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Parameterised Distance to Local Irregularity

Authors Foivos Fioravantes , Nikolaos Melissinos , Theofilos Triommatis

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ACM Subject Classification
  • Mathematics of computing → Graph algorithms
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • Locally irregular
  • largest induced subgraph
  • FPT
  • W-hardness

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Abstract

A graph G is locally irregular if no two of its adjacent vertices have the same degree. The authors of [Fioravantes et al. Complexity of finding maximum locally irregular induced subgraph. SWAT, 2022] introduced and provided some initial algorithmic results on the problem of finding a locally irregular induced subgraph of a given 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 called 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 subset of the edges of G; in this case, S is denoted as an optimal edge-irregulator of G. We prove that computing an optimal vertex-irregulator of a graph G is in FPT when parameterised by various structural parameters of G, while it is W[1]-hard when parameterised by the feedback vertex set number or the treedepth of G. Moreover, computing an optimal edge-irregulator of a graph G 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 6, 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.

Cite As Get BibTex

Foivos Fioravantes, Nikolaos Melissinos, and Theofilos Triommatis. Parameterised Distance to Local Irregularity. In 19th International Symposium on Parameterized and Exact Computation (IPEC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 321, pp. 18:1-18:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.IPEC.2024.18

Author Details

Foivos Fioravantes
  • Department of Theoretical Computer Science, FIT, Czech Technical University in Prague, Czech Republic
Nikolaos Melissinos
  • Department of Theoretical Computer Science, FIT, Czech Technical University in Prague, Czech Republic
Theofilos Triommatis
  • School of Electrical Engineering, Electronics and Computer Science University of Liverpool, UK

Funding

This work was co-funded by the European Union under the project Robotics and advanced industrial production (reg. no. CZ.02.01.01/00/22_008/0004590). Foivos Fioravantes and Nikolaos Melissinos are also supported by the CTU Global postdoc fellowship program.

Acknowledgements

The authors would like to thank Dániel Marx for his important contribution towards rendering the proof of Theorem 7 more elegant.

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