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

TL;DR

This paper advances the understanding of the computational complexity of the list switch homomorphism problem for signed graphs, proposing a conjecture and proving it in the reflexive case to identify polynomial-time solvable instances.

Contribution

It introduces a conjecture characterizing signed graphs with polynomial-time list switch homomorphism problems and proves this characterization for reflexive signed graphs.

Findings

Conjecture proposed for signed graphs switch homomorphism complexity

Proved the conjecture for reflexive signed graphs

Identified structural properties leading to polynomial-time solvability

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.