Petersen's theorem, one of the earliest results in graph theory, states that any bridgeless cubic multigraph contains a perfect matching. While the original proof was neither constructive nor algorithmic, Biedl, Bose, Demaine, and Lubiw [J. Algorithms 38(1)] showed how to implement a later constructive proof by Frink in $\mathcal{O}(n\log^{4}n)$ time using a fully dynamic 2-edge-connectivity structure. Then, Diks and Sta\'nczyk [SOFSEM 2010] described a faster approach that only needs a fully dynamic connectivity structure and works in $\mathcal{O}(n\log^{2}n)$ time. Both algorithms, while reasonable simple, utilize non-trivial (2-edge-)connectivity structures. We show that this is not necessary, and in fact a structure for maintaining a dynamic tree, e.g. link-cut trees, suffices to obtain a simple $\mathcal{O}(n\log n)$ time algorithm.

翻译：佩特森定理是图论中最早的结果之一，指出任何无桥三次多重图均包含完美匹配。原始证明既非构造性也非算法性，Biedl、Bose、Demaine和Lubiw [J. Algorithms 38(1)] 展示了如何利用完全动态2-边连通性结构实现Frink后来提出的构造性证明，时间复杂度为 $\mathcal{O}(n\log^{4}n)$。随后，Diks和Stańczyk [SOFSEM 2010] 提出一种更高效的方法，仅需完全动态连通性结构，时间复杂度为 $\mathcal{O}(n\log^{2}n)$。这两种算法虽然相对简单，但均需借助非平凡的（2-边）连通性结构。我们证明此非必要——实际上，仅维护动态树（如link-cut树）即可实现简单的 $\mathcal{O}(n\log n)$ 时间算法。