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List Covering of Regular Multigraphs

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Combinatorial Algorithms (IWOCA 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13270))

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Abstract

A graph covering projection, also known as a locally bijective homomorphism, is a mapping between vertices and edges of two graphs which preserves incidencies and is a local bijection. This notion stems from topological graph theory, but has also found applications in combinatorics and theoretical computer science.

It has been known that for every fixed simple regular graph H of valency greater than 2, deciding if an input graph covers H is NP-complete. In recent years, topological graph theory has developed into heavily relying on multiple edges, loops, and semi-edges, but only partial results on the complexity of covering multigraphs with semi-edges are known so far. In this paper we consider the list version of the problem, called List-\(H\text {-}\textsc {Cover}\), where the vertices and edges of the input graph come with lists of admissible targets. Our main result reads that the List-\(H\text {-}\textsc {Cover}\) problem is NP-complete for every regular multigraph H of valency greater than 2 which contains at least one semi-simple vertex (i.e., a vertex which is incident with no loops, with no multiple edges and with at most one semi-edge). Using this result we almost show the NP-co/polytime dichotomy for the computational complexity of List-\(H\text {-}\textsc {Cover}\) of cubic multigraphs, leaving just five open cases.

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Notes

  1. 1.

    The proofs of statements marked with \((\spadesuit )\) will appear in the journal version of the paper.

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Acknowledgments

Jan Bok and Nikola Jedličková were supported by research grant GAČR 20-15576S of the Czech Science Foundation and by SVV–2020–260578 and GAUK 1580119. Jiří Fiala and Jan Kratochvíl were supported by research grant GAČR 20-15576S of the Czech Science Foundation. Paweł Rzążewski was supported by the Polish National Science Centre grant no. 2018/31/D/ST6/00062. The last author is grateful to Karolina Okrasa and Marta Piecyk for fruitful and inspiring discussions.

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Authors and Affiliations

  1. Computer Science Institute, Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic

    Jan Bok

  2. Department of Applied Mathematics, Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic

    Jiří Fiala, Nikola Jedličková & Jan Kratochvíl

  3. Warsaw University of Technology, Warsaw, Poland

    Paweł Rzążewski

  4. University of Warsaw, Warsaw, Poland

    Paweł Rzążewski

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  1. Jan Bok
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  2. Jiří Fiala
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  4. Jan Kratochvíl
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  5. Paweł Rzążewski
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Correspondence to Jan Bok .

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Editors and Affiliations

  1. Université Paris-Dauphine, Paris, France

    Cristina Bazgan

  2. Universität Trier, Trier, Germany

    Henning Fernau

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Bok, J., Fiala, J., Jedličková, N., Kratochvíl, J., Rzążewski, P. (2022). List Covering of Regular Multigraphs. In: Bazgan, C., Fernau, H. (eds) Combinatorial Algorithms. IWOCA 2022. Lecture Notes in Computer Science, vol 13270. Springer, Cham. https://doi.org/10.1007/978-3-031-06678-8_17

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  • DOI: https://doi.org/10.1007/978-3-031-06678-8_17

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