# Twin-width VIII: delineation and win-wins

@article{Bonnet2022TwinwidthVD, title={Twin-width VIII: delineation and win-wins}, author={{\'E}douard Bonnet and Dibyayan Chakraborty and Eun Jung Kim and Noleen K{\"o}hler and Raul Lopes and St{\'e}phan Thomass{\'e}}, journal={ArXiv}, year={2022}, volume={abs/2204.00722} }

We introduce the notion of delineation. A graph class C is said delineated by twin-width (or simply, delineated ) if for every hereditary closure D of a subclass of C , it holds that D has bounded twin-width if and only if D is monadically dependent. An effective strengthening of delineation for a class C implies that tractable FO model checking on C is perfectly understood: On hereditary closures D of subclasses of C , FO model checking is fixed-parameter tractable (FPT) exactly when D has…

## 10 Citations

### 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…

### First Order Logic and Twin-Width in Tournaments and Dense Oriented Graphs

- MathematicsArXiv
- 2022

. We characterise the classes of tournaments with tractable ﬁrst-order model checking. For every hereditary class of tournaments T , ﬁrst-order model checking either is ﬁxed parameter tractable, or…

### Twin-width and Limits of Tractability of FO Model Checking on Geometric Graphs

- MathematicsArXiv
- 2022

The complexity of the problem of deciding properties expressible in FO logic on graphs – the FO model checking problem (parameterized by the respective FO formula), is well-understood on so-called…

### Twin-width VII: groups

- MathematicsArXiv
- 2022

. Twin-width is a recently introduced graph parameter with applications in algorithmics, combinatorics, and ﬁnite model theory. For graphs of bounded degree, ﬁniteness of twin-width is preserved by…

### Component twin-width as a parameter for BINARY-CSP and its semiring generalisations

- Computer ScienceArXiv
- 2022

An algebraic approach relying on semirings that unifies different generalizations of BINARY-CSPs (such as the counting, the list, and the weighted versions) is explored, and shows that it improves the best complexity upper bounds in the literature for several well-known problems.

### Computation of twin-width of graphs

- Mathematics
- 2022

: Twin-width is a recently introduced graph parameter. In this arti-cle, we compute twin-width of various ﬁnite graphs. In particular, we prove that the twin-widths of ﬁnite graphs with 4 and 5…

### Fixed-parameter tractability of Directed Multicut with three terminal pairs parameterized by the size of the cutset: twin-width meets flow-augmentation

- Computer ScienceArXiv
- 2022

Fixed-parameter tractability of the Directed Multicut problem with three terminal pairs is shown and an irrelevant vertex rule is shown, if any of the matrices in the said encoding has a large grid minor, which can be proclaimed irrelevant — not contained in the sought solution — and thus reduced.

### Approximating Highly Inapproximable Problems on Graphs of Bounded Twin-Width

- Mathematics, Computer ScienceArXiv
- 2022

This is the first in-depth study of the approximability of problems in graphs of bounded twin-width and generalizes the algorithms for Max Independent Set and Max Induced Matching to the independent (induced) packing of any fixed connected graph H.

### Twin-width can be exponential in treewidth

- Computer ScienceArXiv
- 2022

This work builds a graph with a vertex deletion set of size t to a tree, and twin-width greater than 2 (1 − ε ) t, and shows that the twin- width is sometimes exponential in the treewidth, in the so-called oriented twin-Width and grid number.

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