奇数支配及其推广的参数复杂性
Parameterized Complexity of Odd Domination and its Generalization
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中文总结 AI 辅助
研究奇数支配及其推广奇偶支配的参数复杂性，按解大小\(k\)参数化时在受限图上建立\(W[1]\) - 难性及与围长相关边界，还研究按结构图参数参数化时的情况。
中文摘要 AI 辅助
在“奇数支配”问题中，给定一个图\(G\)和一个正整数\(k\)，任务是确定\(G\)中是否存在顶点子集\(D\)，使得\(G\)中每个顶点的闭邻域包含\(D\)中的奇数个顶点。本文研究该问题的计算复杂性。当按解的大小\(k\)参数化时，在一些受限图上建立了\(W[1]\) - 难性，以及关于输入图围长的固定参数可处理性和\(W[1]\) - 难性之间的清晰边界。然后研究了按几个结构图参数参数化时的问题。此外，还研究了“奇偶支配”（奇数支配的推广）的参数复杂性。
英文摘要
In the \textsc{Odd Domination} problem, given a graph $G$ and a positive integer $k$, the task is to determine whether there exists a vertex subset $D$ of $G$ such that the closed neighborhood of each vertex in $G$ contains an odd number of vertices from $D$. In this paper, we investigate the computational complexity of the problem. When parameterized by the solution size $k$, we establish W[1]-hardness on some restricted graphs and a sharp boundary between fixed-parameter tractability and W[1]-hardness with respect to the girth of the input graph. Then, we address the problem when parameterized by several structural graph parameters. Furthermore, we investigate the parameterized complexity of \textsc{Parity Domination}, which is a generalization of \textsc{Odd Domination}.
