For a graph G of order n , if either G orḠ has diameter one, then i s (GḠ) = n.
Proof. If either G orḠ has diameter one, then GḠ is the corona K n • K 1 . Therefore, by Lemma 2.2 and Observation 2.1(a), the proof is immediate.