Induced subgraphs and tree decompositions XII. Grid theorem for pinched graphs

Given <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>c</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>ℕ</mml:mi> </mml:mrow> </mml:math> , we say a graph <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>G</mml:mi> </mml:math> is <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>c</mml:mi> </mml:math> -pinched if <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>G</mml:mi> </mml:math> does not contain an induced subgraph consisting of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>c</mml:mi> </mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>c</mml:mi> </mml:math> -pinched graphs? For instance, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mn>1</mml:mn> </mml:math> -pinched graphs are exactly graphs of treewidth <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mn>1</mml:mn> </mml:math> . However, bounded treewidth for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>c</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mn>2</mml:mn> </mml:math> -pinched graphs with arbitrarily large treewidth. There is even a fifth obstruction for larger values of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>c</mml:mi> </mml:math> , discovered by Pohoata and later independently by Davies, consisting of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mn>3</mml:mn> </mml:math> -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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>c</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>ℕ</mml:mi> </mml:mrow> </mml:math> , a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>c</mml:mi> </mml:math> -pinched graph <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>G</mml:mi> </mml:math> has large treewidth if and only if <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>G</mml:mi> </mml:math> 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.