A faster edge splitting algorithm in multigraphs and its application to the edge-connectivity augmentation problem

Abstract

This paper first shows that, given a multigraph G and a vertex s with even degree, all edges incident to s can be split off (i.e., if G is k-edge-connected, then the resulting multigraph is also k-edge-connected) in O(mn ²+n ² log n) time, where n and m are the numbers of vertices and edges in G, respectively. This algorithm is unique in the sense that it does not rely on the maximum flow computations. Based on this, we then show that, given a positive integer k, the problem of making a multigraph G k-edge-connected by adding the smallest number of new edges can be solved in O(m+minen ²+n ³ log n, kn ³) time, where e (≤n ²) is the number of pairs of vertices between which G has an edge.