On the metric dimension of generalized broken fan graph and edge corona product of path with star graph

Abstract

Let G be a connected nontrivial graph with vertex set V(G) and edge set E(G). The distance d(u, v) between two vertices u, v∈V (G) is shortest path from u to v. For an ordered set W = {w1, w2, …, wk}, the representation r(v | W ) of v with respect to W is ordered pair r(v | W ) = (d(v, w1), d(v, w2), …, d(v, wk)). A set W is resolving set of G if r(u | W ) ≠ r(v | W ) for two distinct vertices u, v in G. A resolving set with minimum cardinality is called metric basis of G. The metric dimension of a graph G, denoted by dim(G) is defined as the cardinality of metric basis. In this paper we determine the metric dimension of generalized broken fan BF (a1, a2, …, an) graph and edge corona graph Pm ◊ Sn. We obtained the metric dimension of generalized broken fan graph is for xi = 0 if ∀aj ≡ 0, 2, 4 (mod 5), j < i, and xi = 2 if ∃aj ≡ 1, 3 (mod 5), j < i. The metric dimension of edge corona graph Pm ◊ Sn is dim(Pm ◊ Sn) = n + 1 for m = 2 and 1 ≤ n ≤ 2, dim(Pm ◊ Sn) = 2 for m ≥ 3 and n = 1, dim(Pm ◊ Sn) = m + 1 for m ≥ 3 and n = 2, and dim(Pm ◊ Sn) = n + (m − 2)(n − 2) for m ≥ 2 and n ≥ 3.