On resolving domination number of special family of graphs

Abstract

Abstract Let G be a simple, finite, and connected graph. A dominating set D is a set of vertices such that each vertex of G is either in D or has at least one neighbor in D . The minimum cardinality of such a set is called the domination number of G, denoted by γ ( G ). For an ordered set W = { w 1 , w 2 , …, w k } 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 k )), 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 , denoted by dim ( G ). A resolving domination number, denoted by γ r ( G ), is the minimum cardinality of the resolving dominating set. In this paper, we study the existence of resolving domination number of special graph and its line graph L ( G ), middle graph M ( G ), total graph T ( G ), and central graph C ( G ) of Star graph and fan graph. We have found the minimum cardinality of those special graphs.