Edges and Vertices in a Unique Signed Circle in a Signed Graph

TL;DR

This paper investigates the structural conditions in signed graphs that allow for edges or vertices to be uniquely contained in either a positive or negative circle, providing characterizations based on graph bridges.

Contribution

It introduces a characterization of the underlying graph structures needed for edges and vertices to be uniquely contained in a signed circle, differentiating positive and negative cases.

Findings

Characterization of underlying graph structures supporting unique signed circles.

Conditions for edges in a unique signed circle based on graph bridges.

Extension of edge results to vertex cases in signed graphs.

Abstract

We examine the conditions under which a signed graph contains an edge or a vertex that is contained in a unique negative circle or a unique positive circle. For an edge in a unique signed circle, the positive and negative case require the same structure on the underlying graph, but the requirements on the signature are different. We characterize the structure of the underlying graph necessary to support such an edge in terms of bridges of a circle. We then use the results from the edge version of the problem to help solve the vertex version.