The effect of vertex or edge deletion on the metric dimension of graphs

Abstract

The metric dimension dim(G)dim⁡(G) of a graph GG is the minimum cardinality of a set of vertices such that every vertex of GG is uniquely determined by its vector of distances to the chosen vertices. Let vv and ee respectively denote a vertex and an edge of a graph GG. We show that, for any integer kk, there exists a graph GG such that dim(G−v)−dim(G)=kdim⁡(G−v)−dim⁡(G)=k. For an arbitrary edge ee of any graph GG, we prove that dim(G−e)≤dim(G)+2dim⁡(G−e)≤dim⁡(G)+2. We also prove that dim(G−e)≥dim(G)−1dim⁡(G−e)≥dim⁡(G)−1 for GG belonging to a rather general class of graphs. Moreover, we give an example showing that dim(G)−dim(G−e)dim⁡(G)−dim⁡(G−e) can be arbitrarily large. Keywords