AI 中文总结
研究顶点分裂操作，证明余图顶点分裂、(P_t)-自由顶点分裂、弦图顶点分裂及单位区间顶点分裂是NP完全的，结果扩展到排他和浅变体，在指数时间假设下给出这些问题不存在特定时间复杂度算法的结论。
AI 中文摘要
顶点分裂将一个顶点(v)替换为两个不相邻的顶点，其邻域合并起来等于N(v)。若这些邻域不相交，则分裂是“排他的”；若没有新创建的顶点再次被分裂，则分裂是“浅的”。对于图属性(Π)，(Π)-顶点分裂问题询问最多(k)次分裂能否将图(G)转换为满足(Π)的图。我们继续对该操作进行系统研究并解决了几个开放问题。首先，我们证明了余图顶点分裂即使在围长至少为5 的图上也是NP完全的，解决了Firbas和Sorge(ISAAC 2024)的一个问题。更一般地，对于每个固定的(t≥4)，(P_t)-自由顶点分裂是NP完全的。我们还证明了弦图顶点分裂和单位区间顶点分裂是NP完全的，解决了Abu-Khzam、Chakraborty、Isenmann和Oijid(IWOCA 2026)的两个问题。我们的难度结果扩展到了排他和浅变体。假设指数时间假设，这些问题都不存在运行时间为(2^{o(k)}n^{O(1)})的算法；此外，除了单位区间情况外，都不存在运行时间为(2^{o(n)})的算法。
英文摘要
Vertex splitting replaces a vertex (v) by two nonadjacent vertices whose neighborhoods together equal (N(v)). A split is \emph{exclusive} if these neighborhoods are disjoint and \emph{shallow} if no newly created vertex is split again. For a graph property (Π), \textsc{(Π)-Vertex Splitting} asks whether at most (k) splits can transform a graph (G) into one satisfying (Π). We continue the systematic study of this operation and settle several open problems. First, we prove that \textsc{Cograph Vertex Splitting} is \textsf{NP}-complete, even on graphs of girth at least 5, resolving a question of Firbas and Sorge (ISAAC 2024). More generally, \textsc{(P_t)-free Vertex Splitting} is \textsf{NP}-complete for every fixed (t\geq 4). We also prove that \textsc{Chordal Vertex Splitting} and \textsc{Unit-Interval Vertex Splitting} are \textsf{NP}-complete, resolving two questions of Abu-Khzam, Chakraborty, Isenmann, and Oijid (IWOCA 2026). Our hardness results extend to the exclusive and shallow variants. Assuming the Exponential Time Hypothesis, none of these problems admits an algorithm running in (2^{o(k)}n^{O(1)}) time; moreover, except for the unit-interval cases, none admits an algorithm running in (2^{o(n)}) time.
CommentsAbstract shortened for arxiv submission
