The (σ + 1)-Edge-Connectivity Augmentation Problem without Creating Multiple Edges of a Graph

Abstract

The unweighted k-edge-connectivity augmentation problem (kECA for short) is defined by ”Given a σ-edge-connected graph G = (V, E), find an edge set E′ of minimum cardinality such that G′ = (V, E∪ E′) is (σ+δ)-edge-connected and σ+δ = k”, where E′ is called a solution to the problem. Let kECA(S,SA) denote kECA such that both G and G′ are simple. The subject of the present paper is (σ + 1)ECA(S,SA) (or kECA(S,SA) with k = σ + 1). Let M be any maximum matching of a certain graph R(G) whose vertex set VR consists of vertices representing all leaves of G. From M we obtain an edge set E′ 0, with |E′ 0| = |M|, such that each edge connects vertices in distinct leaves of G. Let L1 be the set of leaves to be created by adding E′ 0 to G, and K1 the set of remaining leaves of G. The main result is to propose two O(σ2|V | log(|V |/σ)+ |E|+ |VR|) time algorithms for finding the following solutions: (1) an optimum solution if G has at least 2σ + 6 leaves or if |L1| ≤ |K1| and G has less than 2σ + 6 leaves; (2) a 2 -approximate solution if |L1| > |K1| and G has less than 2σ + 6 leaves.

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Cite this paper

@inproceedings{Taoka2000The, title={The (σ + 1)-Edge-Connectivity Augmentation Problem without Creating Multiple Edges of a Graph}, author={Satoshi Taoka and Toshimasa Watanabe}, year={2000} }