Vertex Intersection Graphs of Paths on a Grid

Abstract

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 B_k-VPG graphs, k ≥ 0. In chip manufacturing, circuit layout is modeled as paths (wires) on a grid, where it is natural to constrain the number of bends per wire for reasons of feasibility and to reduce the cost of the chip. If the number k of bends is not restricted, then the VPG graphs are 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. The recognition and coloring problems for B₀-VPG graphs, however, are NP-complete. We 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 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 and the circle graphs are strictly contained in B₁-VPG. We prove the strict containment of B₀-VPG into B₁-VPG, and we conjecture that, in general, this strict containment continues for all values of k. We present a graph which is not in 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, although it is not known if this is best possible.