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The celebrated Erdős-Pósa Theorem, in one formulation, asserts that for every c ∈ N, graphs with no subgraph (or equivalently, minor) isomorphic to the disjoint union of c cycles have bounded treewidth. What can we say about the treewidth of graphs containing no induced subgraph isomorphic to the disjoint union of c cycles? Let us call these graphs c-perforated. While 1-perforated graphs have treewidth one, complete graphs and complete bipartite graphs are examples of 2-perforated graphs with arbitrarily large treewidth. But there are sparse examples, too: Bonamy, Bonnet, Déprés, Esperet, Geniet, Hilaire, Thomassé and Wesolek constructed 2-perforated graphs with arbitrarily large treewidth and no induced subgraph isomorphic to K3 or K3,3; we call these graphs occultations. Indeed, it turns out that a mild (and inevitable) adjustment of occultations provides examples of 2-perforated graphs with arbitrarily large treewidth and arbitrarily large girth, which we refer to as full occultations. Our main result shows that the converse also holds: for every c ∈ N, a c-perforated graph has large treewidth if and only if it contains, as an induced subgraph, either a large complete graph, or a large complete bipartite graph, or a large full occultation. This distinguishes c-perforated graphs, among graph classes purely defined by forbidden induced subgraphs, as the first to admit a grid-type theorem incorporating obstructions other than subdivided walls and their line graphs. More generally, for all c, o ∈ N, we establish a full characterization of induced subgraph obstructions to bounded treewidth in graphs containing no induced subgraph isomorphic to the disjoint union of c cycles, each of length at least o + 2. 

A clock is a graph consisting of an induced cycle and a vertex not in with at least two non-adjacent neighbours in . We show that every clock-free graph of large treewidth contains a “basic obstruction” of large treewidth as an induced subgraph: a complete graph, a subdivision of a wall, or the line graph of a subdivision of a wall. 

Given c ∈ N, we say a graph G is c-pinched if G does not contain an induced subgraph consisting of c cycles, all going through a single common vertex and otherwise pairwise disjoint and with no edges between them. What can be said about the structure of c-pinched graphs? For instance, 1-pinched graphs are exactly graphs of treewidth 1. However, bounded treewidth for c > 1 is immediately seen to be a false hope because complete graphs, complete bipartite graphs, subdivided walls and line graphs of subdivided walls are all examples of 2-pinched graphs with arbitrarily large treewidth. There is even a fifth obstruction for larger values of c, discovered by Pohoata and later independently by Davies, consisting of 3-pinched graphs with unbounded treewidth and no large induced subgraph isomorphic to any of the first four obstructions. We fuse the above five examples into a grid-type theorem fully describing the unavoidable induced subgraphs of pinched graphs with large treewidth. More precisely, we prove that for every c ∈ N, a c-pinched graph G has large treewidth if and only if G contains one of the following as an induced subgraph: a large complete graph, a large complete bipartite graph, a subdivision of a large wall, the line graph of a subdivision of a large wall, or a large graph from the Pohoata-Davies construction. Our main result also generalizes to an extension of pinched graphs where the lengths of excluded cycles are lower-bounded. 

We prove that the tree independence number of every even-hole-free graph is at most polylogarithmic in its number of vertices. More explicitly, we prove that there exists a constant c > 0 such that for every integer n > 1 every n-vertex even-hole-free graph has a tree decomposition where each bag has stability (independence) number at most clog10 n. This implies that the Maximum Weight Independent Set problem, as well as several other natural algorithmic problems that are known to be NP-hard in general, can be solved in quasipolynomial time if the input graph is even-hole-free. The quasi-polynomial complexity will remain the same even if the exponent of the logarithm is reduced to 1 (which would be asymptotically best possible). 

Unlike minors, the induced subgraph obstructions to bounded treewidth come in a large variety, including, for every t ∈ N, the t-basic obstructions: the graphs Kt+1 and Kt,t, along with the subdivisions of the t-by-t wall and their line graphs. But this list is far from complete. The simplest example of a “non-basic” obstruction is due to Pohoata and Davies (independently). For every n ∈ N, they construct certain graphs of treewidth n and with no 3-basic obstruction as an induced subgraph, which we call n-arrays. Let us say a graph class G is clean if the only obstructions to bounded treewidth in G are in fact the basic ones. It follows that a full description of the induced subgraph obstructions to bounded treewidth is equivalent to a characterization of all families H of graphs for which the class of all H-free graphs is clean (a graph G is H-free if no induced subgraph of G is isomorphic to any graph in H). This remains elusive, but there is an immediate necessary condition: if H-free graphs are clean, then there are only finitely many n ∈ N such that there is an n-array which is H-free. The above necessary condition is not sufficient in general. However, the situation turns out to be different if H is finite: we prove that for every finite set H of graphs, the class of all H-free graphs is clean if and only if there is no H-free n-array except possibly for finitely many values of n. 

Given a graph H, we prove that every (theta, prism)-free graph of sufficiently large treewidth contains either a large clique or an induced subgraph isomorphic to H, if and only if H is a forest. 

Fix k>0, and let G be a graph, with vertex set partitioned into k subsets (`blocks') of approximately equal size. An induced subgraph of G is transversal (with respect to this partition) if it has exactly one vertex in each block (and therefore it has exactly k vertices). A pure pair in G is a pair X,Y of disjoint subsets of V(G) such that either all edges between X,Y are present or none are; and in the present context we are interested in pure pairs (X,Y) where each of X,Y is a subset of one of the blocks, and not the same block. This paper collects several results and open questions concerning how large a pure pair must be present if various types of transversal subgraphs are excluded.