On (2k,k)-connected graphs

TL;DR

This paper introduces a construction method for (2k, k)-connected graphs when k is even, generalizes previous work, and solves the minimum augmentation problem to achieve such connectivity.

Contribution

It provides a new construction for (2k, k)-connected graphs for even k and addresses the minimum edge addition problem for achieving this connectivity.

Findings

Constructed a family of (2k, k)-connected graphs for even k.

Solved the minimum edge augmentation problem for (2k, k)-connectivity.

Developed a new splitting-off theorem for these graphs.

Abstract

A graph G is called (2k, k)-connected if G is 2k-edge-connected and G-v is k-edge-connected for every vertex v. The study of (2k, k)-connected graphs is motivated by a conjecture of Frank which states that a graph has a 2-vertex-connected orientation if and only if it is (4, 2)-connected. In this paper, we provide a construction of the family of (2k, k)-connected graphs for k even which generalizes the construction given by Jord\'an for k = 2. We also solve the corresponding connectivity augmentation problem: given a graph G and an integer k \geq 2, what is the minimum number of edges to be added to make G (2k, k)-connected. Both these results are based on a new splitting-off theorem for (2k, k)-connected graphs.