A contraction sequence of a graph consists of iteratively merging two of its vertices until only one vertex remains. The recently introduced twin-width graph invariant is based on contraction sequences. More precisely, if one puts error edges, henceforth red edges, between two vertices representing non-homogeneous subsets, the twin-width is the minimum integer d such that a contraction sequence exists that keeps red degree at most d. By changing the condition imposed on the trigraphs (i.e., graphs with some edges being red) and possibly slightly tweaking the notion of contractions, we show how to characterize the well-established bounded rankwidth, tree-width, linear rank-width, path-width -usually defined in the framework of branch-decompositions-, and proper minor-closed classes by means of contraction sequences.Contraction sequences hold a crucial advantage over branch-decompositions: While one can scale down contraction sequences to capture classical width notions, the more general bounded twin-width goes beyond their scope, as it contains planar graphs in particular, a class with unbounded rank-width. As an application we give a transparent alternative proof of the celebrated Courcelle's theorem (actually of its generalization by Courcelle, Makowsky, and Rotics), that MSO2 (resp. MSO1) model checking on graphs with bounded tree-width (resp. bounded rank-width) is fixed-parameter tractable in the size of the input sentence. We are hopeful that our characterizations can help in other contexts.We then explore new avenues along the general theme of contraction sequences both in order to refine the landscape between bounded tree-width and bounded twin-width (via spanning twin-width) and to capture more general classes than bounded twin-width. To this end, we define 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. Surprisingly, classes of bounded oriented twin-width coincide with those of bounded twin-width. This greatly simplifies the task of showing that a class has bounded twin-width. As an example, using a lemma by Norine, Seymour, Thomas, and Wollan, we give a 5-line proof that Kt-minor free graphs have bounded twin-width. Without oriented twin-width, this fact was shown by a somewhat intricate 4page proof in the first paper of the series. Finally we explore the concept of partial contraction sequences, instead of terminating on a single-vertex graph, the sequence ends when reaching a particular target class. We show that FO model checking (resp. ∃FO model checking) is fixed-parameter tractable on classes with partial contraction sequences to a class of bounded degree (resp. bounded expansion), provided such a sequence is given. Efficiently finding such partial sequences could turn out simpler than finding a (complete) sequence.

No abstract

In this work, we generalize several results on graph recolouring to digraphs. Given two k-dicolourings of a digraph D, we prove that it is PSPACE-complete to decide whether we can transform one into the other by recolouring one vertex at each step while maintaining a dicolouring at any step even for k = 2 and for digraphs with maximum degree 5 or oriented planar graphs with maximum degree 6.A digraph is said to be k-mixing if there exists a transformation between any pair of k-dicolourings. We show that every digraph D is k-mixing for all k ≥ δ * min (D) + 2, generalizing a result due to Dyer et al. We also prove that every oriented graph G is k-mixing for all k ≥ δ * max ( G) + 1 and for all k ≥ δ * avg ( G) + 1. Here δ * min , δ * max , and δ * avg denote the min-degeneracy, the max-degeneracy, and the average-degeneracy respectively. We pose as a conjecture that, for every digraph D, the dicolouring graph of D on k ≥ δ * min (D) + 2 colours has diameter at most O(|V (D)| 2 ). This is the analogue of Cereceda's conjecture for digraphs. We generalize to digraphs two results supporting Cereceda's conjecture. We first prove that the dicolouring graph of any digraph D on k ≥ 2δ * min (D) + 2 colours has linear diameter, extending a result from Bousquet and Perarnau. We also prove that the analogue of Cereceda's conjecture is true when k ≥ 3 2 (δ * min (D) + 1), which generalizes a result from Bousquet and Heinrich.Restricted to the special case of oriented graphs, we prove that the dicolouring graph of any subcubic oriented graph on k ≥ 2 colours is connected and has diameter at most 2n. We conjecture that every non 2-mixing oriented graph has maximum average degree at least 4, and we provide some support for this conjecture by proving it on the special case of 2-freezable oriented graphs. More generally, we show that every k-freezable oriented graph on n vertices must contain at least kn + k(k − 2) arcs, and we give a family of k-freezable oriented graphs that reach this bound. In the general case, we prove as a partial result that every non 2-mixing oriented graph has maximum average degree at least 7 2 .

We study the existence of polynomial kernels, for parameterized problems without a polynomial kernel on general graphs, when restricted to graphs of bounded twin-width. Our main result is that a polynomial kernel for k-Dominating Set on graphs of twin-width at most 4 would contradict a standard complexity-theoretic assumption. The reduction is quite involved, especially to get the twin-width upper bound down to 4, and can be tweaked to work for Connected k-Dominating Set and Total k-Dominating Set (albeit with a worse upper bound on the twin-width). The k-Independent Set problem admits the same lower bound by a much simpler argument, previously observed [ICALP '21], which extends to k-Independent Dominating Set, k-Path, k-Induced Path, k-Induced Matching, etc.On the positive side, we obtain a simple quadratic vertex kernel for Connected k-VertexCover and Capacitated k-Vertex Cover on graphs of bounded twin-width. Interestingly the kernel applies to graphs of Vapnik-Chervonenkis density 1, and does not require a witness sequence.We also present a more intricate O(k 1.5 ) vertex kernel for Connected k-Vertex Cover. Finally we show that deciding if a graph has twin-width at most 1 can be done in polynomial time, and observe that most optimization/decision graph problems can be solved in polynomial time on graphs of twin-width at most 1.