Vertex resolvability of convex polytopes with n-paths of length p. Issue 2 (3rd April 2022) - Sharma, Sahil; Bhat, Vijay Kumar
Abstract : Let G = ( V, E ) be a simple, connected, and undirected graph. The distance between two vertices u, v ∈ V, denoted by d ( u, v ), is the length of the shortest path connecting u and v . A subset of vertices R G is said to be a resolving set for G if for any two distinct vertices u 1, u 2 ∈ V, there exist a vertex α ∈ R G such that d ( u 1, α ) ≠ d ( u 2, α ) . A minimal resolving set is called a metric basis, and the cardinality of the basis set is called the metric dimension of G, denoted by β ( G ) . In this article, we find the metric dimension for two infinite families of plane graphs Υ n and Π n, p, where Υ n is obtained by the combination of 2 n copies of bipartite graphs ( K 1, 3 ), and Π n, p is obtained by the combination of double antiprism graph with antiprism graph and then adding n -paths of length p .