参数化局部不规则距离
Parameterised distance to local irregularity
AI总结：
研究了局部不规则图的参数化复杂性，证明了在不同参数下计算最优顶点或边不规则消除器的复杂性。
摘要：
一个图G被称为局部不规则图，如果其相邻顶点的度数不相同。在[Fioravantes等人. 最大局部不规则诱导子图的复杂性. SWAT, 2022]中，作者引入并研究了寻找给定图G的最大局部不规则诱导子图的问题，或者等价地，计算一个子集S of V(G)的最小顺序，其删除后使G成为局部不规则图；S被称为G的最优顶点不规则消除器。在本工作中，我们深入分析了计算给定图G的最优顶点不规则消除器的参数化复杂性。此外，我们引入并研究了该问题的一个变种，其中S是G的边子集；在这种情况下，S被称为G的最优边不规则消除器。特别是，我们证明了计算图G的最优顶点不规则消除器在参数化为顶点完整性、邻域多样性或聚类删除数时属于FPT，而在参数化为反馈顶点集数或树深度时为W[1]-难。在计算图G的最优边不规则消除器的情况下，我们证明该问题在参数化为G的顶点完整性时属于FPT，但即使当G是最大度为4的平面二分图时，该问题仍为NP难，且当参数化为解的大小、反馈顶点集或树深度时为W[1]-难。我们的结果描绘了这两个问题的可处理性的全面图景，考虑了大多数标准图结构参数。
英文摘要：
A graph $G$ is \emph{locally irregular} if no two of its adjacent vertices have the same degree. In [Fioravantes et al. Complexity of finding maximum locally irregular induced subgraph. {\it SWAT}, 2022], the authors introduced and studied the problem of finding a locally irregular induced subgraph of a given a graph $G$ of maximum order, or, equivalently, computing a subset $S$ of $V(G)$ of minimum order, whose deletion from $G$ results in a locally irregular graph; $S$ is denoted as an \emph{optimal vertex-irregulator of $G$}. In this work we provide an in-depth analysis of the parameterised complexity of computing an optimal vertex-irregulator of a given graph $G$. Moreover, we introduce and study a variation of this problem, where $S$ is a substet of the edges of $G$; in this case, $S$ is denoted as an \emph{optimal edge-irregulator of $G$}. In particular, we prove that computing an optimal vertex-irregulator of a graph $G$ is in FPT when parameterised by the vertex integrity, neighborhood diversity or cluster deletion number of $G$, while it is $W[1]$-hard when parameterised by the feedback vertex set number or the treedepth of $G$. In the case of computing an optimal edge-irregulator of a graph $G$, we prove that this problem is in FPT when parameterised by the vertex integrity of $G$, while it is NP-hard even if $G$ is a planar bipartite graph of maximum degree $4$, and $W[1]$-hard when parameterised by the size of the solution, the feedback vertex set or the treedepth of $G$. Our results paint a comprehensive picture of the tractability of both problems studied here, considering most of the standard graph-structural parameters.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
