Mixed metric dimension of graphs with edge disjoint cycles.

In a connected graph G , the cardinality of the smallest ordered set of vertices that distinguishes every element of V (G) ∪ E (G) is called the mixed metric dimension of G. In this paper we first establish the exact value of the mixed metric dimension of a unicyclic graph G which is derived from the structure of G. We further consider graphs G with edge disjoint cycles, where for each cycle C i of G we define a unicyclic subgraph G i of G in which C i is the only cycle. Applying the result for unicyclic graph to the subgraph G i of every cycle C i then yields the exact value of the mixed metric dimension of such a graph G. The obtained formulas for the exact value of the mixed metric dimension yield a simple sharp upper bound on the mixed metric dimension, and we conclude the paper conjecturing that the analogous bound holds for general graphs with prescribed cyclomatic number. [ABSTRACT FROM AUTHOR]