Twin-width and permutations

TL;DR

This paper establishes a fundamental link between bounded twin-width and FO-transductions of proper permutation classes for binary relational structures. By introducing twin-models and their ranked variants, it provides a structural bridge between contraction sequences and tree-like representations, enabling a permutation-based encoding of bounded twin-width. The main result shows that every class with bounded twin-width is a FO-transduction of a proper permutation class, and conversely that FO-transductions of such permutation classes have bounded twin-width; as a corollary, the class has at most c^n non-isomorphic n-vertex structures. The work connects model-theoretic transductions, Gaifman graph properties (including star chromatic number and bounded expansion), and permutation classes, yielding a powerful framework for analyzing sparsity and algorithmic tractability in binary relational structures.

Abstract

Inspired by a width invariant on permutations defined by Guillemot and Marx, Bonnet, Kim, Thomassé, and Watrigant introduced the twin-width of graphs, which is a parameter describing its structural complexity. This invariant has been further extended to binary structures, in several (basically equivalent) ways. We prove that a class of binary relational structures (that is: edge-colored partially directed graphs) has bounded twin-width if and only if it is a first-order transduction of a~proper permutation class. As a by-product, we show that every class with bounded twin-width contains at most $2^{O(n)}$ pairwise non-isomorphic $n$-vertex graphs.

Figures (9)

• Figure 1: A 2-sequence witnessing that the initial graph has twin-width at most 2.
• Figure 2: Relations between the classes of structures involved in the proof of the main result. The interpretation $\mathsf S$ is defined in \ref{['def:tS']}, the transduction pairing $(\mathsf L,\mathsf O)$ in \ref{['lem:otm']}, the transduction pairing $(\widehat{\mathsf L},\widehat{\mathsf O})$ as a remark just after \ref{['def:clOY']}, the transduction pairing $(\mathsf G,\mathsf U)$ in \ref{['lem:toGaifman']}, and the transduction pairing $(\widehat{\mathsf G},\widehat{\mathsf U})$ as a remark just after \ref{['def:clGo']}.
• Figure 3: From a graph $G$ to a permutation $\sigma$, and back.
• Figure 4: A contraction sequence, a so-called block representation of the contractions, and a twin-model.
• Figure 5: A graph $G$ and a ranked twin-model of $G$. The boundary $\partial_4Y$ is the set $\{5,g,c,4\}$ (internal vertices labeled by $\tau$), which can be represented as the set of the yellow zones. The relations of $\mathbf L_4$ are depicted as dotted heavy lines (black for $R$, red for $R^\ast$). The width of this twin-model is $2$.

Theorems & Definitions (59)

• Theorem
• Lemma 2.1
• proof
• Theorem 2.2: twin-width1
• Lemma 3.1
• proof
• Claim 3.2
• proof : Proof of the claim.
• Theorem 3.3: twin-width2
• Theorem 3.4: POMNI