Paired domination is a relatively interesting concept introduced by Teresa W. Haynes [9] recently with the following application in mind. If we think of each vertex s ∈ S, as the location of a guard capable of protecting each vertex dominated by S, then for a paired domination the guards location must be selected as adjacent pairs of vertices so that each guard is assigned one other and they ar...

Given a graph G = (V,E), the domination problem is to find a minimum size vertex subset S ⊆ V (G) such that every vertex not in S is adjacent to a vertex in S. A dominating set S of G is called a paired-dominating set if the induced subgraph G[S] contains a perfect matching. The paired-domination problem involves finding a paired-dominating set S of G such that the cardinality of S is minimized...

Let G = (V, E) be a graph without isolated vertices. A set D ⊆ V is a d-distance paired-dominating set of G if D is a d-distance dominating set of G and the induced subgraph 〈D〉 has a perfect matching. The minimum cardinality of a d-distance paired-dominating set for graph G is the d-distance paired-domination number, denoted by γd p(G). In this paper, we study the ddistance paired-domination n...

In this paper, we continue the study of the total domination game in graphs introduced in [Graphs Combin. 31(5) (2015), 1453–1462], where the players Dominator and Staller alternately select vertices of G. Each vertex chosen must strictly increase the number of vertices totally dominated, where a vertex totally dominates another vertex if they are neighbors. This process eventually produces a t...

We settle two conjectures on domination-search, a game proposed by Fomin et.al. [1], one in affirmative and the other in negative. The two results presented here are (1) domination search number can be greater than domination-target number, (2) domination search number for asteroidal-triple-free graphs is at most 2.

The domination number γ(G), the independent domination number ι(G), the connected domination number γc(G), and the paired domination number γp(G) of a graph G (without isolated vertices, if necessary) are related by the simple inequalities γ(G) ≤ ι(G), γ(G) ≤ γc(G), and γ(G) ≤ γp(G). Very little is known about the graphs that satisfy one of these inequalities with equality. I.E. Zverovich and V...