Note on minimally k-connected graphs

TL;DR

This paper presents two efficient algorithms for deriving minimally k-connected graphs from k-trees, exploiting properties of vertex degrees to ensure minimal connectivity.

Contribution

It introduces two O(n^2) algorithms for transforming k-trees into minimally k-connected graphs, expanding understanding of graph connectivity structures.

Findings

Algorithms run in quadratic time (O(n^2)).

Identifies degree properties of insensitive edges in k-trees.

Provides a method to reduce k-trees to minimally k-connected graphs.

Abstract

A k-tree is either a complete graph on (k+1) vertices or given a k-tree G' with n vertices, a k-tree G with (n+1) vertices can be constructed by introducing a new vertex v and picking a k-clique Q in G' and then joining each vertex u in Q. A graph G is k-edge-connected if the graph remains connected even after deleting fewer edges than k from the graph. A k-edge-connected graph G is said to be minimally k-connected if G \ {e} is no longer k-edge-connected for any edge e belongs to E(G) where E(G) denotes the set of edges of G. In this paper we find two separate O (n2) algorithms so that a minimally 2-connected graph can be obtained from a 2-tree and a minimally k-connected graph can be obtained from a k-tree. In a k-tree (k \geq 2) we find the edges which are insensitive to the k-connectivity have both their end vertices of degrees greater than or equal to k+1.This property is fully exploited to find an algorithm which reduces any k-tree to a minimally k-connected graph.