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