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List Homomorphism Problems for Signed Graphs

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

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Author Details

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

Funding

The second and fourth authors were supported by their NSERC Canada Discovery Grants. The work of the first and fifth author was also partially supported by the grant SVV–2020–260578.
  • Bok, Jan: The first author received funding from the European Union’s Horizon 2020 project H2020-MSCA-RISE-2018: Research and Innovation Staff Exchange and from Charles University Grant Agency project 1580119.
  • Jedličková, Nikola: The fifth author was partially supported by the fourth author’s NSERC Canada Discovery Grant, and partially supported by Charles University Grant Agency project 1198419.

Cite AsGet BibTex

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)
https://doi.org/10.4230/LIPIcs.MFCS.2020.20

Abstract

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
Keywords
  • complexity
  • dichotomy
  • graph homomorphism
  • signed graph

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