Towards the computation of the convex hull of a configuration from its corresponding separating matrix

In this paper, we cope with the following problem: compute the size of the convex hull of a configuration C, where the given data is the number of separating lines between any two points of the configuration (where the lines are generated by pairs of other points of the configuration). We give an algorithm for the case that the convex hull is of size 3, and a partial algorithm and some directions for the case that the convex hull is of size bigger than 3.Comment: 10 pages, 3 figures; To appear in the Australasian Journal of Combinatoric

The Orchard crossing number of an abstract graph

We introduce the Orchard crossing number, which is defined in a similar way to the well-known rectilinear crossing number. We compute the Orchard crossing number for some simple families of graphs. We also prove some properties of this crossing number. Moreover, we define a variant of this crossing number which is tightly connected to the rectilinear crossing number, and compute it for some simple families of graphs.Comment: 17 pages, 10 figures. Totally revised, new material added. Submitte

The Maximum Rectilinear Crossing Number of the Wheel Graph

We find and prove the maximum rectilinear crossing number of the wheel graph. First, we illustrate a picture of the wheel graph with many crossings to prove a lower bound. We then prove that this bound is sharp. The treatment is divided into two cases for n even and n odd

The Maximum Rectilinear Crossing Number of the Petersen Graph

We prove that the maximum rectilinear crossing number of the Petersen graph is 49. First, we illustrate a picture of the Petersen graph with 49 crossings to prove the lower bound. We then prove that this bound is sharp by carefully analyzing the ten Cs\u27s which occur in the Petersen graph and their properties

The Minimum of the Maximum Rectilinear Crossing Numbers of Small Cubic Graphs

Here we consider the minimum of the maximum rectilin­ear crossing numbers for all d-regular graphs of order n. The case of connected graphs only is investigated also. For d = 3 exact values are determined for n are less than or equal to 12 and some estimations are given in general