Killing a Vortex

TL;DR

This work resolves the complexity of counting perfect matchings (#PM) on minor-closed graph classes by establishing a sharp dichotomy governed by shallow vortex minors ${\mathcal{S}}$. It introduces a vortex-free refinement of the Graph Minors Structure Theorem (GMST) and a new parametric obstruction family ${\mathscr{S}}_{t}$, showing that ${\mathscr{S}}_{t}$-minor-free graphs admit clique-sums of apex-bounded-genus graphs after removing a bounded vertex set; vortices become necessary precisely when the excluded minor is not contained in any ${\mathscr{S}}_{t}$. The authors design a polynomial-time algorithm to compute the generating function $\mathsf{PerfMatch}(G,\mathbf{w})$ on ${\mathscr{S}}_{t}$-minor-free graphs via dynamic programming on near-embeddings and Pfaffian orientations, yielding a sharp boundary: if a finite minor-excluding family ${\mathcal{F}}$ intersects ${\mathcal{S}}$, PM is poly-time solvable on Excl$(\mathcal{F})$, otherwise PM is #P-hard. By combining these structural and algorithmic insights, the paper fully characterizes minor-closed classes where the number of perfect matchings is polynomially computable and provides a concrete framework for countingPM in broad graph classes. The results connect deep graph-minors theory with learning and combinatorial counting, delivering a precise dichotomy and practical DP-based methods for a wide range of graphs.

Abstract

The Graph Minors Structure Theorem of Robertson and Seymour asserts that, for every graph $H,$ every $H$-minor-free graph can be obtained by clique-sums of ``almost embeddable'' graphs. Here a graph is ``almost embeddable'' if it can be obtained from a graph of bounded Euler-genus by pasting graphs of bounded pathwidth in an ``orderly fashion'' into a bounded number of faces, called the \textit{vortices}, and then adding a bounded number of additional vertices, called \textit{apices}, with arbitrary neighborhoods. Our main result is a {full classification} of all graphs $H$ for which the use of vortices in the theorem above can be avoided. To this end we identify a (parametric) graph $\mathscr{S}_{t}$ and prove that all $\mathscr{S}_{t}$-minor-free graphs can be obtained by clique-sums of graphs embeddable in a surface of bounded Euler-genus after deleting a bounded number of vertices. We show that this result is tight in the sense that the appearance of vortices cannot be avoided for $H$-minor-free graphs, whenever $H$ is not a minor of $\mathscr{S}_{t}$ for some $t\in\mathbb{N}.$ Using our new structure theorem, we design an algorithm that, given an $\mathscr{S}_{t}$-minor-free graph $G,$ computes the generating function of all perfect matchings of $G$ in polynomial time. Our results, combined with known complexity results, imply a complete characterization of minor-closed graph classes where the number of perfect matchings is polynomially computable: They are exactly those graph classes that do not contain every $\mathscr{S}_{t}$ as a minor. This provides a \textit{sharp} complexity dichotomy for the problem of counting perfect matchings in minor-closed classes.

Figures (20)

• Figure 1: Timeline of the results on the complexity of the $\hbox{#Perfect Matching}$ problem.
• Figure 2: The shallow vortex grid $H_{6}$. The additional six pairs of crossed edges are depicted in red. The two blue cycles are the two extremal cycles of the $(6,24)$-cylindrical grid.
• Figure 3: The vga-hierarchy of parameters and the position of the parameters ${\mathsf{apex}}$ and ${\mathsf{genus}}$ in it. If ${\mathsf{p}}$ and ${\mathsf{p}}'$ are parameters in the above diagram then ${\mathsf{p}}\preceq {\mathsf{p}}'$ if and only if there is a path between ${\mathsf{p}}$ and ${\mathsf{p}}'$ that is "above" ${\mathsf{p}}'.$ The two green/pink-colored areas indicate the complexity of $\hbox{#Perfect Matching}$ when restricted to graphs where each of the depicted parameters is bounded. The lower dark green area indicates the current state of the art on polynomial algorithms for $\hbox{#Perfect Matching}.$
• Figure 4: A society $(G,\Omega)$ with a cross. The magenta vertices are the vertices of $V(\Omega).$
• Figure 5: A vortex-free rendition of a society $(G,\Omega)$ in the disk.

Theorems & Definitions (82)

• Definition 1: Shallow vortex grids
• Theorem 1
• Definition 2: Treewidth
• Definition 3
• Proposition 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Definition 4: Society