Planar graphs as VPG-graphs

Abstract

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 B 2-VPG. We also show that the 4-connected planar graphs are a subclass of the intersection graphs of Z-shapes (i.e., a special case of B 2-VPG). Additionally, we demonstrate that a B2-VPG representation of a planar graph can be constructed in O(n3/2) 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 gain a new proof that bipartite planar graphs are a subclass of 2-DIR. © 2013 Springer-Verlag.