Twin-width VI: the lens of contraction sequences

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 rank-width, 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 4-page proof in the first paper of the series. Finally we explore the concept of partial contraction sequences, where, 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. 2012 ACM Subject Classification Theory of computation → Graph algorithms analysis; Theory of computation → Fixed parameter tractability