Twin-Width Meets Feedback Edges and Vertex Integrity

TL;DR

The paper advances the understanding of fixed-parameter approximability for twin-width by tying it to stronger graph parameters: the feedback edge number $k$ and vertex integrity $p$. It proves a tight square-root bound $tww(G)=\mathcal{O}(\sqrt{k})$ (with matching lower bounds), and delivers a significantly improved FP-approximation for twin-width parameterized by $k$ that reduces instances to size $\mathcal{O}(k^2)$ and computes a contraction sequence of width at most $tww(G)+1$ in time $2^{\mathcal{O}(k^2\log k)}$; it also presents a fixed-parameter $2$-approximation algorithm parameterized by $p$, via a bounded-reduction of twin-blocks and a provably effective extension from the reduced graph to the original. The approach refines prior work and introduces the tidy $(H,\mathcal{P})$-graph framework to streamline kernelization and sequence extension, offering concrete structural handles and kernel bounds. Overall, the results sharpen the boundary between tractable and intractable regimes for approximating twin-width and open pathways for fixed-parameter strategies in related width parameters.

Abstract

The approximate computation of twin-width has attracted significant attention already since the moment the parameter was introduced. A recently proposed approach (STACS 2024) towards obtaining a better understanding of this question is to consider the approximability of twin-width via fixed-parameter algorithms whose running time depends not on twin-width itself, but rather on parameters which impose stronger restrictions on the input graph. The first step that article made in this direction is to establish the fixed-parameter approximability of twin-width (with an additive error of 1) when the runtime parameter is the feedback edge number. Here, we make several new steps in this research direction and obtain: - An asymptotically tight bound between twin-width and the feedback edge number; - A significantly improved fixed-parameter approximation algorithm for twin-width under the same runtime parameter (i.e., the feedback edge number) which circumvents many of the technicalities of the original result and simultaneously avoids its formerly non-elementary runtime dependency; - An entirely new fixed-parameter approximation algorithm for twin-width when the runtime parameter is the vertex integrity of the graph.

Figures (2)

• Figure 1: A contraction sequence of width 2 for the leftmost graph, consisting of $6$ trigraphs.
• Figure 3: An illustration of Definition \ref{['def:gtidy']} when $m = 3$. The depicted $G$-tidy trigraph $G'$ consists of $H'$: vertices colored in grey, $F'$: vertices colored in blue (degree-3 vertices in darker shade), and the remaining vertices are colored in green. The edges inside of $H'$ are not depicted (there can be both red and black edges). Note that instead of each pair of green vertices, there should be at least 12 of them (because each path in $\mathcal{P}$ should contain at least $8m = 24$ vertices).

Theorems & Definitions (27)

• Definition 1
• Definition 2
• Definition 3
• Theorem 6: general-bounds
• Lemma 7
• Theorem 8
• Proposition 9
• Theorem 10: BalabanGR24
• Definition 11
• Theorem 12: BalabanGR24, Theorem 17 + Corollary 20