Characterizing 1-Metric Antidimensional Trees and Unicyclic Graphs

Abstract

Let G=(V,E) be a simple connected graph and S={w1, wt} V an ordered subset of vertices. The metric representation of a vertex u V with respect to S is the t-vector r(u|S)=(dG(u,w1), dG(u,wt)), where dG(u,v) represents the length of a shortest u-v path in G. A set S is a k-antiresolving set if k is the largest positive integer such that for every vertex v V-S there exist other k-1 different vertices v1, vk-1 V-S such that v,v1, vk-1 have the same metric representation with respect to S. The k-metric antidimension of G is the minimum cardinality among all the k-antiresolving sets for G, and G is k-metric antidimensional if k is the largest integer such that G contains a k-antiresolving set. In this article, we provide characterizations for 1-metric antidimensional trees and unicyclic graphs, together with computationally efficient algorithms to decide whether these types of graphs are 1-metric antidimensional.