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List Covering of Regular Multigraphs with Semi-edges

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In line with the recent development in topological graph theory, we are considering undirected graphs that are allowed to contain multiple edges, loops, and semi-edges. A graph is called simple if it contains no semi-edges, no loops, and no multiple edges. A graph covering projection, also known as a locally bijective homomorphism, is a mapping between vertices and edges of two graphs which preserves incidences and which is a local bijection on the edge-neighborhood of every vertex. 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. Graphs with semi-edges have been considered in this context only recently and only partial results on the complexity of covering such graphs are known so far. In this paper we consider the list version of the problem, called List-H-Cover, where the vertices and edges of the input graph come with lists of admissible targets. Our main result reads that the List-H-Cover problem is NP-complete for every regular graph 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 show the NP-co/polytime dichotomy for the computational complexity of List-H-Cover for cubic graphs.

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Acknowledgements

Jan Bok and Nikola Jedličková were supported by research grant GAČR 20-15576 S of the Czech Science Foundation and by SVV–2020–260578 and GAUK 1580119. Jiří Fiala and Jan Kratochvíl were supported by research grant GAČR 20-15576 S 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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Bok, J., Fiala, J., Jedličková, N. et al. List Covering of Regular Multigraphs with Semi-edges. Algorithmica 86, 782–807 (2024). https://doi.org/10.1007/s00453-023-01163-7

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