Decomposition of cubic graphs with cyclic connectivity 5

TL;DR

This paper proves that cyclically 5-connected cubic graphs can be decomposed into components that can be completed to cyclically 5-connected graphs by adding three vertices, extending previous results for cyclic connectivity 4.

Contribution

It extends the decomposition results for cyclically 4-connected cubic graphs to the case of cyclically 5-connected graphs, providing new structural insights.

Findings

Each component can be completed to a cyclically 5-connected cubic graph by adding three vertices.

The result applies unless the component is a cycle of length five.

Extends earlier work on cyclic connectivity 4 to cyclic connectivity 5.

Abstract

Let $G$ be a cyclically $5$-connected cubic graph with a $5$-edge-cut separating $G$ into two cyclic components $G_1$ and $G_2$. We prove that each component $G_i$ can be completed to a cyclically $5$-connected cubic graph by adding three vertices, unless $G_i$ is a cycle of length five. Our work extends similar results by Andersen et al. for cyclic connectivity $4$ from 1988.