Non-isolated resolving number of graphs with homogeneous pendant edges

Abstract

All graphs in this paper are a simple, nontrivial and connected graph G. A set W = {w1, w2, w3, …, wk} of vertex set of G, r(v |W) = (d(v, w1), d(v, w2), …, d(v, wk)) is a representation of vertex v to W, the distance between the vertex v and the vertex w, denoted by d(v, w). A set W is called a resolving set of G if every vertices of G have different representation. The minimum cardinality of resolving set W is metric dimension, denoted by dim(G). Furthermore, the resolving set W of G, is called the non-isolated resolving set if there does not for all v ∈ W induced by the non-isolated vertex. A non-isolated resolving number, denoted by nr(G), is minimum cardinality of non-isolated resolving set in G. In this research, we obtain the lower bound of the non isolated resolving number of graphs with homogeneous pendant edges, nr(G⊙ mK1) ≥ |V(G)|m for m ≥ 2 and these results of the non isolated resolving number of graphs with homogeneous pendant edges with G with a path Pn, complete graph Kn and cycle Cn.