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Min Orderings and List Homomorphism Dichotomies for Graphs and Signed Graphs

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Abstract

Since the CSP dichotomy conjecture has been established, a number of other dichotomy questions have attracted interest, including one for list homomorphism problems of signed graphs. Signed graphs arise naturally in many contexts, including for instance nowhere-zero flows for graphs embedded in non-orientable surfaces. The dichotomy classification is known for homomorphisms without list restrictions, so it is surprising that it is not known, or even conjectured, if lists are present since this usually makes the classifications easier to obtain. There is however a conjectured classification, due to Kim and Siggers, in the special case of “semi-balanced” signed graphs. These authors confirmed their conjecture for the class of reflexive signed graphs. As our main result we verify the conjecture for irreflexive signed graphs. For this purpose, we prove an extension result for two-directional ray graphs which is of independent interest and which leads to an analogous extension result for interval graphs. Moreover, we offer an alternative proof for the class of reflexive signed graphs, and a direct polynomial-time algorithm in the polynomial cases where the previous algorithms used algebraic methods of general CSP dichotomy theorems. For both reflexive and irreflexive cases the dichotomy classification depends on a result linking the absence of certain structures to the existence of a special ordering. The structures are used to prove the NP-completeness and the ordering is used to design polynomial algorithms.

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Acknowledgements

J. Bok and N. Jedličková were supported by GAUK 370122 and European Union’s Horizon 2020 Project H2020-MSCA-RISE-2018: Research and Innovation Staff Exchange. R. Brewster and P. Hell gratefully acknowledge support from the NSERC Canada Discovery Grant programme. A. Rafiey gratefully acknowledges support from the Grant NSF1751765. J. Bok further acknowledges a partial support by the ANR Project GRALMECO (ANR-21-CE48-0004). We thank Reza Naserasr and Mark Siggers for helpful discussions. We are also grateful to two anonymous referees for their constructive comments that helped us to improve the manuscript.

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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 Mathematics and Statistics, Thompson Rivers University, Kamloops, BC, Canada

    Richard C. Brewster

  3. School of Computing Science, Simon Fraser University, Vancouver, BC, Canada

    Pavol Hell

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

    Nikola Jedličková

  5. Mathematics and Computer Science, Indiana State University, Terre Haute, IN, USA

    Arash Rafiey

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

    Jan Bok

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Bok, J., Brewster, R.C., Hell, P. et al. Min Orderings and List Homomorphism Dichotomies for Graphs and Signed Graphs. Algorithmica (2024). https://doi.org/10.1007/s00453-024-01228-1

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