Independence free graphs and vertex connectivity augmentation

Bill Jackson, T. Jordán

Research output: Contribution to journalArticle

40 Citations (Scopus)

Abstract

Given an undirected graph G and a positive integer k, the k-vertex-connectivity augmentation problem is to find a smallest set F of new edges for which G + F is k-vertex-connected. Polynomial algorithms for this problem have been found only for k ≤ 4 and a major open question in graph connectivity is whether this problem is solvable in polynomial time in general. In this paper, we develop an algorithm which delivers an optimal solution in polynomial time for every fixed k. In the case when the size of an optimal solution is large compared to k, our algorithm is polynomial for all k. We also derive a min-max formula for the size of a smallest augmenting set in this case. A key step in our proofs is a complete solution of the augmentation problem for a new family of graphs which we call k-independence free graphs. We also prove new splitting off theorems for vertex connectivity.

Original languageEnglish
Pages (from-to)31-77
Number of pages47
JournalJournal of Combinatorial Theory. Series B
Volume94
Issue number1
DOIs
Publication statusPublished - May 2005

Fingerprint

Graph Connectivity
Vertex Connectivity
Augmentation
Polynomials
Polynomial time
Optimal Solution
Polynomial Algorithm
Graph in graph theory
Min-max
Undirected Graph
Polynomial
Integer
Independence
Vertex of a graph
Theorem

Keywords

  • Algorithms
  • Connectivity of graphs
  • Vertex-connectivity augmentation

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Theoretical Computer Science

Cite this

Independence free graphs and vertex connectivity augmentation. / Jackson, Bill; Jordán, T.

In: Journal of Combinatorial Theory. Series B, Vol. 94, No. 1, 05.2005, p. 31-77.

Research output: Contribution to journalArticle

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