Planar Graphs as VPG-Graphs

A graph is Bk-VPG when it has an intersection representation by paths in a rectangular grid with at most k bends (turns). It is known that all planar graphs are B3-VPG and this was conjectured to be tight. We disprove this conjecture by showing that all planar graphs are B2-VPG. We also show that the 4-connected planar graphs constitute a subclass of the intersection graphs of Z-shapes (i.e., a special case of B2-VPG). Additionally, we demonstrate that a B2-VPG representation of a planar graph can be constructed in O(n) time. We further show that the triangle-free planar graphs are contact graphs of: L-shapes, Γ-shapes, vertical segments, and horizontal segments (i.e., a special case of contact B1-VPG). From this proof we obtain a new proof that bipartite planar graphs are a subclass of 2-DIR. Submitted: December 2012 Reviewed: March 2013 Revised: April 2013 Accepted: May 2013 Final: July 2013 Published: July 2013 Article type: Regular paper Communicated by: W. Didimo and M. Patrignani An extended abstract of this paper was presented at the 20th International Symposium on Graph Drawing, in Redmond, USA, in September 2012 [7]. The research of the first author was supported by NSERC and partially by GraDR EUROGIGA project No. GIG/11/E023. The research of the second author was supported by GraDR EUROGIGA project No. GIG/11/E023. E-mail addresses: [email protected] (Steven Chaplick) [email protected] (Torsten Ueckerdt) 476 Chaplick and Ueckerdt Planar Graphs as VPG-Graphs