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 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.
This research was carried out while the first named author was visiting the Department of Operations Research, Eötvös University, Budapest, supported by an Erdős visiting professorship from the Alfréd Rényi Mathematical Institute of the Hungarian Academy of Sciences.
Supported by the Hungarian Scientific Research Fund, grant no. T029772, T030059, F034930 and FKFP grant no. 0607/1999.
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Jackson?, B., Jordná??, T. (2001). Independence Free Graphs and Vertex connectivity Augmentation. In: Aardal, K., Gerards, B. (eds) Integer Programming and Combinatorial Optimization. IPCO 2001. Lecture Notes in Computer Science, vol 2081. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45535-3_21
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DOI: https://doi.org/10.1007/3-540-45535-3_21
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