Iteration of the mincut graph operator

TL;DR

This work analyzes the mincut operator $X$ on graphs, showing that a graph is fixed by $X$ precisely when it is $r$-regular and super-$\lambda$, and proving that no graph diverges under iteration. It establishes that all graphs converge under repeated application of $X$, yielding fixed graphs, cycles, or the null graph $K_0$, and it characterizes scenarios leading to convergence or periodic behavior. The paper also develops a detailed structural understanding of mincuts (nested and crossing) and their implications for when $G\cong X(G)$, providing both necessary and sufficient conditions and several illustrative examples such as complete graphs, complete bipartite graphs, and line graphs. These results contribute to the broader study of graph operators and their dynamics, with implications for fixed points, convergence, and periodicity in graph iterations.

Abstract

A graph operator is a mapping $φ$ which maps every graph $G$ from some class of graphs to a new graph $φ(G)$. 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.

Figures (4)

• Figure 1: Mincut graphs of some well-known classes of graphs.
• Figure 2: Nested mincuts $Z,\, V,\, Y \textrm{ and } W$ of a graph $G$, where $\lambda$ is odd.
• Figure 3: Nested and overlapping mincuts $Z,\, V,\, U, \, Y \textrm{ and } W$ of a graph $G$, where $\lambda$ is even.
• Figure 4: Effect of the iteration of the mincut operator on some graphs.

Theorems & Definitions (40)

• Definition 1.1
• Definition 1.2
• Example 1.3
• Definition 1.4
• Definition 2.1
• Definition 2.2
• Proposition 2.3
• Proposition 2.4
• proof
• Corollary 2.5