Dual Circumference and Collinear Sets

Abstract

We show that, if an n-vertex triangulation T of maximum degree ∆ has a dual that contains a cycle of length , then T has a non-crossing straight-line drawing in which some set, called a collinear set, of Ω( /∆ 4 ) vertices lie on a line. Using the current lower bounds on the length of longest cycles in 3-regular 3-connected graphs, this implies that every n-vertex planar graph of maximum degree ∆ has a collinear set of size Ω(n 0.8 /∆ 4 ). Very recently, Dujmović et al. (SODA 2019) showed that, if S is a collinear set in a triangulation T then, for any point set X ⊂ R 2 with |X| = |S|, T has a non-crossing straight-line drawing in which the vertices of S are drawn on the points in X. Because of this, collinear sets have numerous applications in graph drawing and related areas.