# Twin-width II: small classes

@inproceedings{Bonnet2020TwinwidthIS, title={Twin-width II: small classes}, author={{\'E}douard Bonnet and Colin Geniet and Eun Jung Kim and St{\'e}phan Thomass{\'e} and R{\'e}mi Watrigant}, booktitle={ACM-SIAM Symposium on Discrete Algorithms}, year={2020} }

The twin-width of a graph $G$ is the minimum integer $d$ such that $G$ has a $d$-contraction sequence, that is, a sequence of $|V(G)|-1$ iterated vertex identifications for which the overall maximum number of red edges incident to a single vertex is at most $d$, where a red edge appears between two sets of identified vertices if they are not homogeneous in $G$. We show that if a graph admits a $d$-contraction sequence, then it also has a linear-arity tree of $f(d)$-contractions, for some…

## 53 Citations

### Twin-width III: Max Independent Set and Coloring

- Mathematics, Computer ScienceArXiv
- 2020

A polynomial-time algorithm is presented that properly colors the vertices of a graph with relatively few colors, establishing that bounded twin-width classes are $\chi$-bounded, which significantly extends the $\chi- boundedness of bounded rank- width classes, and does so with a very concise proof.

### Twin-width I: tractable FO model checking

- Mathematics, Computer Science2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS)
- 2020

It is proved that bounded twin-width is preserved by FO interpretations and transductions (allowing operations such as squaring or complementing a graph) and unifies and significantly extends the knowledge on fixed-parameter tractability of FO model checking on non-monotone classes, such as the FPT algorithm on bounded-width posets.

### From $\chi$- to $\chi_p$-bounded classes

- Mathematics
- 2020

$\chi$-bounded classes are studied here in the context of star colorings and more generally $\chi_p$-colorings. This leads to natural extensions of the notion of bounded expansion class and to…

### The Implicit Graph Conjecture is False

- Mathematics2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS)
- 2022

The Implicit Graph Conjecture states that, conversely, every hereditary graph family with at most factorial speed of growth admits an efficient implicit representation, and this paper proves this conjecture by establishing the existence of hereditary graph families with factorialSpeed of growth that require codes of length n.

### Bounds on the Twin-Width of Product Graphs

- MathematicsArXiv
- 2022

Twin-width is a graph width parameter recently introduced by Bonnet, Kim, Thomassé & Watrigant. Given two graphs G and H and a graph product ⋆ , we address the question: is the twin-width of G ⋆ H…

### Twin-width III: Max Independent Set, Min Dominating Set, and Coloring

- Computer Science, MathematicsICALP
- 2021

This paper presents 2 O ( k ) n -time algorithms for k -Independent Set, r -Scattered Set, k -Clique, and k -Dominating Set when an O (1)-sequence of the graph is given in input, and shows how breadth-first search can be mimicked, when replacing “traversing an edge” by “Traversing a biclique all at once”.

### Deciding twin-width at most 4 is NP-complete

- MathematicsICALP
- 2022

It is shown that determining if an n-vertex graph has twin-width at most 4 is NP-complete, and requires time 2Ω(n/ log n) unless the Exponential-Time Hypothesis fails, and how to encode trigraphs H into graphs G, in the sense that every d-sequence inevitably creates H as an induced subtrigraph.

### Twin-width VI: the lens of contraction sequences

- MathematicsSODA
- 2022

This paper defines an oriented version of twin-width, where appearing red edges are oriented away from the newly contracted vertex, and the mere red out-degree should remain bounded, and explores the concept of partial contraction sequences, where, instead of terminating on a single-vertex graph, the sequence ends when reaching a particular target class.

### Twin-width and Transductions of Proper k-Mixed-Thin Graphs

- MathematicsWG
- 2022

, Abstract. The new graph parameter twin-width, introduced by Bonnet, Kim, Thomass´e and Watrigant in 2020, allows for an FPT algorithm for testing all FO properties of graphs. This makes classes of…

### Bounds for the Twin-width of Graphs

- MathematicsSIAM Journal on Discrete Mathematics
- 2022

This work shows that conference graphs of order n (when such graphs exist) have twinwidth at least pn ́ 1q{2, and shows that Paley graphs achieve this lower bound, and calculates the twin-width of random graphs Gpn, pq with p ď c{n for a constant c ă 1.

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A polynomial-time algorithm is presented that properly colors the vertices of a graph with relatively few colors, establishing that bounded twin-width classes are $\chi$-bounded, which significantly extends the $\chi- boundedness of bounded rank- width classes, and does so with a very concise proof.

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It is proved that bounded twin-width is preserved by FO interpretations and transductions (allowing operations such as squaring or complementing a graph) and unifies and significantly extends the knowledge on fixed-parameter tractability of FO model checking on non-monotone classes, such as the FPT algorithm on bounded-width posets.

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