Min orderings and list homomorphism dichotomies for signed and unsigned graphs

TL;DR

This paper classifies the complexity of list homomorphism problems for signed graphs, confirming a conjecture for certain cases and providing new polynomial algorithms and theoretical results.

Contribution

It confirms Kim and Siggers' conjecture for reflexive and irreflexive weakly balanced signed graphs, extending previous work and introducing new algorithms and theoretical insights.

Findings

Confirmed conjecture for reflexive signed graphs

Established new polynomial algorithms for specific cases

Derived a theorem on min orderings of bipartite graphs

Abstract

The CSP dichotomy conjecture has been recently established, but a number of other dichotomy questions remain open, including the dichotomy classification of list homomorphism problems for signed graphs. Signed graphs arise naturally in many contexts, including for instance nowhere-zero flows for graphs embedded in non-orientable surfaces. For a fixed signed graph $\widehat{H}$, the list homomorphism problem asks whether an input signed graph $\widehat{G}$ with lists $L(v) \subseteq V(\widehat{H}), v \in V(\widehat{G}),$ admits a homomorphism $f$ to $\widehat{H}$ with all $f(v) \in L(v), v \in V(\widehat{G})$. Usually, a dichotomy classification is easier to obtain for list homomorphisms than for homomorphisms, but in the context of signed graphs a structural classification of the complexity of list homomorphism problems has not even been conjectured, even though the classification of the complexity of homomorphism problems is known. Kim and Siggers have conjectured a structural classification in the special case of ``weakly balanced" signed graphs. We confirm their conjecture for reflexive and irreflexive signed graphs; this generalizes previous results on weakly balanced signed trees, and weakly balanced separable signed graphs \cite{separable,trees}. In the reflexive case, the result was first presented in \cite{KS}, with the proof using some of our results included in this paper. In fact, here we present our full proof, as an alternative to the proof in \cite{KS}. In particular, we provide direct polynomial algorithms where previously algorithms relied on general dichotomy theorems. The irreflexive results are new, and their proof depends on first deriving a theorem on extensions of min orderings of (unsigned) bipartite graphs, which is interesting on its own. [shortened, full abstract in PDF]