Bend-Bounded Path Intersection Graphs: Sausages, Noodles, and Waffles on   a Grill
  In this paper we study properties of intersection graphs of k-bend paths in the rectangular grid. A k-bend path is a path with at most k 90 degree turns. The class of graphs representable by intersections of k-bend paths is denoted by B_k-VPG. We show here that for every fixed k, B_k-VPG is a proper subset of B_{k+1}-VPG and that recognition of graphs from B_k-VPG is NP-complete even when the input graph is given by a B_{k+1}-VPG representation. We also show that the class B_k-VPG (for k>0) is in no inclusion relation with the class of intersection graphs of straight line segments in the plane. 
Steven Chaplick, V\'it Jel\'inek, Jan Kratochv\'il, Tom\'a\v{s}   Vysko\v{c}il
cs.CG cs.DM math.CO