A characterization of some graphs with metric dimension two

TL;DR

This paper characterizes graphs with metric dimension two, focusing on k-paths and 2-trees, and proves that k-paths have metric dimension k, providing insights into their resolving sets.

Contribution

It offers a new characterization of 2-trees with metric dimension two and establishes that k-paths have a metric dimension equal to k, advancing understanding of graph resolving sets.

Findings

k-paths have metric dimension equal to k

Characterization of all 2-trees with metric dimension two

Provides a formula for the metric dimension of k-paths

Abstract

A set W \subseteq V (G) is called a resolving set, if for each pair of distinct vertices u,v \in V (G) there exists t \in W such that d(u,t) \neq 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 dim_M(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 \leq 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 dim_M(G) = k. Moreover, we provide a characterization of all 2-trees with metric dimension two.