Successful Pressing Sequences in Simple Pseudo-Graphs

Motivated by the study of genomes evolving by reversals, the primary topic of this thesis is “successful pressing sequences” in simple pseudo-graphs. Pressing sequences where first introduced by Hannenhali and Pevzner in 1999 where they showed that sorting signed permutation problem can be solved in polynomial time, therefore demonstrating that the length of a most parsimonious solution to the genome in- version only rearrangement problem can be determined efficiently. A signed permutation is an integer permutation where each entry is given a sign: plus or minus. A reversal in a signed permutation is the operation of reversing a subword and flipping the signs of the subword’s entries. The primary computational problem of sorting signed permutations by reversals is to find the minimum number of reversals needed to transform a signed permutation into the positive identity per- mutation. Hannenhalli and Pevzner showed that the signed sorting problem can be solved in polynomial-time in contrast to the problem of sorting unsigned permuta- tions, which is known to be NP-hard in general. At the core of the argument given by Hannenhali and Pevzner is the study of successful pressing sequences on vertex 2-colored graphs. The connection between permutation sorting and phylogenetics dates back to at least the 1930’s, when two biologists, Dobzhansky and Sturtevant, wrote a series of papers in which they argued that the relationships between possible gene arrange- ments within a given chromosome encode critical information about the evolutionary history of species containing those genomes. In particular, they introduced the idea that the degree of disorder between the genes in two genomes is an indicator of the evolutionary distance between two organisms. This has inspired extensive work in the fields of computational biology, bio-informatics and phylogenetics. In particular, researchers have pursued the question of how a common ancestral genome may have been transformed by evolutionary events into distinct, yet homologous, genomes. In mathematics and computer science, we often represent genomes as signed permuta- tions (signed since DNA is oriented between two strands) and evolutionary events are encoded as operations on signed permutations. Among the most studied operations are block transpositions, prefix-reversals, and reversals, all of which correspond to common evolutionary mechanisms. In addition to the study of pressing sequences in simple pseudo-graphs, in this thesis we discuss related topics such as Cholesky factorizations of matrices over finite- fields, a sampling algorithm to generate simple pseudo-graphs uniformly at random, and the complexity of the “pressing space” of a simple pseudo-graph (the space of all successful pressing sequences of a simple pseudo-graph). This work includes collab- orative work with Dr. Joshua Cooper (Mathematics, University of South Carolina), M.S. graduate Erin Hanna (Mathematics, University of South Carolina), and M.S. candidate Peter Gartland (Mathematics, University of South Carolina)

Signed degree sets in signed graphs

The set D of distinct signed degrees of the vertices in a signed graph G is called its signed degree set. In this paper, we prove that every non-empty set of positive (negative) integers is the signed degree set of some connected signed graph and determine the smallest possible order for such a signed graph. We also prove that every non-empty set of integers is the signed degree set of some connected signed graph

On signed diagonal flip sequences

Eliahou \cite{2} and Kryuchkov \cite{9} conjectured a proposition that Gravier and Payan \cite{4} proved to be equivalent to the Four Color Theorem. It states that any triangulation of a polygon can be transformed into another triangulation of the same polygon by a sequence of signed diagonal flips. It is well known that any pair of polygonal triangulations are connected by a sequence of (non-signed) diagonal flips. In this paper we give a sufficient and necessary condition for a diagonal flip sequence to be a signed diagonal flip sequence.Comment: 11 pages, 24 figures, to appear in European Journal of Combinatoric

A Reciprocity Theorem for Monomer-Dimer Coverings

The problem of counting monomer-dimer coverings of a lattice is a longstanding problem in statistical mechanics. It has only been exactly solved for the special case of dimer coverings in two dimensions. In earlier work, Stanley proved a reciprocity principle governing the number $N(m,n)$ of dimer coverings of an $m$ by $n$ rectangular grid (also known as perfect matchings), where $m$ is fixed and $n$ is allowed to vary. As reinterpreted by Propp, Stanley's result concerns the unique way of extending $N(m,n)$ to $n < 0$ so that the resulting bi-infinite sequence, $N(m,n)$ for $n \in {Z}$, satisfies a linear recurrence relation with constant coefficients. In particular, Stanley shows that $N(m,n)$ is always an integer satisfying the relation $N(m,-2-n) = \epsilon_{m,n}N(m,n)$ where $\epsilon_{m,n} = 1$ unless $m\equiv$ 2(mod 4) and $n$ is odd, in which case $\epsilon_{m,n} = -1$. Furthermore, Propp's method is applicable to higher-dimensional cases. This paper discusses similar investigations of the numbers $M(m,n)$, of monomer-dimer coverings, or equivalently (not necessarily perfect) matchings of an $m$ by $n$ rectangular grid. We show that for each fixed $m$ there is a unique way of extending $M(m,n)$ to $n < 0$ so that the resulting bi-infinite sequence, $M(m,n)$ for $n \in {Z}$, satisfies a linear recurrence relation with constant coefficients. We show that $M(m,n)$, a priori a rational number, is always an integer, using a generalization of the combinatorial model offered by Propp. Lastly, we give a new statement of reciprocity in terms of multivariate generating functions from which Stanley's result follows.Comment: 13 pages, 12 figures, to appear in the proceedings of the Discrete Models for Complex Systems (DMCS) 2003 conference. (v2 - some minor changes