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Computational Complexity of Covering Colored Mixed Multigraphs with Degree Partition Equivalence Classes of Size at Most Two (Extended Abstract)

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Graph-Theoretic Concepts in Computer Science (WG 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14093))

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Abstract

The notion of graph covers (also referred to as locally bijective homomorphisms) plays an important role in topological graph theory and has found its computer science applications in models of local computation. For a fixed target graph H, the H-Cover problem asks if an input graph G allows a graph covering projection onto H. Despite the fact that the quest for characterizing the computational complexity of H-Cover had been started more than 30 years ago, only a handful of general results have been known so far.

In this paper, we present a complete characterization of the computational complexity of covering colored graphs for the case that every equivalence class in the degree partition of the target graph has at most two vertices. We prove this result in a very general form. Following the lines of current development of topological graph theory, we study graphs in the most relaxed sense of the definition - the graphs are mixed (they may have both directed and undirected edges), may have multiple edges, loops, and semi-edges. We show that a strong P/NP-co dichotomy holds true in the sense that for each such fixed target graph H, the H-Cover problem is either polynomial time solvable for arbitrary inputs, or NP-complete even for simple input graphs.

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Acknowledgments

All co-authors were supported by research grant GAčR 20-15576S of the Czech Science Foundation, Nikola Jedličková was further supported by SVV–2020–260578 and GAUK 370122. Jan Bok was partially financed by the ANR project GRALMECO (ANR-21-CE48-0004) and the French government IDEX-ISITE initiative 16-IDEX-0001 (CAP 20-25).

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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 & Michaela Seifrtová

  3. Université Clermont Auvergne, CNRS, Clermont Auvergne INP, Mines Saint-Étienne, LIMOS, 63000, Clermont-Ferrand, France

    Jan Bok

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  1. Jan Bok
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  2. Jiří Fiala
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  3. Nikola Jedličková
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  4. Jan Kratochvíl
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  5. Michaela Seifrtová
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Correspondence to Jan Bok or Jan Kratochvíl .

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Editors and Affiliations

  1. Durham University, Durham, UK

    Daniël Paulusma

  2. University of Fribourg, Fribourg, Fribourg, Switzerland

    Bernard Ries

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Bok, J., Fiala, J., Jedličková, N., Kratochvíl, J., Seifrtová, M. (2023). Computational Complexity of Covering Colored Mixed Multigraphs with Degree Partition Equivalence Classes of Size at Most Two (Extended Abstract). In: Paulusma, D., Ries, B. (eds) Graph-Theoretic Concepts in Computer Science. WG 2023. Lecture Notes in Computer Science, vol 14093. Springer, Cham. https://doi.org/10.1007/978-3-031-43380-1_8

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  • DOI: https://doi.org/10.1007/978-3-031-43380-1_8

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