Parameterised distance to local irregularity
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866914204512944128 |
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| author | Fioravantes, Foivos Melissinos, Nikolaos Triommatis, Theofilos |
| author_facet | Fioravantes, Foivos Melissinos, Nikolaos Triommatis, Theofilos |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_04583 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Parameterised distance to local irregularity Fioravantes, Foivos Melissinos, Nikolaos Triommatis, Theofilos Computational Complexity Discrete Mathematics Data Structures and Algorithms 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. |
| title | Parameterised distance to local irregularity |
| topic | Computational Complexity Discrete Mathematics Data Structures and Algorithms |
| url | https://arxiv.org/abs/2307.04583 |