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.

翻译：图 $G$ 称为**局部不规则图**，若其任意相邻顶点均不具有相同度数。在 [Fioravantes 等. 寻找最大局部不规则诱导子图的复杂度. {\it SWAT}, 2022] 中，作者引入并研究了在给定图 $G$ 中寻找最大阶的局部不规则诱导子图问题，等价于计算 $V(G)$ 的最小阶子集 $S$，使得从 $G$ 中删除 $S$ 后得到局部不规则图；$S$ 称为 **$G$ 的最优顶点不规则调节器**。本文深入分析了计算给定图 $G$ 的最优顶点不规则调节器的参数化复杂度。此外，我们引入并研究了该问题的变体，其中 $S$ 为 $G$ 的边子集；此时 $S$ 称为 **$G$ 的最优边不规则调节器**。特别地，我们证明：当以图 $G$ 的顶点完整性、邻域多样性或团簇删除数为参数时，计算 $G$ 的最优顶点不规则调节器属于 FPT；但当以反馈顶点集数或树深度为参数时，该问题为 $W[1]$-难。对于计算 $G$ 的最优边不规则调节器，我们证明当以 $G$ 的顶点完整性为参数时该问题属于 FPT，但即使 $G$ 是最大度为 $4$ 的平面二部图，该问题仍是 NP-难的，且当以解规模、反馈顶点集或树深度为参数时为 $W[1]$-难。我们的结果全面揭示了本文所研究的两个问题的可解性，涵盖了大多数标准图结构参数。