String graphs of k-bend paths on a grid

We investigate the class of vertex intersection graphs of paths on a grid, and specifically consider the subclasses that are obtained when each path in the representation has at most k bends (turns). We call such a subclass the Bk-VPG graphs, k ≥ 0. If the number k of bends is not restricted, then the VPG graphs are shown to be equivalent to the well-known class of string graphs, namely, the intersection graphs of arbitrary curves in the plane. In the case of B0-VPG graphs, we observe that horizontal and vertical segments have strong Helly number 2, and thus the clique problem has polynomial-time complexity, given the path representation. We show that the coloring problem for Bk-VPG graphs, for k ≥ 0, is NP-complete and give a 2-approximation algorithm for coloring B0-VPG graphs. Furthermore, we prove that triangle-free B0-VPG graphs are 4-colorable, and this is best possible. We present a complete hierarchy of VPG graphs relating them to other known families of graphs. The grid intersection graphs are shown to be equivalent to the bipartite B0-VPG graphs. We prove the strict containment of B0-VPG into B1-VPG. Planar graphs are known to be in the class of string graphs, and we prove here that planar graphs are B3-VPG graphs.