Properties of uniformly $3$-connected graphs

TL;DR

This paper refines the characterization of uniformly 3-connected graphs, linking their structure to the number of vertices and construction operations, and explores how crossing numbers and treewidths are affected.

Contribution

It provides a more detailed construction-based characterization of uniformly 3-connected graphs and analyzes their structural properties, including crossing numbers and treewidths.

Findings

Refined the characterization of uniformly 3-connected graphs.

Analyzed how crossing numbers and treewidths change under construction.

Applied results to graphs with minimal vertices and degree.

Abstract

A graph on at least ${{k+1}}$ vertices is uniformly $k$-connected if each pair of its vertices is connected by $k$ and not more than $k$ independent paths. We reinvestigate a recent constructive characterization of uniformly $3$-connected graphs and obtain a more detailed result that relates the number of vertices to the operations involved in constructing a respective uniformly $3$-connected graph. Furthermore, we investigate how crossing numbers and treewidths behave under the mentioned constructions. We demonstrate how these results can be utilized to study the structure and properties of uniformly $3$-connected graphs with minimum number of vertices of minimum degree.