## Sepehr Hajebi gave an online talk on bounding the tree-width of graphs with forbidden induced subgraphs at the Virtual Discrete Math Colloquium

On July 7, 2022, Sepehr Hajebi from the University of Waterloo gave an online talk at the Virtual Discrete Math Colloquium on bounding the tree-width of graphs with forbidden induced subgraphs. The title of his talk was “Holes, hubs and bounded treewidth“.

## Sepehr Hajebi, Holes, hubs and bounded treewidth

A hole in a graph $G$ is an induced cycle of length at least four, and for every hole $H$ in $G$, a vertex $h\in G\setminus H$ is called a $t$-hub for $H$ if $h$ has at least $t$ neighbor in $H$. Sintiari and Trotignon were the first to construct graphs with arbitrarily large treewidth and no induced subgraph isomorphic to the “basic obstructions,” that is, a fixed complete graph, a fixed complete bipartite graph (with parts of equal size), all subdivisions of a fixed wall and line graphs of all subdivisions of a fixed wall. They named their counterexamples “layered wheels” for a good reason: layered wheels contain wheels in abundance, where a wheel means a hole with a $3$-hub. In accordance, one may ask whether graphs with no wheel and no induced subgraph isomorphic to the basic obstructions have bounded treewidth. This was also disproved by a recent construction due to Davies. But holes with a $2$-hub cannot be avoided in graphs with large treewidth: graphs containing no hole with a $2$-hub and no induced subgraph isomorphic to the basic obstructions have bounded treewidth. I will present a proof of this result, and will also give an overview of related works.
Based on joint work with Tara Abrishami, Bogdan Alecu, Maria Chudnovsky, Sophie Spirkl and Kristina Vušković.
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