3-connected graphs and their degree sequences

TL;DR

This paper establishes precise conditions for degree sequences to correspond to 3-connected graphs and introduces a matrix that partitions all such graphs, advancing understanding of their structural properties.

Contribution

It provides necessary and sufficient conditions for degree sequences of 3-connected graphs and introduces a matrix to organize these graphs.

Findings

Characterization of degree sequences for 3-connected graphs

Conditions for sequences to be exclusively realizable as 3-connected graphs

A matrix that partitions the set of 3-connected graphs

Abstract

Necessary and sufficient conditions for a sequence of positive integers to be the degree sequence of a 3-connected simple graph are detailed. Conditions are also given under which such a sequence is necessarily 3-connected i.e. the sequence can only be realised as a 3-connected graph. Finally, a matrix is introduced whose non-empty entries partition the set of 3-connected graphs.