Chordal signed graphs and signed bigraphs
Jing Huang, Ying Ying Ye
TL;DR
The paper extends the classical notion of chordality to signed graphs and signed bigraphs, linking chordal signed graphs to strict chordal digraphs and providing a forbidden-subgraph characterization. It develops a layered analysis by considering complete, non-separable, and separable bigraphs, introducing a comprehensive family of forbidden subgraphs (including F1–F6, cycles C2k, and the S/J+lollipop constructions) and new structural tools (signed simplicial edges, canonical orderings, and edge-colour representations). The main contribution is a unifying theorem: a signed bigraph is chordal if and only if it is free of all graphs in the assembled family ${ t F}$ as induced subgraphs, with proofs built from detailed case analyses and inductive arguments. This characterization advances algorithmic recognition and structural understanding of chordality in the signed setting, and connects to existing work on strict chordal digraphs via the transformation between digraphs and signed graphs.
Abstract
Chordal graphs and chordal bigraphs enjoy beautiful characterizations, in terms of forbidden subgraphs, vertex/edge orderings, vertex/edge separating sets, and tree-like representations. In this paper, we introduce chordal signed graphs and chordal signed bigraphs. Interestingly, chordal signed graphs are equivalent to strict chordal digraphs studied by Hell and Hernández-Cruz. A forbidden subdigraph characterization of strict chordal digraphs can be translated to a forbidden subgraph characterization of chordal signed graphs. We give a forbidden subgraph characterization of chordal signed bigraphs. The forbidden subgraphs for chordal signed bigraphs are analogous to those for chordal signed graphs but the proofs are much more complicated and intriguing.