Random intersection graphs was introduced by SingerCohen [6]. These graphs have recently received considerable attention in the literature and been used in diverse applications [1,2,7–13]. In a general random intersection graph, each node is assigned a set of items in some random manner, and any two nodes establish an undirected edge in between if and only if they have at least a certain number...

As with many min-max characterizations, proving one of the inequalities is straightforward. For any U ⊆ S and I ∈ I1 ∩ I2, we have |I| = |I ∩ U |+ |I ∩ (S \ U)| ≤ r1(U) + r2(S \ U), since I ∩ U is an independent set in I1 and I ∩ (S \ U) is an independent set in I2. Therefore, maxI∈I1∩I2 |I| ≤ minU∈S (r1(U) + r2(S \ U)). The following important examples illustrate some of the applications of th...

A jump system is a set of lattice points satisfying a certain exchange axiom. This notion was introduced by Bouchet and Cunningham 2], as a common generalization of (among others) the sets of bases of a matroid and degree sequences of subgraphs of a graph. We prove, under additional assumptions, a min-max formula for the distance of a lattice point from a jump system. The conditions are met in ...

We develop wavelet methods for the multiresolution representation of parametric curves and surfaces. To support the representation, we construct a new family of compactly supported symmetric biorthogonal wavelets with interpolating scaling functions. The wavelets in these biorthogonal pairs have properties better suited for curves and surfaces than many commonly used lters. We also give example...

Just as the Bernstein basis and Be Âzier points give us tools to manipulate polynomials, the Poisson basis and control points provide a new framework for investigating analytic functions. Using the Poisson representation, two different subdivision algorithms for analytic functions can be derived as extensions of the Be Âzier polynomial subdivision algorithm. The ®rst procedure generates an appr...

Weighted finite-state machines with n tapes describe n-ary rational string relations. The join n-ary relation is very important regarding to applications. It is shown how to compute it via a more simple operation, the auto-intersection. Join and auto-intersection generally do not preserve rationality. We define a class of triples 〈A, i, j〉 such that the auto-intersection of the machine A w.r.t....

A directed path graph is the intersection graph of a family of directed subpaths of a directed tree. A rooted path graph is the intersection graph of a family of directed subpaths of a rooted tree. Rooted path graphs are directed path graphs. Several characterizations are known for directed path graphs: one by forbidden induced subgraphs and one by forbidden asteroids. It is an open problem to ...

A chordal graph is the intersection graph of a family of subtrees of a tree, or, equivalently, it is the (edge-)intersection graph of leaf-generated subtrees of a full binary tree. In this paper, a generalization of chordal graphs from this viewpoint is studied: a graph G=(V; E) is representable if there is a family of subtrees {Sv}v∈V of a binary tree, such that uv ∈ E if and only if |Su ∩ Sv|...

Let P be a collection of nontrivial simple paths in a 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 the corresponding members of P share a 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 grap...