On the Cycle Rank Conjecture About Metric Dimension and Zero Forcing Number in Graphs

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. The zero forcing number Z(G) of G is the minimum cardinality of a subset S of black vertices of G such that all the vertices will be turned black after applying finitely many times the following rule: a non black vertex is turned black if it is the only non black neighbor of a black vertex.