Braces of Perfect Matching Width 2

TL;DR

This paper develops the theory of perfect matching width ($\operatorname{pmw}$) for graphs with perfect matchings and links it to matching minors and the tight-cut decomposition into bricks and braces. It proves tight bounds: for any matching minor $H$ of a matching-covered $G$, $\operatorname{pmw}(H) \le 2\,\operatorname{pmw}(G)$, and if $H$ is a brick/brace of $G$ maximizing $\operatorname{pmw}(H)$, then $\tfrac{1}{2}\,\operatorname{pmw}(H) \le \operatorname{pmw}(G) \le \operatorname{pmw}(H)$; it also strengthens this with $M$-perfect matching width ($\operatorname{pmw}_{M}$) bounds. The core contribution is a complete characterization of braces with width two, via two complementary views: (i) edge-maximal braces forming bipartite ladders, and (ii) elimination-orderings that produce width-2 decompositions, yielding a polynomial-time recognition algorithm. The paper further shows that braces of width two can have arbitrarily large treewidth, connects $\operatorname{pmw}$ to treewidth, and extends the width-2 characterization to $M$-pmw, showing that braces with $\operatorname{pmw}_{M}=2$ are precisely $C_4$ or $K_{3,3}$ when $M$-width is considered. Collectively, these results establish a robust framework for understanding pmw via bricks/braces, tight cuts, and M-decompositions, with practical algorithms for recognizing and constructing width-2 decompositions.

Abstract

Perfect matching width is a treewidth-like parameter designed for graphs with perfect matchings. The concept was originally introduced by Norine for the study of non-bipartite Pfaffian graphs. Additionally, perfect matching width appears to be a useful structural tool for investigating matching minors, a specialised version of minors related to perfect matchings. In this paper we lay the groundwork for understanding the interaction of perfect matching width and matching minors by establishing tight connections between the perfect matching width of any matching covered graph $G$ and the perfect matching width of its bricks and braces (a matching theoretic version of blocks) and proving that perfect matching width is almost monotone under the matching minor relation. As an application, we give several characterisations for braces of perfect matching width two, including one that allows for a polynomial time recognition algorithm.

Figures (9)

• Figure 1: The braces cube and the Möbius ladder $\mathscr{M}_{10}$ of order 10.
• Figure 2: An example for a cubic tree $T$ with its \ref{['spine']} and its odd edges.
• Figure 3: A graph $G$ with the non-trivial tight cut $\partial( Z )$, a perfect matching $M\in\mathcal{M}( G )$, and an $M$-decomposition $(T,\nu)$ of width four. The arrows in $T$ are the edges forming $\vec{T_{{Z}}}$, note that it is free of inconsistencies and has a unique sink $s$.
• Figure 4: There are exactly two vertices in $X_2\setminus X_1$ and they come from different colour classes of $B.$
• Figure 5: The linear structure of a perfect matching decomposition of width $2$ with a claw on each side and two vertices from different colour classes added in each step. The filled vertices in the figure represent the leaves mapped to a vertex in $V_1$ and the empty vertices represent the leaves mapped to vertices in $V_2.$

Theorems & Definitions (79)

• Theorem 1.0
• Theorem 1.0
• Theorem 1.0
• Theorem 1.0: hatzel2019cyclewidth
• Theorem 1.1: lovasz1987matching
• Lemma 1.2: lucchesi2015thin
• Corollary 1.3: lucchesi2015thin
• Lemma 2.2
• proof
• Corollary 2.3