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

The main result of this paper is a new perfect matching algorithm for biconnected cubic graphs. The algorithm runs in time O(n log(2) n). It is also possible, by applying randomized data structures, to get O(n log n log log(3) 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 log(4) n). We use a similar approach. However, thanks to exploring some properties of biconnected cubic graphs we are Ale 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.