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Towards a dichotomy for the switch list homomorphism problem for signed graphs

Hyobin Kim, Mark Siggers

Abstract

We make advances towards a structural characterisation of the signed graphs $H$ for which the list switch $H$-colouring problem $\operatorname{LSwHom}(H)$ problem is polynomial time solvable. We conjecture a characterisation for signed graphs that can be switched to graphs such that every negative edge is also positive, and prove the characterisation in the case that the signed graph is reflexive.

Towards a dichotomy for the switch list homomorphism problem for signed graphs

Abstract

We make advances towards a structural characterisation of the signed graphs for which the list switch -colouring problem problem is polynomial time solvable. We conjecture a characterisation for signed graphs that can be switched to graphs such that every negative edge is also positive, and prove the characterisation in the case that the signed graph is reflexive.

Paper Structure

This paper contains 12 sections, 16 theorems, 15 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Structures, CSP, and List Homomorphism
  4. The switching graph
  5. Definition of List Switch-Homomorphism
  6. Collapses of Conservative Polymorphisms
  7. Dichotomy for List $\cS_{H}$-colouring
  8. Reduction to bipartite graphs and bipartite-min orderings
  9. The bipartite resolution and the constant parity subgraph
  10. bipartite-min orderings
  11. Bipartite-min orderings and purple-red graphs
  12. Possible and impossible polymorphism collapses

Key Result

Theorem 1.1

For a $\mathrm{br}$-graph $H$, the problem $\sHom(H)$ is in $\P$ if the switch-core of $H$ has at most two edges, and is otherwise in $\NPC$.

Figures (3)

  • Figure 1: A $\mathrm{br}$-graph $H$ and its switching graph $\cS_{H}$.
  • Figure 2: A $b$-excluder for $(a,b,c)$
  • Figure 3: The tree $T$, the switch-core of $\cP_{T}$, and its red subgraph $R$

Theorems & Definitions (39)

  • Theorem 1.1: BS
  • Conjecture 1.2
  • Conjecture 1.3
  • Conjecture 1.4
  • Theorem 2.1: The $\CSP$-dichotomy, Bu17Zh17
  • Corollary 2.2
  • Definition 2.3
  • Proposition 2.4: BFHN
  • proof
  • Definition 2.6
  • ...and 29 more