List homomorphisms to separable signed graphs

TL;DR

This work provides a complete dichotomy for the complexity of List $\widehat{H}$-Colour-ing on two important families of irreflexive signed graphs: path-separable and cycle-separable. The authors develop a rich structural framework—blocks, segments, sources, and leaning patterns—that yields polynomial-time algorithms when the unicoloured edges form segmented structures and NP-hardness when segmentation fails due to chains, alternating 4-cycles, or invertible pairs. For path-separable graphs, segmentation (right/left or left-right) characterizes tractable cases via a special bipartite min ordering, while non-segmented instances induce NP-complete problems through chain-based reductions. In cycle-separable graphs, a small set of switch-equivalent configurations (including $\widehat{H_0}$, $\widehat{H_1}$, and $\widehat{H_\ell}$ with odd $\ell$) yields polynomial-time solvability, with NP-hardness recovered through gadget-based reductions otherwise. Overall, the paper advances the understanding of signed-graph homomorphisms and provides structural insights likely to inform a broader classification of irreflexive signed graphs.

Abstract

The complexity of the list homomorphism problem for signed graphs appears difficult to classify. Existing results focus on special classes of signed graphs, such as trees and reflexive signed graphs. Irreflexive signed graphs are in a certain sense the heart of the problem, as noted by a recent paper of Kim and Siggers. We focus on a special class of irreflexive signed graphs, namely those in which the unicoloured edges form a spanning path or cycle, which we call separable signed graphs. We classify the complexity of list homomorphisms to these separable signed graphs; we believe that these signed graphs will play an important role for the general resolution of the irreflexive case. We also relate our results to a conjecture of Kim and Siggers concerning the special case of semi-balanced irreflexive signed graphs; we have proved the conjecture in another paper, and the present results add structural information to that topic.

Figures (6)

• Figure 1: A signed graph (on the left) together with a chain (on the right). The upper walk is $U$, the lower walk is $D$; the dotted blue edges must be absent.
• Figure 2: The family $\cal F$ of signed graphs yielding NP-complete problems, and a chain in each. (The figure appeared first in mfcs.)
• Figure 3: The graph $F_1$, with an invertible pair.
• Figure 4: An example of a left-right-segmented signed graph. The additional bicoloured edges from all white vertices before $v_{12}$ to all black vertices after $v_{15}$ are not shown.
• Figure 5: The cycle-separable signed graphs $\widehat{H}_0$, $\widehat{H}_1$, and $\widehat{H}_\ell$ with $\ell \geq 3$ odd.

Theorems & Definitions (16)

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