We combine the known notion of the edge intersection graphs of paths in a tree with a VLSI grid layout model to introduce the edge intersection graphs of paths on a grid. Let P be a collection of nontrivial simple paths on a grid G. We define the edge intersection graph EPG(P) of P to have vertices which correspond to the members of P, such that two vertices are adjacent in EPG(P) if the corresponding paths in P share an edge in G. An undirected graph G is called an edge intersection graph of paths on a grid (EPG) if G = EPG(P) for some P and G, and P, G is an EPG representation of G. We prove that every graph is an EPG graph. A turn of a path at a grid point is called a bend. We consider here EPG representations in which every path has at most a single bend, called B 1 -EPG representations and the corresponding graphs are called B 1 -EPG graphs. We prove that any tree is a B 1 -EPG graph. Moreover, we give a structural property that enables one to generate non B 1 -EPG graphs. Furthermore, we characterize the representation of cliques and chordless 4-cycles in B 1 -EPG graphs. We also prove that single bend paths on a grid have Strong Helly number 3.

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 B 0 -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 B 0 -VPG graphs, however, are NPcomplete. We give a 2-approximation algorithm for coloring B 0 -VPG graphs. Furthermore, we prove that triangle-free B 0 -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 B 0 -VPG graphs and the circle graphs are strictly contained in B 1 -VPG. We prove the strict containment of B 0 -VPG into B 1 -VPG, and we conjecture that, in general, this strict containment continues for all values of k. We present a graph which is not in B 1 -VPG. Planar graphs are known to be in the class of string graphs, and we prove here that planar graphs are B 3 -VPG graphs, although it is not known if this is best possible.

Edge-intersection graphs of paths in grids are graphs that can be represented such that vertices are paths in a grid and edges between vertices of the graph exist whenever two grid paths share a grid edge. This type of graphs is motivated by applications in conflict resolution of paths in grid networks.In this paper, we continue the study of edge-intersection graphs of paths in a grid, which was initiated by Golumbic, Lipshteyn and Stern. We show that for any k, if the number of bends in each path is restricted to be at most k, then not all graphs can be represented. Then we study some graph classes that can be represented with k-bend paths, for small k.We show that every planar graph has a representation with 5-bend paths, every outerplanar graph has a representation with 3-bend paths, and every planar bipartite graph has a representation with 2-bend paths. We also study line graphs, graphs of bounded pathwidth, and graphs with κ-regular edge orientations.