Parameterised distance to local irregularity

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.