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 the first 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, we also give a min-max formula for the size of a smallest augmenting set. 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.
CITATION STYLE
Jackson, B., & Jordaá, T. (2001). Independence free graphs and vertex connectivity augmentation. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 2081, pp. 264–279). Springer Verlag. https://doi.org/10.1007/3-540-45535-3_21
Mendeley helps you to discover research relevant for your work.