Parameterised distance to local irregularity

TL;DR

This paper analyzes the parameterized complexity of finding minimal modifications to graphs to make them locally irregular, introducing vertex and edge irregularity problems and exploring their computational tractability under various graph parameters.

Contribution

It provides a detailed complexity classification of vertex and edge irregularity problems, including FPT, NP-hard, and W[1]-hard results based on multiple graph parameters.

Findings

Vertex-irregulator is FPT with respect to vertex integrity, neighborhood diversity, cluster deletion.

Vertex-irregulator is W[1]-hard with respect to feedback vertex set and treedepth.

Edge-irregulator is FPT with respect to vertex integrity, NP-hard on planar bipartite graphs, W[1]-hard by solution size, feedback vertex set, treedepth.

Abstract

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.