Resolving independent domination number of some special graphs

Abstract

Abstract Dominating set is a set D of vertices of graph G ( V, E ) and every vertex u ∈ V ( G ) − D is adjacent to some vertex υ ∈ D . The set D is called independent set if no two vertices in D are adjacent. Independent domination number of G is the minimum cardinality of D and denoted by γ i ( G ). The metric representation of vertex υ in connected graph G with respect to an ordered set W = w 1 , w 2 , w 3 ,…, w k of vertices is the k -vector r ( υ | W ) = ( d ( υ | w 1 ), d ( υ | w 2 ), d ( υ | w 3 ),…, d ( υ | w k )), where d ( υ, w ) represents the distance between the vertices υ and w . The set W is resolving independent dominating set for G if W is independent in G , and distinct vertices of G have distinct representations with respect to W . The minimum cardinality of resolving independent dominating set is called resolving independent domination number and denoted by γ ri ( G ). In this paper, we analyze the resolving independent domination number of path graph, cycle graph, friendship graph, helm graph, and fan graph.