On cyclically 4-connected cubic graphs

TL;DR

This paper characterizes the construction of cyclically 4-connected cubic graphs, introducing the concept of cycle spread to identify minimal edge pairs needed for their generation, and provides an algorithmic approach for their enumeration.

Contribution

It generalizes Wormald's result by introducing cycle spread, reducing the set of edge pairs needed to generate all such graphs, and distinguishes between planar and non-planar cases.

Findings

Non-planar cyclically 4-connected cubic graphs (except certain known graphs) are generated from Q8 by bridging edges with cycle spread at least (1,2).

Planar cyclically 4-connected cubic graphs (except ladders) are generated similarly from ladders.

The set of edge pairs to consider is smaller than all non-adjacent pairs, simplifying graph generation.

Abstract

For $k \ge 4$, let $Q_{2k}$ and $V_{2k}$ denote the ladder and M\"obius ladder on $2k$ vertices, respectively. We prove results that build on a result by Wormald that states that any cyclically $4$-connected cubic graph other than $Q_8$ or $V_8$ is obtained from a smaller cyclically $4$-connected cubic graph by bridging a pair of non-adjacent edges. We introduce the concept of cycle spread, which generalizes the edge pair distance defined by Wormald, and show that the set of pairs of edges that needs to be considered in order to obtain all cyclically $4$-connected cubic graphs is smaller than the set of all pairs of non-adjacent edges. We prove that all non-planar cyclically $4$-connected cubic graphs with at least $10$ vertices, other than the M\"obius ladders and the Petersen graph, are obtained from $Q_8$ by bridging pairs of edges with cycle spread at least $(1,2)$. Moreover every graph obtained in this way is non-planar, cyclically $4$-connected, and cubic. All planar cyclically $4$-connected cubic graphs with at least $10$ vertices except for the ladders are obtained from the ladders by bridging pairs of edges with cycle spread at least $(1,2)$. We implemented an algorithm based on these results using McKay's nauty system for isomorphism checking.