Vertex and edge metric dimensions of cacti

TL;DR

This paper extends the understanding of vertex and edge metric dimensions from unicyclic graphs to cactus graphs, providing new bounds and confirming the cycle rank conjecture for this class.

Contribution

It generalizes previous results to cactus graphs by defining unicyclic subgraphs for each cycle and applying known approaches, leading to new bounds and a proof of the cycle rank conjecture.

Findings

Determined the metric dimensions for cactus graphs.

Proved the cycle rank conjecture for cacti.

Provided a simple upper bound on metric dimensions.

Abstract

In a graph G; a vertex (resp. an edge) metric generator is a set of vertices S such that any pair of vertices (resp. edges) from G is distinguished by at least one vertex from S: The cardinality of a smallest vertex (resp. edge) metric generator is the vertex (resp. edge) metric dimension of G: In [19] we determined the vertex (resp. edge) metric dimension of unicyclic graphs and that it takes its value from two consecutive integers. Therein, several cycle configurations were introduced and the vertex (resp. edge) metric dimension takes the greater of the two consecutive values only if any of these configurations is present in the graph. In this paper we extend the result to cactus graphs i.e. graphs in which all cycles are pairwise edge disjoint. We do so by defining a unicyclic subgraph of G for every cycle of G and applying the already introduced approach for unicyclic graphs which involves the configurations. The obtained results enable us to prove the cycle rank conjecture for cacti. They also yield a simple upper bound on metric dimensions of cactus graphs and we conclude the paper by conjecturing that the same upper bound holds in general.