Twin-width of graphs on surfaces

TL;DR

This work establishes an asymptotically optimal upper bound on the twin-width of graphs embeddable in surfaces: $tw(G)\le 18\sqrt{47g}+O(1)$ for Euler genus $g$, and shows this bound is tight up to a constant factor. Central to the approach is a strengthened Product Structure Theorem: every such graph is a subgraph of the strong product of a path and a graph whose rooted tree-decomposition has all bags size at most $8$ except a single root bag of size at most $\max\{8,32g-27\}$. The authors give a constructive, quadratic-time algorithm to produce a witnessing contraction sequence, leveraging the structural decomposition. They also prove a matching albeit weaker lower bound $tw(G)\ge\sqrt{3g/2}-O(g^{3/8})$ for some graphs, via random graph constructions, highlighting the near-optimality of the upper bound and deepening the understanding of twin-width for surface-embeddable graphs.

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 $\max\{8,32g-27\}$.

Figures (9)

• Figure 1: A rooted tree $T_0$ and edges $a_1b_1$ and $a_2b_2$, which are drawn dashed, bounding a part of the torus that is homeomorphic to a disk as in the proof of Lemma \ref{['lm:BFS-paths']}. Possible additional edges of the BFS spanning tree $T$ are drawn dotted. The root of the tree $T_0$ is depicted by an empty circle unlike the other vertices of $T_0$.
• Figure 2: The decomposition of the tree $T_0$ into paths $P_1,\ldots,P_4$ (the case $g=2$ and $k=4$), the illustration of the split into $2g=4$ segments covering the walk $W$ as in the proof of Lemma \ref{['lm:BFS-paths']}, and the collection of $11$ vertical paths covering $W$ obtained in the proof. The root of the tree $T_0$ is depicted by an empty circle unlike the other vertices.
• Figure 3: The three possible cases where the path $C$ can lead to in the induction step in the proof of Lemma \ref{['lm:tw7']} for $k=5$.
• Figure 4: The three possible cases of the values $a$ and $b$ in the induction step in the proof of Lemma \ref{['lm:tw7']} for $k=6$ under the assumption $b-a\ge 2$.
• Figure 5: The initial step of the case $k=7$ in the proof of Lemma \ref{['lm:twout']}; possible paths $A$, $B$ and $C$ are drawn dashed. The two cases when a further argument is needed are depicted in the right.

Theorems & Definitions (21)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6
• Corollary 7
• Lemma 8
• Lemma 9
• proof