Finding perfect matchings in bridgeless cubic multigraphs without dynamic (2-)connectivity

TL;DR

This paper presents a simplified and more efficient algorithm for finding perfect matchings in bridgeless cubic multigraphs, eliminating the need for complex dynamic connectivity structures.

Contribution

It introduces a new algorithm that uses dynamic trees like link-cut trees, achieving an $ ext{O}(n ext{log}n)$ time complexity, simplifying previous methods.

Findings

Achieves $ ext{O}(n ext{log}n)$ time complexity for the problem.

Eliminates the need for complex 2-edge-connectivity data structures.

Provides a simpler approach using dynamic trees like link-cut trees.

Abstract

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.