弦图某些子类的共安全支配问题的细粒度复杂性
A Fine-Grained Complexity of Co-Secure Domination for Some Subclasses of Chordal Graphs
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中文总结 AI 辅助
本文研究弦图子类中2-树的共安全支配数界，证明共安全支配集问题在$r\geq4$的$K_{1,r}$-自由分裂图上NP-完全，在$K_{1,3}$-自由分裂图上可线性时间求解。
中文摘要 AI 辅助
对于连通图$G=(V,E)$，若集合$D \subseteq V$既是$G$的支配集，且对每个顶点$u \in D$，都存在顶点$v \in V \setminus D$使得$uv \in E$，且$(D \setminus \{u\}) \cup \{v\}$也是$G$的支配集，则称$D$为共安全支配集。本文中，我们给出了受限2-树（弦图的一个热门子类）中共安全支配数（最小共安全支配集的大小）的界。我们还证明了一个有趣的二分性：共安全支配集问题在$r\geq4$的$K_{1,r}$-自由分裂图上是NP-完全的，而在$K_{1,3}$-自由分裂图上可在线性时间内求解。
英文摘要
For a connected graph $G = (V, E)$, a set $D \subseteq V$ is a co-secure dominating set if $D$ is a dominating set of $G$ and for each vertex $u \in D$ there exists a vertex $v \in V \setminus D$ such that $uv \in E$ and $(D \setminus \{u\}) \cup \{v\}$ is a dominating set of $G$. In this article, we present bounds on the co-secure domination number (size of minimum co-secure dominating set) in some restricted 2-trees, a popular subclass of chordal graphs. We also show an interesting dichotomy that the co-secure dominating set problem is NP-complete on $K_{1,r}$-free split graphs, $r\ge 4$, whereas it is linear-time solvable on $K_{1,3}$-free split graphs.
