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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. 

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. 

A hole in a graph $$G$$ is an induced cycle of length at least four, and an even hole is a hole of even length. The diamond is the graph obtained from the complete graph $$K_4$$ by removing an edge. A pyramid is a graph consisting of a vertex $$a$$ called the apex and a triangle $$\{b_1, b_2, b_3\}$$ called the base, and three paths $$P_i$$ from $$a$$ to $$b_i$$ for $$1 \leq i \leq 3$$, all of length at least one, such that for $$i \neq j$$, the only edge between $$P_i \setminus \{a\}$$ and $$P_j \setminus \{a\}$$ is $$b_ib_j$$, and at most one of $$P_1$$, $$P_2$$, and $$P_3$$ has length exactly one. For a family $$\mathcal{H}$$ of graphs, we say a graph $$G$$ is $$\mathcal{H}$$-free if no induced subgraph of $$G$$ is isomorphic to a member of $$\mathcal{H}$$. Cameron, da Silva, Huang, and Vušković proved that (even hole, triangle)-free graphs have treewidth at most five, which motivates studying the treewidth of even-hole-free graphs of larger clique number. Sintiari and Trotignon provided a construction of (even hole, pyramid, $$K_4$$)-free graphs of arbitrarily large treewidth. Here, we show that for every $$t$$, (even hole, pyramid, diamond, $$K_t$$)-free graphs have bounded treewidth. The graphs constructed by Sintiari and Trotignon contain diamonds, so our result is sharp in the sense that it is false if we do not exclude diamonds. Our main result is in fact more general, that treewidth is bounded in graphs excluding certain wheels and three-path-configurations, diamonds, and a fixed complete graph. The proof uses “non-crossing decompositions” methods similar to those in previous papers in this series. In previous papers, however, bounded degree was a necessary condition to prove bounded treewidth. The result of this paper is the first to use the method of “non-crossing decompositions” to prove bounded treewidth in a graph class of unbounded maximum degree. 

A _theta_ is a graph consisting of two non-adjacent vertices and three internally disjoint paths between them, each of length at least two. For a family $$\mathcal{H}$$ of graphs, we say a graph $$G$$ is $$\mathcal{H}$$-_free_ if no induced subgraph of $$G$$ is isomorphic to a member of $$\mathcal{H}$$. We prove a conjecture of Sintiari and Trotignon, that there exists an absolute constant $$c$$ for which every (theta, triangle)-free graph $$G$$ has treewidth at most $$c\log (|V(G)|)$$. A construction by Sintiari and Trotignon shows that this bound is asymptotically best possible, and (theta, triangle)-free graphs comprise the first known hereditary class of graphs with arbitrarily large yet logarithmic treewidth.Our main result is in fact a generalization of the above conjecture, that treewidth is at most logarithmic in $|V(G)|$ for every graph $$G$$ excluding the so-called _three-path-configurations_ as well as a fixed complete graph. It follows that several NP-hard problems such as Stable Set, Vertex Cover, Dominating Set and $$k$$-Coloring (for fixed $$k$$) admit polynomial time algorithms in graphs excluding the three-path-configurations and a fixed complete graph.