Drawing Outer 1-planar Graphs with Few Slopes

A graph is outer 1-planar if it admits a drawing where each vertex is on the outer face and each edge is crossed by at most another edge. Outer 1-planar graphs are a superclass of the outerplanar graphs and a subclass of the planar partial 3-trees. We show that an outer 1-planar graph G of bounded degree ∆ admits an outer 1-planar straight-line drawing that uses O(∆) different slopes, which generalizes a previous result by Knauer et al. about the outerplanar slope number of outerplanar graphs [18]. We also show that O(∆) slopes suffice to construct a crossing-free straight-line drawing of G; the best known upper bound on the planar slope number of planar partial 3-trees of bounded degree ∆ is O(∆) as proved by Jeĺınek et al. [16]. Submitted: October 2014 Reviewed: February 2015 Revised: March 2015 Reviewed: August 2015 Revised: September 2015 Accepted: October 2015 Final: October 2015 Published: November 2015 Article type: Regular paper Communicated by: C. Duncan and A. Symvonis Research supported in part by the MIUR project AMANDA “Algorithmics for MAssive and Networked DAta”, prot. 2012C4E3KT 001. E-mail addresses: [email protected] (Emilio Di Giacomo) [email protected] (Giuseppe Liotta) [email protected] (Fabrizio Montecchiani) 708 Di Giacomo et al. Drawing Outer 1-planar Graphs with Few Slopes