Planar Graphs as VPG-Graphs

Abstract

A graph is $B_k-VPG$ 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 $B_3-VPG$ 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 $B_2-VPG$ representation of a planar graph can be constructed in $O(n^{3/2} )$ 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 $B_1-VPG$). From this proof we gain a new proof that bipartite planar graphs are a subclass of 2-DIR