Minimal and Disjoint Negation Sets in Signed Graphs

TL;DR

This paper studies negation sets in signed graphs, providing algorithms and properties related to minimality, disjointness, and bipartiteness, advancing understanding of graph balance and negation configurations.

Contribution

It introduces algorithms for finding maximum disjoint negation sets and characterizes bipartite negation sets in certain classes of graphs.

Findings

Tests for minimal, minimum, and unique negation sets

Disjoint negation sets must be bipartite

Algorithms for maximum disjoint negation sets

Abstract

A signed graph is a graph with a function that assigns a label of positive or negative to each edge. The sign of a circle is the product of the signs of its edges; a graph is balanced if all of its circles are positive. A set of edges whose negation yields a balanced graph is a negation set. Results: tests to determine whether a negation set is minimal, minimum, or the unique minimum; any two disjoint negation sets must be bipartite; two classes of graphs are shown to have bipartite negation sets (in general, existence is an unsolved problem); I give an algorithm which finds a maximum family of disjoint negation sets that includes a given negation set.