Reconstructibility of matroid polytopes

We specify what is meant for a polytope to be reconstructible from its graph or dual graph. And we introduce the problem of class reconstructibility, i.e., the face lattice of the polytope can be determined from the (dual) graph within a given class. We provide examples of cubical polytopes that are not reconstructible from their dual graphs. Furthermore, we show that matroid (base) polytopes are not reconstructible from their graphs and not class reconstructible from their dual graphs; our counterexamples include hypersimplices. Additionally, we prove that matroid polytopes are class reconstructible from their graphs, and we present a $O(n^3)$ algorithm that computes the vertices of a matroid polytope from its $n$-vertex graph. Moreover, our proof includes a characterisation of all matroids with isomorphic basis exchange graphs.Comment: 22 pages, 5 figure

Flat Embeddings of Genetic and Distance Data

The idea of displaying data in the plane is very attractive in many different fields of research. This thesis will focus on distance-based phylogenetics and multidimensional scaling (MDS). Both types of method can be viewed as a high-dimensional data reduction to pairwise distances and visualization of the data based on these distances. The difference between phylogenetics and multidimensional scaling is that the first one aims at finding a network or a tree structure that fits the distances, whereas MDS does not fix any structure and objects are simply placed in a low-dimensional space so that distances in the solution fit distances in the input as good as possible. Chapter 1 provides an introduction to the phylogenetics and multidimensional scaling. Chapter 2 focuses on the theoretical background of flat split systems (planar split networks). We prove equivalences between flat split systems, planar split networks and loop-free acyclic oriented matroids of rank three. The latter is a convenient mathematical structure that we used to design the algorithm for computing planar split networks that is described in Chapter 3. We base our approach on the well established agglomerative algorithms Neighbor-Joining and Neighbor-Net. In Chapter 4 we introduce multidimensional scaling and propose a new method for computing MDS plots that is based on the agglomerative approach and spring embeddings. Chapter 5 presents several case studies that we use to compare both of our methods and some classical agglomerative approaches in the distance-based phylogenetics

Cocircuit Graphs and Efficient Orientation Reconstruction in Oriented Matroids

We consider the cocircuit graph GM of an oriented matroid M, which is the 1-skeleton of the cell complex formed by the span of the cocircuits of M. As a result of Cordovil, Fukuda, and Guedes de Oliveira, the isomorphism class of M is not determined by GM, but it is determined if M is uniform and the vertices in GM are paired if they are associated to negative cocircuits; furthermore the reorientation class of an oriented matroid M with rank(M) 2 is determined by GM if every vertex in GM is labeled by the zero support of the associated cocircuit. In this paper we show that the isomorphism class of a uniform oriented matroid is determined by the cocircuit graph, and we present polynomial algorithms which provide constructive proofs to all these results. Furthermore it is shown that the correctness of the input of the algorithms can be verified in polynomial time