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

Paweł Gawrychowski, Mateusz Wasylkiewicz

TL;DR

The paper tackles finding a perfect matching in bridgeless cubic multigraphs, a problem guaranteed solvable by Petersen's theorem. It introduces a streamlined, self-contained approach that implements Frink's reductions using only a dynamic tree data structure (link-cut trees), achieving a deterministic $O(n\log n)$ time without relying on fully dynamic 2-edge-connectivity. The key innovation is maintaining a spanning tree $T_i$ of each reduced graph $G_i$ and a cover edge for every $e\in E(T_i)$, enabling reductions and reversals via a swap primitive and carefully managed invariants. This simpler, robust method broadens applicability (e.g., to cubic multigraphs with up to two bridges) and avoids complex connectivity structures, with potential impact on related combinatorial problems and decomposition tasks in cubic graphs.

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 Stańczyk [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.

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

TL;DR

The paper tackles finding a perfect matching in bridgeless cubic multigraphs, a problem guaranteed solvable by Petersen's theorem. It introduces a streamlined, self-contained approach that implements Frink's reductions using only a dynamic tree data structure (link-cut trees), achieving a deterministic time without relying on fully dynamic 2-edge-connectivity. The key innovation is maintaining a spanning tree of each reduced graph and a cover edge for every , enabling reductions and reversals via a swap primitive and carefully managed invariants. This simpler, robust method broadens applicability (e.g., to cubic multigraphs with up to two bridges) and avoids complex connectivity structures, with potential impact on related combinatorial problems and decomposition tasks in cubic graphs.

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 time using a fully dynamic 2-edge-connectivity structure. Then, Diks and Stańczyk [SOFSEM 2010] described a faster approach that only needs a fully dynamic connectivity structure and works in 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 time algorithm.
Paper Structure (14 sections, 5 theorems, 8 figures, 1 algorithm)

This paper contains 14 sections, 5 theorems, 8 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Link-cut trees
  4. Outline of previous algorithms
  5. $\mathcal{O}(n^2)$ time algorithm based on Frink's proof
  6. $\mathcal{O}(n\log^4{n})$ time algorithm with fully dynamic 2-edge-connectivity
  7. $\mathcal{O}(n\log^2{n})$ time algorithm with fully dynamic connectivity
  8. Outline of our algorithm
  9. Details
  10. Swap
  11. Reductions of type I
  12. Reductions of type II
  13. Complexity analysis
  14. Conclusions

Key Result

Theorem 1

Let $G$ be any bridgeless cubic multigraph and $\{v,w\}$ any single edge of $G$. Let $\{a,v\}$ and $\{b,v\}$ be other edges of $G$ incident to $v$. Let $\{c,w\}$ and $\{d,w\}$ be other edges of $G$ incident to $w$. Define multigraphs $H_1=((G\setminus v)\setminus w)\cup\{a,c\}\cup\{b,d\}$ and $H_2=(

Figures (8)

  • Figure 1: Straight and crossing reduction of type I on single edge $\{v,w\}$.
  • Figure 2: Reverting reduction of type I. The matched edges are marked by wavy lines.
  • Figure 3: Reduction of type II (and its reverting). The matched edges are marked by thick lines.
  • Figure 4: Proof of Lemma \ref{['lem:swap']}. The edges of $T_i$ are marked by thick lines and edges of, correspondingly, $R$ and $R'$ are marked by red lines.
  • Figure 5: The case when $|A_i\cap E(T_i)|=3$. The edges of $T_i$ and $T_{i+1}$ are marked by thick lines.
  • ...and 3 more figures

Theorems & Definitions (8)

  • Theorem 1: Frink
  • Lemma 1: BiedlEtAl2001
  • proof
  • Lemma 2: BiedlEtAl2001
  • Lemma 3
  • proof
  • Lemma 4
  • proof