Vertex Intersection Graphs of 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. 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 NPcomplete. 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. Submitted: December 2010 Reviewed: September 2011 Revised: October 2011 Reviewed: October 2011 Revised: November 2011 Accepted: November 2011 Final: December 2011 Published: January 2012 Article type: Regular paper Communicated by: T. Warnow This research took place while the authors, Andrei Asinowski and Vincent Limouzy, were postdoctoral fellows at the Ceasarea Rothschild Institute at the University of Haifa Elad Cohen was partially supported by the Israel Science Foundation grant 347/09 E-mail addresses: [email protected] (Andrei Asinowski) [email protected] (Elad Cohen) [email protected] (Martin Charles Golumbic) [email protected] (Vincent Limouzy) [email protected] (Marina Lipshteyn) [email protected] (Michal Stern) 130 Asinowski et al. Vertex Intersection Graphs of Paths on a Grid