圆凸、三叉凸二分图与 $P_4$-整齐图上的罗马控制问题
Roman Domination on Circular-Convex, Triad-Convex Bipartite Graphs and $P_4$-Tidy Graphs
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中文总结 AI 辅助
本文针对圆凸二分图、三叉凸二分图和$P_4$-整齐图，分别给出$O(n^6)$、$O(n^7)$和$O(n+m)$时间的罗马控制数精确算法，扩展了该问题的可解图类边界。
中文摘要 AI 辅助
图 $G=(V,E)$ 上的罗马控制问题（RDP）要求寻找一个标记函数 $f:V\rightarrow\{0,1,2\}$，使得每个被赋值为 $0$ 的顶点都邻接于某个被赋值为 $2$ 的顶点。目标是最小化总权重 $\sum_{v\in V} f(v)$；该最小值称为 $G$ 的罗马控制数，记为 $\gamma_R(G)$。本文研究了由凸性和诱导 $P_4$ 结构所启发的图类上的 RDP。首先，我们考虑圆凸二分图，它是凸二分图的自然超类，已知 RDP 在该类上可多项式时间求解。假设给定一个圆凸表示，我们通过切割圆序、分离区间顶点和环绕顶点，并对至多两个赋值为 $2$ 的环绕顶点进行分支，在 $O(n^6)$ 时间内计算出 $\gamma_R(G)$。其次，我们研究三叉凸二分图，它是树凸二分图的一个受限子类，其凸性树是 $K_{1,3}$ 的细分。尽管 RDP 在更广泛的树凸子类（如星凸和梳凸二分图）上是困难的，但我们证明在三叉凸二分图上可以在 $O(n^7)$ 时间内计算出 $\gamma_R(G)$。最后，我们研究 $P_4$-整齐图，它真扩展了余图（cographs）。利用 Giakoumakis 等人对 $P_4$-整齐图的结构分解，我们给出一个直接、精确的算法，在 $O(n+m)$ 时间内计算出 $\gamma_R(G)$。这些结果扩展了罗马控制在基于凸性的二分图和 $P_4$ 结构图类上的算法边界。
英文摘要
The Roman Domination Problem (RDP) on a graph \(G=(V,E)\) asks for a labeling function \(f:V\rightarrow\{0,1,2\}\) such that every vertex assigned value \(0\) is adjacent to a vertex assigned value \(2\). The objective is to minimize the total weight \(\sum_{v\in V} f(v)\); this minimum value is the Roman domination number of \(G\), denoted by \(γ_R(G)\). In this paper, we study RDP on graph classes motivated by convexity and induced-\(P_4\) structure. First, we consider circular-convex bipartite graphs, a natural superclass of convex bipartite graphs, where RDP is already known to be polynomial-time solvable. Assuming that a circular-convex representation is given, we compute \(γ_R(G)\) in \(O(n^6)\) time by cutting the circular order, separating interval and wrap-around vertices, and branching over at most two wrap-around vertices assigned value \(2\). Second, we study triad-convex bipartite graphs, a restricted subclass of tree-convex bipartite graphs whose convexity tree is a subdivision of \(K_{1,3}\). Although RDP is hard on broader tree-convex subclasses such as star-convex and comb-convex bipartite graphs, we show that \(γ_R(G)\) can be computed in \(O(n^7)\) time on triad-convex bipartite graphs. Finally, we study \(P_4\)-tidy graphs, which properly extend cographs. Using the Giakoumakis et al. structural decomposition of \(P_4\)-tidy graphs, we give a direct, exact algorithm that computes \(γ_R(G)\) in \(O(n+m)\) time. These results extend the algorithmic boundary of Roman domination on convexity-based bipartite graphs and \(P_4\)-structured graph classes.
发表机构
- Indian Institute of Technology Guwahati（印度理工学院古瓦哈提分校）
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