Bounds on metric dimensions of graphs with edge disjoint cycles

TL;DR

This paper investigates the bounds on vertex and edge metric dimensions in graphs with edge disjoint cycles, establishing that their difference is limited by the number of cycles, and conjectures a general bound for graphs with any cyclomatic number.

Contribution

The paper provides new bounds on the difference between vertex and edge metric dimensions for graphs with edge disjoint cycles, extending previous results and proposing a general conjecture.

Findings

Vertex and edge metric dimensions differ by at most one in unicyclic graphs.

In graphs with c edge disjoint cycles, the difference is at most c.

Conjecture: the difference is bounded by the cyclomatic number c in general graphs.

Abstract

In a graph G, cardinality of the smallest ordered set of vertices that distinguishes every element of V (G) is the (vertex) metric dimension of G. Similarly, the cardinality of such a set is the edge metric dimension of G, if it distinguishes E(G). In this paper these invariants are considered first for unicyclic graphs, and it is shown that the vertex and edge metric dimensions obtain values from two particular consecutive integers, which can be determined from the structure of the graph. In particular, as a consequence, we obtain that these two invariants can differ for at most one for a same unicyclic graph. Next we extend the results to graphs with edge disjoint cycles showing that the two invariants can differ at most by c, where c is the number of cycles in such a graph. We conclude the paper with a conjecture that generalizes the previously mentioned consequences to graphs with prescribed cyclomatic number c by claiming that the difference of the invariant is still bounded by c.