In a graph G, cardinality of the smallest ordered set of vertices that distinguishes every element of V (G) is the (vertex) metric dimension of G. Similarly, the cardinality of such a set is the edge metric dimension of G, if it distinguishes E(G). In this paper these invariants are considered first for unicyclic graphs, and it is shown that the vertex and edge metric dimensions obtain values from two particular consecutive integers, which can be determined from the structure of the graph. In particular, as a consequence, we obtain that these two invariants can differ for at most one for a same unicyclic graph. Next we extend the results to graphs with edge disjoint cycles showing that the two invariants can differ at most by c, where c is the number of cycles in such a graph. We conclude the paper with a conjecture that generalizes the previously mentioned consequences to graphs with prescribed cyclomatic number c by claiming that the difference of the invariant is still bounded by c.

中文翻译：

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具有边不相交循环的图的度量维度的界限

在图 G 中，区分 V (G) 的每个元素的最小有序顶点集的基数是 G 的（顶点）度量维度。类似地，这样的集合的基数是 G 的边度量维度，如果它区分 E(G)。在本文中，这些不变量首先被考虑用于单环图，并表明顶点和边度量维度从两个特定的连续整数中获得值，这可以从图的结构中确定。特别是，因此，我们得到对于同一个单环图，这两个不变量最多可以有一个不同。接下来，我们将结果扩展到具有边不相交循环的图，表明两个不变量最多可以相差 c，其中 c 是此类图中的循环数。