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.We present a complete hierarchy of VPG graphs relating them to other known families of graphs. String graphs are equivalent to VPG graphs. The grid intersection graphs are shown to be equivalent to the bipartite B₀-VPG graphs. Chordal B₀-VPG graphs are shown to be exactly Strongly Chordal B₀-VPG graphs. We prove the strict containment of B₀-VPG and circle graphs 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.In the case of B₀-VPG graphs, we observe that a set of horizontal and vertical segments have strong Helly number 2. 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.