发表机构
National Institute of Technology Calicut（国家理工学院卡利卡特分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究完美意大利支配问题在分裂图上的复杂性，给出$O(n^4)$算法，证明其可解性边界紧，并在分裂图近邻及受限扩展上建立NP困难与不可近似结果。
AI 中文摘要
完美意大利支配函数为每个顶点分配一个来自集合$\{0,1,2\}$的标签，使得每个标签为零的顶点的邻域内标签之和恰好等于二。尽管相关的判定问题（PID）在弦图上为NP完全，但在分裂图上的复杂性状态仍然开放。我们给出了一个在$n$个顶点的分裂图上运行时间为$O(n^4)$的算法。我们注意到这与罗马支配、意大利支配和完美罗马支配形成对比，这些在分裂图上均为NP完全。我们进一步表明这一可解性边界是紧的：PID在距离分裂图删除距离为一的图上是NP困难的。我们还提供了由附加悬挂星构成的分裂图受限扩展上的PID二分结果，并为相同情况建立了不可近似性结果。
英文摘要
A perfect Italian dominating function assigns a label from $\{0,1,2\}$ to each vertex so that the labels in the neighborhood of every zero-labeled vertex sum to exactly two. Although the associated decision problem (PID) is NP-complete on chordal graphs, the complexity status on split graphs remains open. We give an $O(n^4)$ time algorithm for split graphs on $n$ vertices. We note that this is in contrast to Roman, Italian, and perfect Roman domination, which are NP-complete on split graphs. We further show that this tractability boundary is tight: PID is NP-hard on graphs at deletion distance one from split graphs. We also provide dichotomy results for PID on restricted extensions of split graphs formed by attaching pendant stars, and establish inapproximability results for the same.
CommentsSubmitted to CALDAM 2027
