Parameterised distance to local irregularity
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A graph G is 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. 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 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 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.
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