A characterization of some graphs with metric dimension two

Abstract

A set W ∪ V (G) is called a resolving set, if for each pair of distinct vertices u,v ϵ V (G) there exists t ϵ W such that d(u,t)=d(v,t), where d(x,y) is the distance between vertices x and y. The cardinality of a minimum resolving set for G is called the metric dimension of G and is denoted by dimM(G). A k-tree is a chordal graph all of whose maximal cliques are the same size k + 1 and all of whose minimal clique separators are also all the same size k. A k-path is a k-tree with maximum degree 2k, where for each integer j, k ≤ j < 2k, there exists a unique pair of vertices, u and v, such that deg(u) =deg(v) = j. In this paper, we prove that if G is a k-path, then dimM(G) = k. Moreover, we provide a characterization of all 2-trees with metric dimension two. © 2017 World Scientific Publishing Company.