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.

a b s t r a c tAn (h, s, t)-representation of a graph G consists of a collection of subtrees of a tree T , where each subtree corresponds to a vertex in G, such that (i) the maximum degree of T is at most h, (ii) every subtree has maximum degree at most s, (iii) there is an edge between two vertices in the graph G if and only if the corresponding subtrees have at least t vertices in common in T . The class of graphs that have an (h, s, t)-representation is denoted by [h, s, t].It is well known that the class of chordal graphs corresponds to the class [3, 3, 1]. Moreover, it was proved by Jamison and Mulder that chordal graphs correspond to orthodox-[3, 3, 1] graphs defined below.In this paper, we investigate the class of [h, 2, t] graphs, i.e., the intersection graphs of paths in a tree. The [h, 2, 1] graphs are also known as path graphs [F. Gavril, A recognition algorithm for the intersection graphs of paths in trees, Discrete Math. 23 (1978) 211-227] or VPT graphs [M.C. Golumbic, R.E. Jamison, Edge and vertex intersection of paths in a tree, Discrete Math. 55 (1985) 151-159], and [h, 2, 2] graphs are known as the EPT graphs. We consider variations of [h, 2, t] by three main parameters: h, t and whether the graph has an orthodox representation. We give the complete hierarchy of relationships between the classes of weakly chordal, chordal, [h, 2, t] and orthodox-[h, 2, t] graphs for varied values of h and t.

Let P be a collection of nontrivial simple paths on a host tree T. The edge intersection graph of P, denoted by EPT(P), has vertex set that corresponds to the members of P, and two vertices are joined by an edge if and only if the corresponding members of P share at least one common edge in T. An undirected graph G is called an edge intersection graph of paths in a tree if G = EPT(P) for some P and T. The EPT graphs are useful in network applications. Scheduling undirected calls in a tree network or assigning wavelengths to virtual connections in an optical tree network are equivalent to coloring its EPT graph.An undirected graph G is chordal if every cycle in G of length greater than 3 possesses a chord. Chordal graphs correspond to vertex intersection graphs of subtrees on a tree. An undirected graph G is weakly chordal if every cycle of length greater than 4 in G and in its complement G possesses a chord. It is known that the EPT graphs restricted to host trees of vertex degree 3 are precisely the chordal EPT graphs. We prove a new analogous result that weakly chordal EPT graphs are precisely the EPT graphs with host tree restricted to degree 4. Moreover, this provides an algorithm to reduce a given EPT representation of a weakly chordal EPT graph to an EPT representation on a degree 4 tree. Finally, we raise a number of intriguing open questions regarding related families of graphs.

A graph G = (V, E) is a chordal probe graph if its vertices can be partitioned into two sets, P (probes) and N (non-probes), where N is a stable set and such that G can be extended to a chordal graph by adding edges between non-probes. We give several characterizations of chordal probe graphs, first, in the case of a fixed given partition of the vertices into probes and non-probes, and second, in the more general case where no partition is given. In both of these cases, our results are obtained by introducing new classes, namely, N-triangulatable graphs and cycle-bicolorable graphs. We give polynomial time recognition algorithms for each class. Ntriangulatable graphs have properties similar to chordal graphs, and we characterize them using graph separators and using a vertex elimination ordering. For cycle-bicolorable graphs, which are shown to be perfect, we prove that any cycle-bicoloring of a graph renders it N-triangulatable. The corresponding recognition complexity for chordal probe graphs, given a partition of the vertices into probes and non-probes, is O(|P ||E|), thus also providing an interesting tractable subcase of the chordal graph sandwich problem. If no partition is given in advance, the complexity of our recognition algorithm is O(|E| 2).