Metric dimensions vs. cyclomatic number of graphs with minimum degree at least two

Abstract

The vertex (resp. edge) metric dimension of a connected graph G, denoted by dim(G) (resp. edim(G)), is defined as the size of a smallest set S⊆V(G) which distinguishes all pairs of vertices (resp. edges) in G. Bounds dim(G)≤L(G)+2c(G) and edim(G)≤L(G)+2c(G), where c(G) is the cyclomatic number in G and L(G) depends on the number of leaves in G, are known to hold for cacti and it is conjectured that they hold for general graphs. In leafless graphs it holds that L(G)=0, so for such graphs the conjectured upper bound becomes 2c(G). In this paper, we show that the bound 2c(G) cannot be attained by leafless cacti, so the upper bound for such cacti decreases to 2c(G)−1, and we characterize all extremal leafless cacti for the decreased bound. We conjecture that the decreased bound holds for all leafless graphs, i.e. graphs with minimum degree at least two. We support this conjecture by showing that it holds for all graphs with minimum degree at least three and that it is sufficient to show that it holds for all 2-connected graphs, and we also verify the conjecture for graphs of small order.