The local metric dimension of generalized broken fan graph and edge corona product of star graph and path graph

Abstract

Let G be a connected nontrivial graph. The distance between two vertices u and v in G is the length of the shortest u-v path, denoted by d(u, v). For an ordered set W = {w1, w2, …, wn} of n vertices on G, the representation of a vertex v with respect to W is ordered pair r(v|W ) = (d(v, w1), d(v, w2), …, d(v, wn)). W is local metric set of G if r(u|W ) ≠ r(v|W ) for every pair of adjacent vertices u and v in G. The local metric set W with minimum cardinality is called local metric basis and its cardinality is the local metric dimension of G, denoted by diml(G). In this paper, we determine the local metric dimension of generalized broken fan graph and edge corona product K1,m ◊ Pn graph. We obtained the local metric dimension of generalized broken fan graph is . The local metric dimension of edge corona graph K1,m ◊ Pn is for m = 1 and 1 ≤ n ≤ 5, for m = 1 and n ≥ 6, for m ≥ 2 and 1 ≤ n ≤ 5, and for m ≥ 2 and n ≥ 6.