New concept of connection in signed graphs

TL;DR

This paper introduces a new concept of connectivity in signed graphs based on the existence of both positive and negative chains, and explores its properties and implications, especially in graphs without positive cycles.

Contribution

It proposes a novel definition of connection in signed graphs and analyzes its properties, linking them to matroid theory and special classes of signed graphs.

Findings

Defined sign components, sign articulation vertices, and sign isthmi.

Compared new concepts to classical graph and matroid properties.

Applied results to signed graphs without positive cycles and with all negative edges.

Abstract

In a signed graph each edge has a sign, $+1$ or $-1$. We introduce in the present paper a new definition of connection in a signed graph by the existence of both positive and negative chains between vertices. We prove some results and properties of this definition, such as sign components, sign articulation vertices, and sign isthmi, and we compare them to corresponding graph and signed-graphic matroid properties. We apply our results to signed graphs without positive cycles. For signed graphs in which every edge is negative our properties become parity properties.