We present a near-linear-time algorithm that, given a bridgeless cubic graph, finds a perfect matching intersecting every 3-edge-cut in exactly one edge. This improves over a cubic algorithm of Boyd et al. for the same problem, and over our previous algorithm, which worked only for 3-edge-connected graphs. The main ingredient is a cactus representation of the 2-edge-cuts, together with an efficient update procedure under 2-cut reductions.

翻译：我们提出了一种近线性时间算法，对于给定的无桥三次图，能够找到一个与每个三边割恰交于一条边的完美匹配。该算法改进了Boyd等人针对同一问题的三次时间算法，以及我们先前仅适用于三边连通图的算法。其主要组成部分是二边割的仙人掌表示，以及在此结构下通过二割约简进行高效更新的处理流程。