A constructive characterization of uniformly 4-connected graphs

TL;DR

This paper provides a constructive method to characterize all uniformly 4-connected graphs by starting from basic graphs and applying specific graph operations, offering a new way to understand their structure.

Contribution

It introduces a novel constructive characterization of uniformly 4-connected graphs using graph operations on specific vertex and edge sets.

Findings

All uniformly 4-connected graphs can be generated from C5^2 or C6^2 using Δ1+ or Δ2+ operations.

The characterization simplifies understanding the structure of uniformly 4-connected graphs.

Provides a basis for further structural analysis of highly connected graphs.

Abstract

A constructive characterization of the class of uniformly $4$-connected graphs is presented. The characterization is based on the application of graph operations to appropriate vertex and edge sets in uniformly $4$-connected graphs, that is, any uniformly $4$-connected graph can be obtained from $C_5^2$ or $C_6^2$ by a number of $\Delta_1^+$ or $\Delta_2^+$-operations to quasi $4$-compatible sets.