图的$k$ -度量维数的计算复杂性
On the complexity of computing the $k$-metric dimension of graphs
摘要：
给定一个连通图$G=(V,E)$，一个集合$S\subseteq V$是$G$的$k$ -度量生成元，如果对于任意两个不同的顶点$u,v\in V$，至少存在$k$顶点$w_1,...,w_k\in S$，使得对于每个$i\in \{1,...,k\}$都有$d_G(u,w_i)\ne d_G(v,w_i)$.最小基数的度量生成元称为$k$ -度量基，其基数称为$G$的$k$ -度量维数。研究了一些关于$k$ -度量维数问题的复杂性问题.例如，我们证明了计算图的$k$ -度量维数的问题是$NP$ -完全的.然而，对于树的特定情况，问题在线性时间内解决。
英文摘要：
Given a connected graph $G=(V,E)$, a set $S\subseteq V$ is a $k$-metric generator for $G$ if for any two different vertices $u,v\in V$, there exist at least $k$ vertices $w_1,...,w_k\in S$ such that $d_G(u,w_i)\ne d_G(v,w_i)$ for every $i\in \{1,...,k\}$. A metric generator of minimum cardinality is called a $k$-metric basis and its cardinality the $k$-metric dimension of $G$. We study some problems regarding the complexity of some $k$-metric dimension problems. For instance, we show that the problem of computing the $k$-metric dimension of graphs is $NP$-Complete. However, the problem is solved in linear time for the particular case of trees.
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