Michal Stern

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.

Equivalences and the complete hierarchy of intersection graphs of paths in a tree

An (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.

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.

Edge-Intersection Graphs of k-Bend Paths in Grids

Edge-intersection graphs of paths in grids are graphs that can be represented with vertices as paths in grids and edges between the 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 bipartite planar graph has a representation with 2-bend paths.

Smallest Odd Holes in Claw-Free Graphs (Extended Abstract)

In this paper, we give general structure properties of a smallest odd hole in a claw-free graph that lead to a polynomial time algorithm. The algorithm is based on a modified BFS we call Γ-BFS. For a graph G with n vertices and m edges, the time complexity of the algorithm is O(nm 2). The algorithm is very easy to implement. We conclude the paper with a suggestion for an extension of our approach in order to detect an odd hole in a general graph.

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.

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] .

Single bend paths on a grid have strong helly number 4: errata atque emendationes ad “edge intersection graphs of single bend paths on a grid”

In this note, we prove that a collection of single bend paths on a grid, the so-called B1 representations, have strong Helly number 4. This corrects a false claim from Ref. [Golumbic et al., Networks, 54 (2009), 130–138; and other errata are also corrected.

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].

Student poster session

During WG 2012 there was a Student Poster Session, where the following posters were presented (alphabetically ordered by students last name).