Density theorems for intersection graphs of t-monotone curves

A curve \gamma in the plane is t-monotone if its interior has at most t-1 vertical tangent points. A family of t-monotone curves F is \emph{simple} if any two members intersect at most once. It is shown that if F is a simple family of n t-monotone curves with at least \epsilon n^2 intersecting pairs (disjoint pairs), then there exists two subfamilies F_1,F_2 \subset F of size \delta n each, such that every curve in F_1 intersects (is disjoint to) every curve in F_2, where \delta depends only on \epsilon. We apply these results to find pairwise disjoint edges in simple topological graphs

Lombardi Drawings of Graphs

We introduce the notion of Lombardi graph drawings, named after the American abstract artist Mark Lombardi. In these drawings, edges are represented as circular arcs rather than as line segments or polylines, and the vertices have perfect angular resolution: the edges are equally spaced around each vertex. We describe algorithms for finding Lombardi drawings of regular graphs, graphs of bounded degeneracy, and certain families of planar graphs.Comment: Expanded version of paper appearing in the 18th International Symposium on Graph Drawing (GD 2010). 13 pages, 7 figure

Simultaneous Embeddability of Two Partitions

We study the simultaneous embeddability of a pair of partitions of the same underlying set into disjoint blocks. Each element of the set is mapped to a point in the plane and each block of either of the two partitions is mapped to a region that contains exactly those points that belong to the elements in the block and that is bounded by a simple closed curve. We establish three main classes of simultaneous embeddability (weak, strong, and full embeddability) that differ by increasingly strict well-formedness conditions on how different block regions are allowed to intersect. We show that these simultaneous embeddability classes are closely related to different planarity concepts of hypergraphs. For each embeddability class we give a full characterization. We show that (i) every pair of partitions has a weak simultaneous embedding, (ii) it is NP-complete to decide the existence of a strong simultaneous embedding, and (iii) the existence of a full simultaneous embedding can be tested in linear time.Comment: 17 pages, 7 figures, extended version of a paper to appear at GD 201

A computational approach to Conway's thrackle conjecture

A drawing of a graph in the plane is called a thrackle if every pair of edges meets precisely once, either at a common vertex or at a proper crossing. Let t(n) denote the maximum number of edges that a thrackle of n vertices can have. According to a 40 years old conjecture of Conway, t(n)=n for every n>2. For any eps>0, we give an algorithm terminating in e^{O((1/eps^2)ln(1/eps))} steps to decide whether t(n)2. Using this approach, we improve the best known upper bound, t(n)<=3/2(n-1), due to Cairns and Nikolayevsky, to 167/117n<1.428n.Comment: 16 pages, 7 figure

Unit Grid Intersection Graphs: Recognition and Properties

It has been known since 1991 that the problem of recognizing grid intersection graphs is NP-complete. Here we use a modified argument of the above result to show that even if we restrict to the class of unit grid intersection graphs (UGIGs), the recognition remains hard, as well as for all graph classes contained inbetween. The result holds even when considering only graphs with arbitrarily large girth. Furthermore, we ask the question of representing UGIGs on grids of minimal size. We show that the UGIGs that can be represented in a square of side length 1+epsilon, for a positive epsilon no greater than 1, are exactly the orthogonal ray graphs, and that there exist families of trees that need an arbitrarily large grid