Constructing 2- and 3-connected graphs

TL;DR

This paper explores methods for constructing 2- and 3-connected graphs, introducing new concepts like graph equivalence relations and cores, and extends the discussion to higher connectivity levels.

Contribution

It provides a novel construction framework for 2- and 3-connected graphs, including the development of the $ ext{sim}_2$-core and addressing higher connectivity cases.

Findings

Introduces a new construction method for 2-connected graphs.

Develops the concept of $ ext{sim}_2$-core for 3-connected graphs.

Addresses the case of $k$-connected graphs for $k \\geq 4$.

Abstract

This work re-examines a classical construction of a 2-connected (simple) graph where every intermediate graph is 2-connected before detailing an analogous construction for 3-connected graphs which requires a graph equivalence relation $\sim_2$ and a related concept of the $\sim_2$-core of a graph. The case of $k$-connected graphs for $k\geq 4$ is also addressed.