$1$-String $B_2$-VPG Representation of Planar Graphs
  In this paper, we prove that every planar graph has a 1-string $B_2$-VPG representation---a string representation using paths in a rectangular grid that contain at most two bends. Furthermore, two paths representing vertices $u,v$ intersect precisely once whenever there is an edge between $u$ and $v$. 
Therese Biedl, Martin Derka
cs.CG cs.DS