Treewidth, Circle Graphs, and Circular Drawings
Robert Hickingbotham, Freddie Illingworth, Bojan Mohar, David R. Wood · SIAM Journal on Discrete Mathematics · 2024
Abstract. A circle graph is an intersection graph of a set of chords of a circle. We 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. Using the same tools, we also study the treewidth of graphs [Formula: see text] that have a circular drawing whose crossing graph is well-behaved in some way. In this setting, we show that if the crossing graph is [Formula: see text]-minor-free, then [Formula: see text] has treewidth at most [Formula: see text] and has no [Formula: see text]-topological minor. On the other hand, we show that there are graphs with arbitrarily large Hadwiger number that have circular drawings whose crossing graphs are 2-degenerate.