Perfect Matching for Biconnected Cubic Graphs in O(n log2 n) Time

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Abstract

The main result of this paper is a new perfect matching algorithm for biconnected cubic graphs. The algorithm runs in time O(n log2 n). It is also possible, by applying randomized data structures, to get O(n logn loglog3 n) average time. Our solution improves the one given by T. Biedl et al. [3]. The algorithm of Biedl et al. runs in time O(n log4 n). We use a similar approach. However, thanks to exploring some properties of biconnected cubic graphs we are able to replace complex fully-dynamic biconnectivity data structure with much simpler, dynamic graph connectivity and dynamic tree data structures. Moreover, we present a significant modification of the new algorithm which makes application of a decremental dynamic graph connectivity data structure possible, instead of one supporting the fully dynamic graph connectivity. It gives hope for further improvements.