Independence free graphs and vertex connectivity augmentation

  • Authors:

  • Bill Jackson;Tibor Jordán

  • Affiliations:

  • School of Mathematical Sciences, Queen Mary, University of London, Mile End Road, London E1 4NS, England, UK;Department of Operations Research, Eötvös University, Pázmány Péter sétány 1/C, 1117 Budapest, Hungary

  • Venue:
  • Journal of Combinatorial Theory Series B
  • Year:
  • 2005

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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.