The multiset dimension of graphs

We introduce a variation of the metric dimension, called the multiset dimension. The representation multiset of a vertex $v$ with respect to $W$ (which is a subset of the vertex set of a graph $G$), $r_m (v|W)$, is defined as a multiset of distances between $v$ and the vertices in $W$ together with their multiplicities. If $r_m (u |W) \neq r_m(v|W)$ for every pair of distinct vertices $u$ and $v$, then $W$ is called a resolving set of $G$. If $G$ has a resolving set, then the cardinality of a smallest resolving set is called the multiset dimension of $G$, denoted by $md(G)$. If $G$ does not contain a resolving set, we write $md(G) = \infty$. We present basic results on the multiset dimension. We also study graphs of given diameter and give some sufficient conditions for a graph to have an infinite multiset dimension.