A set S of vertices is a total dominating set of a graph G if every vertex of G is adjacent to some vertex in S . A paired-dominating set of G is a dominating set whose induced subgraph has a perfect matching. The minimum cardinality of a total dominating set (respectively, a paired-dominating set) is the total domination number γt(G) (respectively, the paired-domination number γpr(G) ). We giv...

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...

a subset $s$ of vertices in a graph $g$ is called a geodetic set if every vertex not in $s$ lies on a shortest path between two vertices from $s$‎. ‎a subset $d$ of vertices in $g$ is called dominating set if every vertex not in $d$ has at least one neighbor in $d$‎. ‎a geodetic dominating set $s$ is both a geodetic and a dominating set‎. ‎the geodetic (domination‎, ‎geodetic domination) number...

Let γ(G), i(G), γS(G) and iS(G) denote the domination number, the independent domination number, the strong domination number and the independent strong domination number of a graph G, respectively. A graph G is called γi-perfect (domination perfect) if γ(H) = i(H), for every induced subgraph H of G. The classes of γγS-perfect, γSiS-perfect, iiS-perfect and γiS-perfect graphs are defined analog...

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...

A dominating set of a graph is a vertex subset that any vertex belongs to or is adjacent to. Among the many well-studied variants of domination are the so-called paired-dominating sets. A paired-dominating set is a dominating set whose induced subgraph has a perfect matching. In this paper, we continue their study. We focus on graphs that do not contain the net-graph (obtained by attaching a pe...