[Paper Review] Twin-Width of Graphs on Surfaces

Daniel Král͏̌, Kristýna Pekárková+1 more|arXiv (Cornell University)|Jul 11, 2023

Advanced Graph Theory Research1 citations

TL;DR

This paper establishes an asymptotically optimal upper bound of $18\sqrt{47g} + O(1)$ on the twin-width of graphs embeddable in surfaces of Euler genus $g$, using a strengthened Product Structure Theorem that decomposes such graphs as subgraphs of the strong product of a path and a graph with nearly bounded tree-width. The result yields a quadratic-time algorithm to compute a witnessing contraction sequence.

ABSTRACT

Twin-width is a width parameter introduced by Bonnet, Kim, Thomassé and Watrigant [FOCS'20, JACM'22], which has many structural and algorithmic applications. We prove that the twin-width of every graph embeddable in a surface of Euler genus $g$ is $18\sqrt{47g}+O(1)$, which is asymptotically best possible as it asymptotically differs from the lower bound by a constant multiplicative factor. Our proof also yields a quadratic time algorithm to find a corresponding contraction sequence. To prove the upper bound on twin-width of graphs embeddable in surfaces, we provide a stronger version of the Product Structure Theorem for graphs of Euler genus $g$ that asserts that every such graph is a subgraph of the strong product of a path and a graph with a tree-decomposition with all bags of size at most eight with a single exceptional bag of size $\max\{8,32g-27\}$.

Motivation & Objective

• To establish tight asymptotic upper bounds on the twin-width of graphs embeddable in surfaces of Euler genus $g$.
• To improve upon prior exponential and double-exponential bounds for twin-width in minor-closed graph classes.
• To develop a stronger version of the Product Structure Theorem tailored for graphs on surfaces.
• To provide a quadratic-time algorithm for computing a contraction sequence witnessing the twin-width bound.
• To demonstrate that the upper bound is asymptotically optimal up to a constant multiplicative factor.

Proposed method

• Introduce a modified Product Structure Theorem stating that every graph of Euler genus $g$ is a subgraph of the strong product of a path and a graph with a tree-decomposition where all bags have size at most 8, except one of size $\max\{8, 32g - 27\}$.
• Use a layered contraction process dividing the algorithm into three phases: initial contraction of vertex sets, path-based contraction, and final merging of remaining vertices.
• Control red degrees during contraction by bounding neighbor counts in each layer and across neighboring layers using combinatorial arguments.
• Track red degrees through each phase, showing they never exceed $\max\{6(s+1), 3 \cdot 2^{25}\}$, where $s$ is a parameter related to the tree-decomposition.
• Leverage structural properties of surface-embedded graphs and random graph twin-width lower bounds to establish asymptotic optimality.
• Apply Chernoff bounds and extremal graph theory to derive a lower bound of $\sqrt{3g/2} - O(g^{3/8})$ on the twin-width of some graphs embeddable in genus $g$ surfaces.

Experimental results

Research questions

• RQ1 What is the tightest possible asymptotic upper bound on the twin-width of graphs embeddable in surfaces of Euler genus $g$?
• RQ2 Can the Product Structure Theorem be strengthened to provide a more structured decomposition for graphs on surfaces?
• RQ3 Is the upper bound on twin-width for surface-embedded graphs asymptotically optimal?
• RQ4 Can a quadratic-time algorithm be designed to compute a contraction sequence achieving the twin-width bound?
• RQ5 How does the twin-width of random graphs like $G_{n,1/2}$ relate to the twin-width of surface-embedded graphs?

Key findings

• The twin-width of any graph embeddable in a surface of Euler genus $g$ is at most $18\sqrt{47g} + O(1) \approx 123.4\sqrt{g} + O(1)$, which is asymptotically optimal up to a constant factor.
• The lower bound of $\sqrt{3g/2} - O(g^{3/8})$ for twin-width of some genus-$g$ graphs shows that the upper bound differs from the best possible by at most a multiplicative factor of approximately 100.76.
• A quadratic-time algorithm exists to compute a contraction sequence achieving the stated twin-width bound.
• The paper proves a strengthened Product Structure Theorem: every such graph is a subgraph of the strong product of a path and a graph with a tree-decomposition having all bags of size at most 8, except one of size $\max\{8, 32g - 27\}$.
• The red degree of any vertex during the contraction process is bounded by $\max\{6(s+1), 3 \cdot 2^{25}\}$, ensuring the twin-width remains within the stated bound.
• The result closes a gap in the understanding of twin-width for minor-closed graph classes, particularly for surface-embedded graphs, and unifies structural and algorithmic insights.