Metric dimension on sparse graphs and its applications to zero forcing sets

TL;DR

This paper investigates the metric dimension of sparse graphs, establishing an amortized bound when adding edges to a spanning tree, and applies this to analyze the relationship between metric dimension and zero forcing number, confirming a weakened conjecture.

Contribution

It proves an amortized bound on metric dimension increase with edge additions and extends a conjecture relating metric dimension and zero forcing number to broader classes of graphs.

Findings

Metric dimension increase can be bounded by 6 times the number of added edges.

A weakened conjecture relating metric dimension, zero forcing number, and edges removed is proven.

The conjecture holds for graphs with edge disjoint cycles, generalizing previous results.

Abstract

The metric dimension dim(G) of a graph $G$ is the minimum cardinality of a subset $S$ of vertices of $G$ such that each vertex of $G$ is uniquely determined by its distances to $S$. It is well-known that the metric dimension of a graph can be drastically increased by the modification of a single edge. Our main result consists in proving that the increase of the metric dimension of an edge addition can be amortized in the sense that if the graph consists of a spanning tree $T$ plus $c$ edges, then the metric dimension of $G$ is at most the metric dimension of $T$ plus $6c$. We then use this result to prove a weakening of a conjecture of Eroh et al. The zero forcing number $Z(G)$ of $G$ is the minimum cardinality of a subset $S$ of black vertices (whereas the other vertices are colored white) of $G$ such that all the vertices will turned black after applying finitely many times the following rule: a white vertex is turned black if it is the only white neighbor of a black vertex. Eroh et al. conjectured that, for any graph $G$, $dim(G)\leq Z(G) + c(G)$, where $c(G)$ is the number of edges that have to be removed from $G$ to get a forest. They proved the conjecture is true for trees and unicyclic graphs. We prove a weaker version of the conjecture: $dim(G)\leq Z(G)+6c(G)$ holds for any graph. We also prove that the conjecture is true for graphs with edge disjoint cycles, widely generalizing the unicyclic result of Eroh et al.