On The Strong Metric Dimension of Lollipop Graph and Generalized Web Graph

Abstract

Let G be a connected graph with vertex set V (G) and edge set E(G). The interval I[u; v] between u and v to be the collection of all vertices that belong to some shortest u v path. A vertex s strongly resolves two vertices u and v if u belongs to a shortest v s path, denoted by u 2 I[v; s] or v belongs to a shortest u s path, denoted by v 2 I[u; s]. A vertex set S of G is a strong resolving set of G if every two distinct vertices of G are strongly resolved by some vertex of S. The strong metric basis of G is a strong resolving set with minimal cardinality. The strong metric dimension sdim(G) of a graph G is defined as the cardinality of strong metric basis. In this paper we determine the strong metric dimension of a lollipop Lm;n graph and a generalized web W B(G; m; n) graph. Lollipop graph Lm;n is the graph obtained by joining a complete graph Km (m 3) to a path graph Pn (n 1) with a bridge. We obtain the strong metric dimension of a lollipop graph Lm;n is m 1. Generalized web graph W B(G; m; n) is the graph obtained from the generalized pyramid graph P (G; m) by taking p copies of Pn (n 2) and merging an end vertex of a di ff erent copy of Pn with each vertex of the furthermost copy of G from the apex. We obtain the strong metric dimension of generalized web graph with G  C and without center vertex is m = m