For a set S of n lines labeled from 1 to n, we say that S supports an n-vertex planar graph G if for every labeling from 1 to n of its vertices, G has a straight-line crossing-free drawing with each vertex drawn as a point on its associated line. It is known from previous work [4] that no set of n parallel lines supports all n-vertex planar graphs. We show that intersecting lines, even if they ...

We consider several classes of intersection graphs of line segments in the plane and prove new equality and separation results between those classes. In particular, we show that: • intersection graphs of grounded segments and intersection graphs of downward rays form the same graph class, • not every intersection graph of rays is an intersection graph of downward rays, and • not every intersect...

Vertices of an affiliation network are linked to features and two vertices are declared adjacent whenever they share a common feature. We introduce a random intersection graph process aimed at modeling sparse evolving affiliation networks. We establish the asymptotic degree distribution and provide explicit asymptotic formulas for assortativity and clustering coefficients showing how these edge...

For a general random intersection graph we show an approximation of the vertex degree distribution by a Poisson mixture. key words: intersection graph, random graph, degree distribution

We study the interplay between the maximum genus of a graph and bases of its cycle space via the corresponding intersection graph. Our main results show that the matching number of the intersection graph is independent of the basis precisely when the graph is upper-embeddable, and completely describe the range of matching numbers when the graph is not upper-embeddable. Particular attention is p...

The intersection graph of a set system S is a graph on the vertex set S, in which two vertices are connected by an edge if and only if the corresponding sets have nonempty intersection. It was shown by Tietze (1905) that every finite graph is the intersection graph of 3-dimensional convex polytopes. The analogous statement is false in any fixed dimension if the polytopes are allowed to have onl...

Random s-intersection graphs have recently received much interest in a wide range of application areas. Broadly speaking, a random s-intersection graph is constructed by first assigning each vertex a set of items in some random manner, and then putting an undirected edge between all pairs of vertices that share at least s items (the graph is called a random intersection graph when s = 1). A spe...

In this paper, we study different classes of intersection graphs of maximal hypercubes of median graphs. For a median graph G and k ≥ 0, the intersection graph Qk(G) is defined as the graph whose vertices are maximal hypercubes (by inclusion) in G, and two vertices Hx and Hy in Qk(G) are adjacent whenever the intersection Hx ∩ Hy contains a subgraph isomorphic to Qk. Characterizations of clique...