A set S  V is a induced -paired dominating set if S is a dominating set of G and the induced subgraph is a perfect matching. The induced paired domination number ip(G) is the minimum cardinality taken over all paired dominating sets in G. The minimum number of colours required to colour all the vertices so that adjacent vertices do not receive the same colour and is denoted by (G). The a...

In this paper we continue the study of paired-domination in graphs. A paired–dominating set, abbreviated PDS, of a graph G with no isolated vertex is a dominating set of vertices whose induced subgraph has a perfect matching. The paired-domination number of G, denoted by γp(G), is the minimum cardinality of a PDS of G. The upper paired–domination number of G, denoted by Γp(G), is the maximum ca...

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...

‎a total dominating set of a graph $g$ is a set $d$ of vertices of $g$ such that every vertex of $g$ has a neighbor in $d$‎. ‎the total domination number of a graph $g$‎, ‎denoted by $gamma_t(g)$‎, ‎is~the minimum cardinality of a total dominating set of $g$‎. ‎chellali and haynes [total and paired-domination numbers of a tree, akce international ournal of graphs and combinatorics 1 (2004)‎, ‎6...

A paired dominating set $P$ is a with the additional property that has perfect matching. While maximum cardainality of minimal in graph $G$ called upper domination number $G$, denoted by $\Gamma(G)$, cardinality $\Gamma_{pr}(G)$. By Henning and Pradhan (2019), we know $\Gamma_{pr}(G)\leq 2\Gamma(G)$ for any without isolated vertices. We focus on graphs satisfying equality $\Gamma_{pr}(G)= 2\Gam...

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 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...