In a connected graph G; a set of vertices S \subseteq V (G) locates all vertices of G if forevery pair of vertices u; v from G there exists at least one vertex s in S such that the distances from u; v to s are distinct. The cardinality of the smallest set of vertices that locates every element of V (G) is the (vertex) metric dimension of G. The motivation for such notion is the problem of finding the smallest possible number of sensors and the locations in a network for them to be instaled, so that every object in a netvork can be located by measuring distances to the sensors. Similarly as with locating vertices, the problem of locating edges is defined and the corresponding edge metric dimension of G. We start with considering metric dimensions of unicyclic graphs, where we establish an upper and lower bound of metric dimensions of such graphs, which imply that they obtain values from two particular consecutive integers. The condition under which the dimensions take each of the two possible values is then established, this condition involves three graph configurations per each dimension. This approach then naturally extends to cacti, i.e. graphs
with edge disjoint cycle. The exact value of metric dimensions for cacti yields a simple upper bound on the dimensions of cacti, and we conjecture that this bound holds also for all connected graphs. We give several results on graphs with minimum degree at least two and 2-connected graphs which support such conjecture. Joint work with Riste Škrekovski.

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