Directionally 2-Signed and Bidirected Graphs

TL;DR

This paper explores the properties of directionally 2-signed and bidirected graphs, establishing a characterization of antibalanced signed graphs through reorientation of edges in bidirected graphs.

Contribution

It extends a theorem by proving that antibalanced signed graphs correspond to bidirected graphs where each vertex is a source or sink after reorientation.

Findings

Antibalanced signed graphs are characterized by all vertices being sources or sinks after reorientation.

Extension of Sriraj and Sampathkumar's theorem to bidirected graphs.

Provides a new perspective on the structure of antibalanced graphs.

Abstract

An edge uv in a graph \Gamma\ is directionally 2-signed (or, (2,d)-signed) by an ordered pair (a,b), a,b in {+,-}, if the label l(uv) = (a,b) from u to v, and l(vu) = (b,a) from v to u. Directionally 2-signed graphs are equivalent to bidirected graphs, where each end of an edge has a sign. A bidirected graph implies a signed graph, where each edge has a sign. We extend a theorem of Sriraj and Sampathkumar by proving that the signed graph is antibalanced (all even cycles and only even cycles have positive edge sign product) if, and only if, in the bidirected graph, after suitable reorientation of edges every vertex is a source or a sink.