Metric Dimension on Sparse Graphs and its Applications to Zero Forcing Sets

The metric dimension dim(G) of a graph $G$ is the minimum cardinality subset $S$ vertices such that each vertex uniquely determined by its distances to $S$. It well-known can be drastically increased modification single edge. Our main result consists in proving increase an edge addition amortized sense if spanning tree $T$ plus $c$ edges, then at most $6c$. We use this prove weakening conjecture Eroh et al. zero forcing number $Z(G)$ black (whereas other are colored white) all will turned after applying finitely many times following rule: white it only neighbor vertex. conjectured that, for any $G$, $dim(G)\leq Z(G) + c(G)$, where $c(G)$ edges have removed from get forest. They proved true trees and unicyclic graphs. weaker version conjecture: Z(G)+6c(G)$ holds graph. also graphs with disjoint cycles, widely generalizing