A Comparison between the Metric Dimension and Zero Forcing Number of Trees and Unicyclic Graphs

TL;DR

This paper compares the metric dimension and zero forcing number in trees and unicyclic graphs, establishing inequalities, characterizations, and proposing a cycle rank conjecture for general graphs.

Contribution

It proves bounds relating metric dimension and zero forcing number for trees and unicyclic graphs, characterizes trees where these parameters are equal, and introduces the cycle rank conjecture.

Findings

Proves im(T) \u2264 Z(T) for trees.

Shows im(G) ss Z(G)+1 for unicyclic graphs.

Establishes bounds on im(T+e) for edges outside the tree.

Abstract

The \emph{metric dimension} $\dim(G)$ of a graph $G$ is the minimum number of vertices such that every vertex of $G$ is uniquely determined by its vector of distances to the chosen vertices. The \emph{zero forcing number} $Z(G)$ of a graph $G$ is the minimum cardinality of a set $S$ of black vertices (whereas vertices in $V(G)\!\setminus\!S$ are colored white) such that $V(G)$ is turned black after finitely many applications of "the color-change rule": a white vertex is converted black if it is the only white neighbor of a black vertex. We show that $\dim(T) \leq Z(T)$ for a tree $T$, and that $\dim(G) \le Z(G)+1$ if $G$ is a unicyclic graph, along the way, we characterize trees $T$ attaining $\dim(T)=Z(T)$. For a general graph $G$, we introduce the "cycle rank conjecture". We conclude with a proof of $\dim(T)-2 \leq \dim(T+e) \le \dim(T)+1$ for $e \in E(\overline{T})$.