论强度量维数的难解性
On the Hardness of Strong Metric Dimension
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中文总结 AI 辅助
该研究聚焦强度量维数问题的计算复杂性，证明该问题在直径为2的图、路径宽度与反馈顶点集数均为常数的图上仍为NP完全问题，深化了对该图论问题难解性边界的认识。
中文摘要 AI 辅助
设G为连通简单无向图。若存在一条从顶点w到u的等距路径（即最短路径）包含v，或存在一条从w到v的等距路径包含u，则称顶点w强分辨顶点对u,v∈V(G)（u与v互不相同）。若G中每一对不同顶点都至少能被子集S⊆V(G)中的一个顶点强分辨，则称S强分辨G。在强度量维数（Strong Metric Dimension）问题中，输入为图G和正整数k，目标是判断是否存在大小至多为k的子集S⊆V(G)可强分辨G。本文证明，即使在(i)直径为2的图，以及(ii)路径宽度和反馈顶点集数均为常数的图上，强度量维数问题仍是NP完全的。
英文摘要
Let \(G\) be a connected simple undirected graph. A vertex \(w\) is said to \emph{strongly resolve} a pair of distinct vertices \(u, v \in V(G)\) if either there exists an isometric path (i.e.~a shortest path) from \(w\) to \(u\) that contains \(v\), or there exists an isometric path from \(w\) to \(v\) that contains \(u\). A subset \(S \subseteq V(G)\) is said to \emph{strongly resolve} \(G\) if every pair of distinct vertices of \(G\) is strongly resolved by at least one vertex in \(S\). In the \textsc{Strong Metric Dimension} problem, the input consists of a graph \(G\) and a positive integer \(k\), and the objective is to determine whether there exists a subset \(S \subseteq V(G)\) of size at most \(k\) that strongly resolves \(G\). In this article, we show that \textsc{Strong Metric Dimension} is \NP-complete even on \((i)\) graphs of diameter two, and \((ii)\) graphs of constant pathwidth and constant feedback vertex set number.
