Resolving Domination Number of Graphs

Abstract

For a set W = { s1,s2,...,sk of vertices of a graph G, the representation multiset of a vertexv of G with respect to W is r(v | W ) = { d(v, s1),d(v, s2),...,d(v, sk) } , where d(v, si) is a distance between of the vertex v and the vertices in W together with their multiplicities. The set W is a resolving set of G if r(v | W )

} = r(u | W ) for every pair u, v of distinct vertices of G. The minimum resolving set W is a multiset basis of G. IfG has a multiset basis, then its cardinality is called multiset dimension, denoted by md(G). A set W of vertices in G is a dominating set for G if every vertex of G that is not in W is adjacent to some vertex of W . The minimum cardinality of the dominating set is a domination number, denoted by γ(G). A vertex set of some vertices in G that is both resolving and dominating set is a resolving dominating set. The minimum cardinality of resolving dominating set is called resolving domination number, denoted by γr (G). In our paper, we investigate and establish sharp bounds of the resolving domination number of G and determine the exact value of some family graphs.