In the literature of graph recognition algorithms, two of the most extensively covered graphs are interval graphs and permutation graphs [6, 2]. One way of generalizing both interval graphs and permutation graphs is the intersection graph of a collection of trapezoids with corner points lying on two parallel lines. Such a graph is called the trapezoid graph [3, 4]; Ma and Spinrad [8] show that ...

In this paper we continue the investigation of the class of edge intersection graphs of a collection of paths in a tree (EPT graphs) where two paths edge intersect if they share an edge. The class of EPT graphs differs from the class known as path graphs, the latter being the class of vertex intersection graphs of paths in a tree. A characterization is presented here showing when a path graph i...

We show that the class of intersection graphs of subtree filaments in a tree is identical to the class of overlap graphs of subtrees in a tree.

We study a variant of intersection representations with unit balls: unit disks in the plane and unit intervals on the line. Given a planar graph and a bipartition of the edges of the graph into near and far edges, the goal is to represent the vertices of the graph by unit-size balls so that the balls for two adjacent vertices intersect if and only if the corresponding edge is near. We consider ...

The boxicity box(H) of a graph H is the smallest integer d such that H is the intersection of d interval graphs, or equivalently, that H is the intersection graph of axis-aligned boxes in R. These intersection representations can be interpreted as covering representations of the complement H of H with co-interval graphs, that is, complements of interval graphs. We follow the recent framework of...

We analyze the component evolution in inhomogeneous random intersection graphs when the average degree is close to 1. As the average degree increases, the size of the largest component in the random intersection graph goes through a phase transition. We give bounds on the size of the largest components before and after this transition. We also prove that the largest component after the transiti...

The intersection number of a graph G is the minimum size of a ground set S such that G is an intersection graph of some family of subsets F ⊆ 2 . The overlap number of G is defined similarly, except that G is required to be an overlap graph of F . In this paper we show two algorithmic aspects concerning both these graph invariants. On the one hand, we show that the corresponding optimization pr...

In this work we investigate the complexity of some problems related to the Simultaneous Embedding with Fixed Edges (SEFE) of k planar graphs and the PARTITIONED k-PAGE BOOK EMBEDDING (PBE-k) problems, which are known to be equivalent under certain conditions. While the computational complexity of SEFE for k = 2 is still a central open question in Graph Drawing, the problem is NP-complete for k ...

Recently, it was proved that triangle-free intersection graphs of n line segments in the plane can have chromatic number as large as (log log n). Essentially the same construction produces (log log n)-chromatic triangle-free intersection graphs of a variety of other geometric shapes—those belonging to any class of compact arcconnected sets in R2 closed under horizontal scaling, vertical scaling...

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 reas...