Free Triangle Orders

We define a new class of ordered sets, called free triangle orders. These are ordered sets represented by a left-to-right ordering on geometric objects contained in a horizontal strip in the plane. The objects are called 'free triangles', and have one vertex on each of the two boundaries of the strip and one vertex in its interior. These ordered sets generalize the classes of trapezoid and triangle orders studied by Bogart, Möhring, and Ryan, represented by trapezoids and triangles respectively, contained within a strip in the plane, and are a special case of the orders of tube dimension 2 introduced by Habib and co-workers, which are represented by any set of convex bodies contained within a strip in the plane. Our main result is that the class of free triangle orders strictly contains the class of trapezoid orders.     

Triangle,Parallelogram, and Trapezoid Orders

In his 1998 paper, Ryan classified the sets of unit, proper, and plain trapezoid and parallelogram orders. We extend this classification to include unit, proper, and plain triangle orders. We prove that there are 20 combinations of these properties that give rise to distinct classes of ordered sets, and order these classes by containment.     

Unit and Proper Tube Orders

In 2005, we defined the n-tube orders, which are the n-dimensional analogue of interval orders in 1 dimension, and trapezoid orders in 2 dimensions. In this paper we consider two variations of n-tube orders: unit n-tube orders and proper n-tube orders. It has been proven that the classes of unit and proper interval orders are equal, and the classes of unit and proper trapezoid orders are not. We prove that the classes of unit and proper n-tube orders are not equal for all n ≥ 3, so the general case follows the situation in 2 dimensions.     

Comparability Invariance Results for Tolerance Orders

We prove comparability invariance results for three classes of ordered sets: bounded tolerance orders (equivalent to parallelogram orders), unit bitolerance orders (equivalent to point-core bitolerance orders) and unit tolerance orders (equivalent to 50% tolerance orders). Each proof uses a different technique and relies on the alternate characterization.     

Obstacle Numbers of Graphs

An obstacle representation of a graph G is a drawing of G in the plane with straight-line edges, together with a set of polygons (respectively, convex polygons) called obstacles, such that an edge exists in G if and only if it does not intersect an obstacle. The obstacle number (convex obstacle number) of G is the smallest number of obstacles (convex obstacles) in any obstacle representation of G. In this paper, we identify families of graphs with obstacle number 1 and construct graphs with arbitrarily large obstacle number (convex obstacle number). We prove that a graph has an obstacle representation with a single convex k-gon if and only if it is a circular arc graph with clique covering number at most k in which no two arcs cover the host circle. We also prove independently that a graph has an obstacle representation with a single segment obstacle if and only if it is the complement of an interval bigraph.     

Subspace intersection graphs

Given a set R of affine subspaces in R^d of dimension e, its intersection graph G has a vertex for each subspace, and two vertices are adjacent in G if and only if their corresponding subspaces intersect. For each pair of positive integers d and e we obtain the class of (d,e)-subspace intersection graphs. We classify the classes of (d,e)-subspace intersection graphs by containment, for e=1 or e=d−1 or d≤4.