Chordal signed graphs and signed bigraphs

TL;DR

This paper introduces and characterizes chordal signed graphs and bigraphs, establishing their equivalence to certain digraphs and providing forbidden subgraph characterizations for these classes.

Contribution

It extends the theory of chordal graphs to signed graphs and bigraphs, offering new characterizations and translating existing digraph results.

Findings

Chordal signed graphs are equivalent to strict chordal digraphs.

Forbidden subgraph characterizations for chordal signed bigraphs are provided.

Proofs for bigraphs are more complex than for 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 Hern\'andez-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.