Iteration of the mincut graph operator

TL;DR

This paper investigates the properties of the mincut graph operator, focusing on its iterative effects, fixed points, and convergence, revealing conditions for stability and posing open questions for future research.

Contribution

It characterizes graphs that remain unchanged under the mincut operator and proves that no graph diverges when the operator is iterated.

Findings

Graphs that are super edge-connected and regular are fixed points.

No graph diverges under repeated application of the mincut operator.

The paper establishes necessary and sufficient conditions for stability.

Abstract

A graph operator is a mapping $\phi$ which maps every graph $G$ from some class of graphs to a new graph $\phi(G)$. In this paper, we introduce and study the properties of the mincut operator, specifically the effects of iteration of the operator. We show that the property of being super edge-connected and regular is both necessary and sufficient for a graph to remain fixed under the mincut operator. Furthermore, we show that no graph diverges under iteration of this operator. We conclude by stating further research questions on the mincut operator.