图族的同时强度量维数
The Simultaneous Strong Metric Dimension of Graph Families
摘要：
设${\cal G}$是定义在公共（标记）顶点集$V$上的图族。一个集合$S\subset V$被称为是${\cal G}$的同时强度量生成元，如果它是这个族的每个图的强度量生成元。${\cal G}$的所有同时强度量生成元中的最小基数记为$Sd_s({\cal G})$，称为${\cal G}$的同时强度量维数.我们得到一般结果$Sd_s({\cal G})$任意族的图，特别强调的情况下，家庭组成的一个图和它的补充。特别是，它表明，找到的同时强度量维数的家庭图的问题是$NP$ -硬，即使是在家庭的树。
英文摘要：
Let ${\cal G}$ be a family of graphs defined on a common (labeled) vertex set $V$. A set $S\subset V$ is said to be a simultaneous strong metric generator for ${\cal G}$ if it is a strong metric generator for every graph of the family. The minimum cardinality among all simultaneous strong metric generators for ${\cal G}$, denoted by $Sd_s({\cal G})$, is called the simultaneous strong metric dimension of ${\cal G}$. We obtain general results on $Sd_s({\cal G})$ for arbitrary families of graphs, with special emphasis on the case of families composed by a graph and its complement. In particular, it is shown that the problem of finding the simultaneous strong metric dimension of families of graphs is $NP$-hard, even when restricted to families of trees.
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