Paired Domination of Certain Nanotubes

Abstract

A vertex subset D of a graph G is a dominating set if every vertex of G is either in D or is adjacent to a vertex in D. The paired-domination problem on G asks for a minimum-cardinality dominating set S of G such that the subgraph induced by S contains a perfect matching. A set S of vertices in G is a total dominating set of G if every vertex of V is adjacent to some vertex in S. In this paper we construct a minimum paired dominating set for H-Naphtalenic Nanotorus, $TUC_{4}C_{8} (S)$ nanotube and V-phenylenic nanotube and hence we determine the total dominating set.