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

• Authors: Satoshi Taoka;Toshimasa Watanabe
• Venue: TCS '00 Proceedings of the International Conference IFIP on Theoretical Computer Science, Exploring New Frontiers of Theoretical Informatics
• Year: 2000

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|2) 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 3/2 -approximate solution if |L1| |K1| and G has less than 2σ + 6 leaves.