Holes in Convex and Simple Drawings

Abstract

Gons and holes in point sets have been extensively studied in the literature. For simple drawings of the complete graph a generalization of the Erd\H{o}s--Szekeres theorem is known and empty triangles have been investigated. We introduce a notion of $k$-holes for simple drawings and survey generalizations thereof, like empty $k$-cycles. We present a family of simple drawings without $4$-holes and prove a generalization of Gerken's empty hexagon theorem for convex drawings. A crucial intermediate step is the structural investigation of pseudolinear subdrawings in convex drawings. With respect to empty $k$-cycles, we show the existence of empty $4$-cycles in every simple drawing of $K_n$ and give a construction that admits only $\Theta(n^2)$ of them.