List Homomorphism Problems for Signed Graphs

Authors Jan Bok , Richard Brewster , Tomás Feder, Pavol Hell , Nikola Jedličková

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Jan Bok
  • Computer Science Institute, Charles University, Prague, Czech Republic
Richard Brewster
  • Department of Mathematics and Statistics, Thompson Rivers University, Kamloops, Canada
Tomás Feder
  • Independent Researcher, Palo Alto, CA, USA
Pavol Hell
  • School of Computing Science, Simon Fraser University, Burnaby, Canada
Nikola Jedličková
  • Department of Applied Mathematics, Charles University, Prague, Czech Republic

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Jan Bok, Richard Brewster, Tomás Feder, Pavol Hell, and Nikola Jedličková. List Homomorphism Problems for Signed Graphs. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 20:1-20:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


We consider homomorphisms of signed graphs from a computational perspective. In particular, we study the list homomorphism problem seeking a homomorphism of an input signed graph (G,σ), equipped with lists L(v) ⊆ V(H), v ∈ V(G), of allowed images, to a fixed target signed graph (H,π). The complexity of the similar homomorphism problem without lists (corresponding to all lists being L(v) = V(H)) has been previously classified by Brewster and Siggers, but the list version remains open and appears difficult. Both versions (with lists or without lists) can be formulated as constraint satisfaction problems, and hence enjoy the algebraic dichotomy classification recently verified by Bulatov and Zhuk. By contrast, we seek a combinatorial classification for the list version, akin to the combinatorial classification for the version without lists completed by Brewster and Siggers. We illustrate the possible complications by classifying the complexity of the list homomorphism problem when H is a (reflexive or irreflexive) signed tree. It turns out that the problems are polynomial-time solvable for certain caterpillar-like trees, and are NP-complete otherwise. The tools we develop will be useful for classifications of other classes of signed graphs, and we mention some follow-up research of this kind; those classifications are surprisingly complex.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph theory
  • complexity
  • dichotomy
  • graph homomorphism
  • signed graph


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  1. Jørgen Bang-Jensen, Pavol Hell, and Gary MacGillivray. The complexity of colouring by semicomplete digraphs. SIAM J. Discrete Math., 1(3):281-298, 1988. Google Scholar
  2. Jan Bok, Richard C. Brewster, Pavol Hell, and Nikola Jedličková. List homomorphisms of signed graphs. In Bordeaux Graph Workshop, pages 81-84, 2019. Google Scholar
  3. Richard C. Brewster. Vertex colourings of edge-coloured graphs. ProQuest LLC, Ann Arbor, MI, 1993. Thesis (Ph.D.)-Simon Fraser University (Canada). Google Scholar
  4. Richard C. Brewster, Florent Foucaud, Pavol Hell, and Reza Naserasr. The complexity of signed graph and edge-coloured graph homomorphisms. Discrete Mathematics, 340(2):223-235, 2017. Google Scholar
  5. Richard C. Brewster and Timothy Graves. Edge-switching homomorphisms of edge-coloured graphs. Discrete Math., 309(18):5540-5546, 2009. Google Scholar
  6. Richard C. Brewster and Mark Siggers. A complexity dichotomy for signed h-colouring. Discrete Mathematics, 341(10):2768-2773, 2018. Google Scholar
  7. Andrei A. Bulatov. A dichotomy theorem for nonuniform CSPs. In 58th Annual IEEE Symposium on Foundations of Computer Science - FOCS 2017, pages 319-330. IEEE Computer Soc., Los Alamitos, CA, 2017. Google Scholar
  8. Tomás Feder and Pavol Hell. List homomorphisms to reflexive graphs. Journal of Combinatorial Theory, Series B, 72(2):236-250, 1998. Google Scholar
  9. Tomás Feder, Pavol Hell, and Jing Huang. Bi-arc graphs and the complexity of list homomorphisms. Journal of Graph Theory, 42(1):61-80, 2003. Google Scholar
  10. Tomás Feder and Moshe Y. Vardi. The computational structure of monotone monadic SNP and constraint satisfaction: a study through Datalog and group theory. In STOC, pages 612-622, 1993. Google Scholar
  11. Florent Foucaud and Reza Naserasr. The complexity of homomorphisms of signed graphs and signed constraint satisfaction. In Latin American Symposium on Theoretical Informatics, pages 526-537. Springer, 2014. Google Scholar
  12. Bertrand Guenin. Packing odd circuit covers: A conjecture. Manuscript, 2005. Google Scholar
  13. Frank Harary. On the notion of balance of a signed graph. Michigan Math. J., 2:143-146 (1955), 1953/54. Google Scholar
  14. Frank Harary and Jerald A. Kabell. A simple algorithm to detect balance in signed graphs. Math. Social Sci., 1(1):131-136, 1980/81. Google Scholar
  15. Pavol Hell and Jaroslav Nešetřil. On the complexity of H-coloring. J. Combin. Theory Ser. B, 48(1):92-110, 1990. Google Scholar
  16. Pavol Hell and Jaroslav Nešetřil. Graphs and homomorphisms, volume 28 of Oxford Lecture Series in Mathematics and its Applications. Oxford University Press, Oxford, 2004. Google Scholar
  17. Peter Jeavons. On the algebraic structure of combinatorial problems. Theoret. Comput. Sci., 200(1-2):185-204, 1998. Google Scholar
  18. Reza Naserasr, Edita Rollová, and Éric Sopena. Homomorphisms of signed graphs. J. Graph Theory, 79(3):178-212, 2015. Google Scholar
  19. Reza Naserasr, Éric Sopena, and Thomas Zaslavsky. Homomorphisms of signed graphs: An update, 2019. URL:
  20. Thomas Zaslavsky. Characterizations of signed graphs. J. Graph Theory, 5(4):401-406, 1981. Google Scholar
  21. Thomas Zaslavsky. Signed graph coloring. Discrete Math., 39(2):215-228, 1982. Google Scholar
  22. Thomas Zaslavsky. Signed graphs. Discrete Appl. Math., 4(1):47-74, 1982. Google Scholar
  23. Thomas Zaslavsky. Is there a matroid theory of signed graph embedding? Ars Combin., 45:129-141, 1997. Google Scholar
  24. Thomas Zaslavsky. A mathematical bibliography of signed and gain graphs and allied areas. Electron. J. Combin., 5:Dynamic Surveys 8, 124, 1998. Manuscript prepared with Marge Pratt. Google Scholar
  25. Dmitriy Zhuk. A proof of CSP dichotomy conjecture. In 58th Annual IEEE Symposium on Foundations of Computer Science - FOCS 2017, pages 331-342. IEEE Computer Soc., Los Alamitos, CA, 2017. Google Scholar
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