Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
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Towards A Dichotomy for the List Switch Homomorphism Problem for Signed Graphs

  • Hyobeen Kim (Kyungpook National University Mathematics Department) ;
  • Mark Siggers (Kyungpook National University Mathematics Department)
  • Received : 2023.02.05
  • Accepted : 2023.07.31
  • Published : 2023.09.30
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Abstract

We make advances towards a structural characterisation of the signed graphs H for which the list switch H-colouring problem List-S-Hom(H) can be solved in polynomial time. We conjecture two different characterisations, the second refining the first, in the case that the graph H can be switched to a graph in which every negative edge is also positive. Using a recent proof of the first characterisations for reflexive signed graphs, by Bok et. al., we prove the second characterisation for reflexive signed graphs. We also provide several tools for reducing the problem to the bipartite case, and prove a full complexity dichotomy for a related problem.

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Acknowledgement

We thank the authors of [1] for early copies of their manuscript, and Pavol Hell in particular for generously directing us towards key ideas on several occasions. We thank an anonymous reviewer for a very detailed reading which revealed essential mistakes in an earlier version of the paper.

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