Weakly Negative Circles Versus Best Clustering in Signed Graphs

TL;DR

This paper investigates the structure of signed graphs, focusing on weakly negative circles that obstruct clustering, and characterizes graphs where the minimum violation set equals the maximum number of disjoint such circles.

Contribution

It provides a characterization of signed graphs where the minimum clustering violations match the maximum number of disjoint weakly negative circles.

Findings

Identifies conditions under which $Q=w$ in signed graphs.

Characterizes graphs with this property.

Links weakly negative circles to clustering obstructions.

Abstract

Clustering a signed graph means partitioning the vertices into sets ("clusters") so that every positive edge, and no negative edge, is within a cluster. Clustering is not always possible; the obstruction is circles with exactly one negative edge ("weakly negative circles"). The correlation clustering problem is to cluster with the minimum number of edges that violate the clustering rule, called $Q$. A lower bound is $w$, the maximum number of edge-disjoint weakly negative circles. If every two such circles are edge disjoint, then $Q=w$. We characterize signed graphs of this kind.