A comparison between the metric dimension and zero forcing number of trees and unicyclic graphs

The 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 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)\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) ≤ Z(T) for a tree T, and that dim(G) ≤ 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 ≤ dim(T + e) ≤ dim(T) + 1 for e ∈ E(T).