# 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*+min*en*^{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.

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