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