Let t(C) be the number of tangent pairs among a set C of n Jordan regions in the plane. Pach, Suk, and Treml [6] showed that if C consists of convex bodies and its intersection graph is bipartite then t(C) ≤ 4n−Θ(1), and, moreover, there are such sets that admit at least 3n−Θ( √ n) tangencies. We close this gap and generalize their result by proving that the correct bound is 3n−Θ(1), even when ...

We prove Ramsey-type results for intersection graphs of geometric objects the plane. In particular, we prove the following bounds, all of which are tight apart from the constant c. There is a constant c > 0 such that for every family F of n ≥ 2 convex sets in the plane, the intersection graph of F or its complement contains a balanced complete bipartite graph of size at least cn. There is a con...

Tolerance graphs have been extensively studied since their introduction, due to their interesting structure and their numerous applications, as they generalize both interval and permutation graphs in a natural way. It has been conjectured by Golumbic, Monma, and Trotter in 1984 that the intersection of tolerance and cocomparability graphs coincides with bounded tolerance graphs. Since cocompara...

An intersection representation of a graph G is a function f : V…G† ! 2 (where S is any set) with the property that uv A E…G† if and only if f …u†V f …v†0h. The size of the representation is jSj. The intersection number of G is the smallest size of an intersection representation of G. The intersection number can be expressed as an integer program, and the value of the linear relaxation of that p...

Abstract Private set intersection (PSI) allows two mutually distrusting parties each with a as input, to learn the of both their sets without revealing anything more about respective input sets. Traditionally, PSI studies static setting where computation is performed only once on parties’ We initiate study updatable private (UPSI), which compute regular basis that also constantly get updated. c...

Random intersection graphs, denoted by G(n,m, p), are introduced in [9, 14] as opposed to classical Erdős-Rényi random graphs. Let us consider a set V with n vertices and another universal set W with m elements. Define a bipartite graph B(n,m, p) with independent vertex sets V and W . Edges between v ∈ V and w ∈ W exist independently with probability p. The random intersection graph G(n,m, p) d...

Let $G$ be a bipartite graph. In this paper we consider the two kind of location problems namely $p$-center and $p$-median problems on bipartite graphs. The $p$-center and $p$-median problems asks to find a subset of vertices of cardinality $p$, so that respectively the maximum and sum of the distances from this set to all other vertices in $G$ is minimized. For each case we present some proper...

The boxicity of a graph G is defined as the minimum integer k such that G is an intersection graph of axis-parallel k-dimensional boxes. Chordal bipartite graphs are bipartite graphs that do not contain an induced cycle of length greater than 4. It was conjectured by Otachi, Okamoto and Yamazaki that chordal bipartite graphs have boxicity at most 2. We disprove this conjecture by exhibiting an ...