Bounds on metric dimensions of graphs with edge disjoint cycles

• In the second revision of paper we have corrected according to suggestions reviewer. This includes slight two proofs remarks 2. and 6. reviewer, while other are mainly mistypes which accordingly corrected. We thank reviewer for careful reading paper. a graph G , cardinality smallest ordered set vertices that distinguishes every element V ( ) is (vertex) metric dimension . Similarly, such edge if it E this these invariants considered first unicyclic graphs, shown vertex dimensions obtain values from particular consecutive integers, can be determined structure graph. particular, as consequence, differ by at most one same Next extend results graphs with disjoint cycles (i.e. cactus graphs) showing c where number in conclude conjecture generalizes previously mentioned consequences prescribed cyclomatic claiming difference invariant still bounded