关于三次图中的 $1$-限制支配与 $(1,2)$-支配

中文摘要 AI 辅助

图的一个支配集 $D$ 被称为 $1$-限制的，如果 $D$ 中每个顶点在 $D$ 外至多有一个邻居；而一个 $(1,2)$-支配集是一个支配集，其中集合中每个顶点在集合内至少有两个邻居。这两个概念在三次图上是一致的。我们证明决策问题 1-Limited Dominating Set 即使限制在 2-连通平面三次图上也是 $\mathsf{NP}$-完全的，从而完成了对所有固定正整数 $k$ 的 $k$-Limited Dominating Set 的已知复杂性结果。我们还确定了整个 Goldberg 族的精确 $1$-限制支配数。这提供了又一个无限的三次图族，支持关于 $(1,2)$-支配和诱导环的若干开放猜想和提出的界。

英文摘要

A dominating set $D$ of a graph is called $1$-limited if every vertex of $D$ has at most one neighbor outside $D$, while a $(1,2)$-dominating set is a dominating set in which every vertex of the set has at least two neighbors within the set. These two notions coincide on cubic graphs. We prove that the decision problem 1-Limited Dominating Set is $\mathsf{NP}$-complete even when restricted to $2$-connected planar cubic graphs, thereby completing the known complexity results for $k$-Limited Dominating Set for all fixed positive integers $k$. We also determine the exact $1$-limited domination number of the entire Goldberg family. This provides a further infinite family of cubic graphs supporting several open conjectures and proposed bounds concerning $(1,2)$-domination and induced cycles.

发表机构

• University of Maribor（马里博尔大学）
• Institute of Mathematics, Physics and Mechanics（数学、物理与力学研究所）

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