Given a connected graph G, two vertices \(u,v\in V(G)\) doubly resolve \(x,y\in V(G)\) if \(d_{G}(x,u)-d_{G}(y,u)\ne d_{G}(x,v)-d_{G}(y,v)\). The doubly metric dimension \(\psi (G)\) of G is the cardinality of a minimum set of vertices that doubly resolves each pair of vertices from V(G). It is well known that deciding the doubly metric dimension of G is NP-hard. In this work we determine the exact values of doubly metric dimensions of unicyclic graphs which completes the known result. Furthermore, we give formulae for doubly metric dimensions of cactus graphs and block graphs.

中文翻译：

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仙人掌图和框图的双度量维度

给定一个连通图G，两个顶点\(u,v\in V(G)\)双重解析\(x,y\in V(G)\)如果\(d_{G}(x,u)-d_ {G}(y,u)\ne d_{G}(x,v)-d_{G}(y,v)\)。G的双重度量维度\(\psi (G)\)是从V ( G )双重解析每对顶点的最小顶点集的基数。众所周知，确定G的双度量维度是 NP 困难的。在这项工作中，我们确定了单环图的双度量维度的精确值，从而完成了已知结果。此外，我们给出了仙人掌图和框图的双度量维度的公式。