Let G = (V ,E) be a graph without isolated vertices. A set S ⊆ V is a paired-dominating set if every vertex in V − S is adjacent to a vertex in S and the subgraph induced by S contains a perfect matching. The paired-domination problem is to determine the paired-domination number, which is the minimum cardinality of a paired-dominating set. Motivated by a mistaken algorithm given by Chen, Kang a...

The concept of induced paired domination number of a graph was introduced by D.S.Studer, T.W. Haynes and L.M. Lawson11, with the following application in mind. In the guard application an induced paired dominating set represents a configuration of security guards in which each guard is assigned one other as a designated backup with in (as in a paired dominating set), but to avoid conflicts (suc...

Let G be a graph without isolated vertices. The total domination number of G is the minimum number of vertices that can dominate all vertices in G, and the paired domination number of G is the minimum number of vertices in a dominating set whose induced subgraph contains a perfect matching. This paper determines the total domination number and the paired domination number of the toroidal meshes...

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...

Let G = (V, E) be a graph without isolated vertices. A matching in G is a set of independent edges in G. A perfect matching M in G is a matching such that every vertex of G is incident to an edge of M . A set S ⊆ V is a paired-dominating set of G if every vertex not in S is adjacent to a vertex in S, and if the subgraph induced by S contains a perfect matching. The paired-domination problem is ...

The inflation GI of a graph G with n(G) vertices and m(G) edges is obtained from G by replacing every vertex of degree d of G by a clique Kd. A set S of vertices in a graph G is a paired dominating set of G if every vertex of G is adjacent to some vertex in S and if the subgraph induced by S contains a perfect matching. The paired domination number γp(G) is the minimum cardinality of a paired d...