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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. 

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. 

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. 

Abstract What are the unavoidable induced subgraphs of graphs with large treewidth? It is well‐known that the answer must include a complete graph, a complete bipartite graph, all subdivisions of a wall and line graphs of all subdivisions of a wall (we refer to these graphs as the “basic treewidth obstructions”). So it is natural to ask whether graphs excluding the basic treewidth obstructions as induced subgraphs have bounded treewidth. Sintiari and Trotignon answered this question in the negative. Their counterexamples, the so‐called “layered wheels,” contain wheels, where awheelconsists of ahole(i.e., an induced cycle of length at least four) along with a vertex with at least three neighbors in the hole. This leads one to ask whether graphs excluding wheels and the basic treewidth obstructions as induced subgraphs have bounded treewidth. This also turns out to be false due to Davies' recent example of graphs with large treewidth, no wheels and no basic treewidth obstructions as induced subgraphs. However, in Davies' example there exist holes and vertices (outside of the hole) with two neighbors in them. Here we prove that a hole with a vertex with at least two neighbors in it is inevitable in graphs with large treewidth and no basic obstruction. Our main result is that graphs in which every vertex has at most one neighbor in every hole (that does not contain it) and with the basic treewidth obstructions excluded as induced subgraphs have bounded treewidth.