On February 15, 2023, Robert Hickingbotham from Monash University gave an online talk at the Virtual Discrete Math Colloquium on the relation between the tree-width and the Hadwiger number of circle graphs. The title of his talk was “Treewidth, Circle Graphs and Circular Drawings“.

## Robert Hickingbotham, Treewidth, Circle Graphs and Circular Drawings

A circle graph is an intersection graph of a set of chords of a circle. In this talk, I will describe the unavoidable induced subgraphs of circle graphs with large treewidth. This includes examples that are far from the `usual suspects’. Our results imply that treewidth and Hadwiger number are linearly tied on the class of circle graphs, and that the unavoidable induced subgraphs of a vertex-minor-closed class with large treewidth are the usual suspects if and only if the class has bounded rank-width. I will also discuss applications of our results to the treewidth of graphs $G$ that have a circular drawing whose crossing graph is well-behaved in some way. In this setting, our results show that if the crossing graph is $K_t$-minor-free, then $G$ has treewidth at most $12t-23$ and has no $K_{2,4t}$-topological minor. On the other hand, I’ll present a construction of graphs with arbitrarily large Hadwiger number that have circular drawings whose crossing graphs are $2$-degenerate. This is joint work with Freddie Illingworth, Bojan Mohar, and David R. Wood