On resolving domination number of friendship graph and its operation

Abstract

Let G = (V, E) be a simple, finite, and connected graph of order n. A dominating set D ⊆ V(G) such every vertex not in D is adjacent to at least one member of D. A dominating set of smallest size is called a minimum dominating set and it is known as the domination number. The domination number is the minimum cardinality of a dominating set and denoted by γ(G). The other hand, for an ordered set W = {w 1, w 2, w 3, …, wk } of vertices and a vertex v in a connected graph G, the (metric) representation of v with respect to W is the k − vector r(v|W) = (d(v, w 1), d(v, w 2), d(v, w 3), …, d(v, wk )), where d(x, y) represents the distance between the vertices x and y. The set W is a resolving set for G if distinct vertices of G have distinct representations with respect to W. A resolving set of minimum cardinality is called a minimum resolving set or a basis and the cardinality of a basis for G is its metric dimension dim(G). A Set of vertices of a graph G that is both resolving and dominating is a resolving dominating set. The minimum cardinality of a resolving dominating set is called resolving domination number γr (G). In this paper, we discussed the resolving domination number of friendship graphs and its operation.