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Given a connected graph $G$, the metric (resp. edge metric) dimension of $G$is the cardinality of the smallest ordered set of vertices that uniquelyidentifies every pair of distinct vertices (resp. edges) of $G$ by means ofdistance vectors to such a set. In this work, we settle three open problems on(edge) metric dimension of graphs. Specifically, we show that for every $r,t\ge2$ with $r\ne t$, there is $n_0$, such that for every $n\ge n_0$ there exists agraph $G$ of order $n$ with metric dimension $r$ and edge metric dimension $t$,which among other consequences, shows the existence of infinitely many graphwhose edge metric dimension is strictly smaller than its metric dimension. Inaddition, we also prove that it is not possible to bound the edge metricdimension of a graph $G$ by some constant factor of the metric dimension of$G$.

Graphs with the edge metric dimension smaller than the metric dimension