Graphs with Large Total Angular Resolution

Abstract

0000−0002−2364−0583] , Matias Korman 2 , Yoshio Okamoto 3[0000−0002−9826−7074] , Irene Parada 1[0000−0003−3147−0083] , Daniel Perz 1[0000−0002−6557−2355] , André van Renssen 4[0000−0002−9294−9947] , and Birgit Vogtenhuber 1[0000−0002−7166−4467]Abstract. The total angular resolution of a straight-line drawing is the minimum angle between two edges of the drawing. It combines two properties contributing to the readability of a drawing: the angular resolution, which is the minimum angle between incident edges, and the crossing resolution, which is the minimum angle between crossing edges. We consider the total angular resolution of a graph, which is the maximum total angular resolution of a straight-line drawing of this graph. We prove that, up to a finite number of well specified exceptions of constant size, the number of edges of a graph with n vertices and a total angular resolution greater than 60 • is bounded by 2n − 6. This bound is tight. In addition, we show that deciding whether a graph has total angular resolution at least 60 • is NP-hard.