A Comparison Between the Zero Forcing Number and the Strong Metric Dimension of Graphs

Abstract

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. The strong metric dimension, \(sdim(G)\), of a graph \(G\) is the minimum among cardinalities of all strong resolving sets: \(W \subseteq V(G)\) is a strong resolving set of \(G\) if for any \(u, v \in V(G)\), there exists an \(x \in W\) such that either \(u\) lies on an \(x-v\) geodesic or \(v\) lies on an \(x-u\) geodesic. In this paper, we prove that \(Z(G) \le sdim(G)+3r(G)\) for a connected graph \(G\), where \(r(G)\) is the cycle rank of \(G\). Further, we prove the sharp bound \(Z(G) \le sdim(G)\) when \(G\) is a tree or a unicyclic graph, and we characterize trees \(T\) attaining \(Z(T)=sdim(T)\). It is easy to see that \(sdim(T+e)-sdim(T)\) can be arbitrarily large for a tree \(T\); we prove that \(sdim(T+e) \ge sdim(T)-2\) and show that the bound is sharp.