Elad Cohen

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, kC 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.

Theory and Practice in Large Carpooling Problems

Abstract We address the carpooling problem as a graph-theoretic problem. If the set of drivers is known in advance, then for any car capacity, the problem is equivalent to the assignment problem in bipartite graphs. Otherwise, when we do not know in advance who will drive their vehicle and who will be a passenger, the problem is NP-hard. We devise and implement quick heuristics for both cases, based on graph algorithms, as well as parallel algorithms based on geometric/algebraic approach. We compare between the algorithms on random graphs, as well as on real, very large, data.

Recognizing some subclasses of vertex intersection graphs of 0-bend paths in a grid

We investigate graphs that can be represented as vertex intersections of horizontal and vertical paths in a grid, known as B0 -VPG graphs. Recognizing these graphs is an NP-hard problem. In light of this, we focus on their subclasses. In the paper, we describe polynomial time algorithms for recognizing chordal B0 -VPG graphs, and for recognizing B0 -VPG graphs that have a representation on a grid with 2 rows.

String graphs of k-bend 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 . 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 [S. Bellantoni, I. Ben-Arroyo Hartman, T. Przytycka, S. Whitesides, Grid intersection graphs and boxicity, Discrete Math. 114, (1993), 41–49; I. Ben-Arroyo Hartman, I. Newman, R. Ziv, On grid intersection graphs, Discrete Math. 87(1), (1991), 41–52] are shown to be equivalent to the bipartite B 0 -VPG graphs. Chordal B 0 -VPG graphs are shown to be exactly Strongly Chordal B 0 -VPG graphs. We prove the strict containment of B 0 -VPG and circle graphs into 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. In the case of B 0 -VPG graphs, we observe that a set of horizontal and vertical segments have strong Helly number 2. We show that the coloring problem for B k -VPG graphs, for k ⩾ 0 , is NP-complete and 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.

Characterizations of cographs as intersection graphs of paths on a grid

A cograph is a graph which does not contain any induced path on four vertices. In this paper, we characterize those cographs that are intersection graphs of paths on a grid in the following two cases: (i) the paths on the grid all have at most one bend and the intersections concern edges (→B1→B1-EPG); (ii) the paths on the grid are not bended and the intersections concern vertices (→B0→B0-VPG). In both cases, we give a characterization by a family of forbidden induced subgraphs. We further present linear-time algorithms to recognize B1B1-EPG cographs and B0B0-VPG cographs using their cotree.

What Is between Chordal and Weakly Chordal Graphs

An (h ,s ,t )-representation of a graph G consists of a collection of subtrees {S v | v *** V (G )} of a tree T , such that (i) the maximum degree of T is at most h , (ii) every subtree has maximum degree at most s , and (iii) there is an edge between two vertices in the graph if and only if the corresponding subtrees in T have at least t vertices in common. For example, chordal graphs correspond to [ *** , *** ,1] = [3,3,1] = [3,3,2] graphs (notation of *** here means that no restriction is imposed). We investigate the complete bipartite graph K 2,n and prove new theorems characterizing those K 2,n graphs that have an (h ,s ,2)-representation and those that have an (h ,s ,3)-representation. We characterize [3,2,4] graphs as equivalent to the 4-flower-free [2,4,4] graphs and give a recognition algorithm for [2,3,4] graphs. Based on these characterizations, we present new results that confirm that weakly chordal graphs, as opposed to chordal graphs, can not be characterized within the [h ,s ,t ] framework. Furthermore, we show a hierarchy of families of graphs between chordal and weakly chordal within the [h ,s ,t ] framework.

Posets and VPG Graphs

We investigate the class of intersection graphs of paths on a grid (VPG graphs), and specifically the relationship between the bending number of a cocomparability graph and the poset dimension of its complement. We show that the bending number of a cocomparability graph G is at most the poset dimension of the complement of G minus one. Then, via Ramsey type arguments, we show our upper bound is best possible.

On the Bi-enhancement of Chordal-bipartite Probe Graphs

Abstract Given a class C of graphs, a graph G = ( V , E ) is said to be a C -probe graph if there exists a stable (i.e., independent) set of vertices X ⊆ V and a set F of pairs of vertices of X such that the graph G ′ = ( V , E ∪ F ) is in the class C . Recently, there has been increasing interest and research on a variety of C -probe graph classes, such as interval probe graphs, chordal probe graphs and chain probe graphs. In this paper we focus on chordal-bipartite probe graphs. We prove a structural result that if B is a bipartite graph with no chordless cycle of length strictly greater than 6, then B is chordal-bipartite probe if and only if a certain “enhanced” graph B ∗ is a chordal-bipartite graph. This theorem is analogous to a result on interval probe graphs in Zhang (1994) [18] and to one on chordal probe graphs in Golumbic and Lipshteyn (2004) [11] .

Tolerance intersection graphs of degree bounded subtrees of a tree with constant tolerance 2

Abstract An ( h , s , t ) -representation of a graph G consists of a collection of subtrees { S v : v ∈ V ( G ) } of a tree T , such that (i) the maximum degree of T is at most h , (ii) every subtree has maximum degree at most s , and (iii) there is an edge between two vertices in the graph if and only if the corresponding subtrees in T have at least t vertices in common. Jamison and Mulder denote the family of graphs that admit such a representation as [ h , s , t ] . Our main theorem shows that the class of weakly chordal graphs is incomparable with the class [ h , s , t ] . We introduce new characterizations of the graph K 2 , n in terms of the families [ h , s , 2 ] and [ h , s , 3 ] . We then present our second main result characterizing the graphs in [4, 3, 2] as being the graphs in [4, 4, 2] avoiding a particular family of substructures, and we give a recognition algorithm for the family [4, 3, 2].