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)

Counting Power Domination Sets in Complete m-ary Trees

Motivated by the question of computing the probability of successful power domination by placing k monitors uniformly at random, in this paper we give a recursive formula to count the number of power domination sets of size k in a labeled complete m-ary tree. As a corollary we show that the desired probability can be computed in exponential with linear exponent time

Analogies between the crossing number and the tangle crossing number

Tanglegrams are special graphs that consist of a pair of rooted binary trees with the same number of leaves, and a perfect matching between the two leaf-sets. These objects are of use in phylogenetics and are represented with straightline drawings where the leaves of the two plane binary trees are on two parallel lines and only the matching edges can cross. The tangle crossing number of a tanglegram is the minimum crossing number over all such drawings and is related to biologically relevant quantities, such as the number of times a parasite switched hosts. Our main results for tanglegrams which parallel known theorems for crossing numbers are as follows. The removal of a single matching edge in a tanglegram with $n$ leaves decreases the tangle crossing number by at most $n-3$, and this is sharp. Additionally, if $\gamma(n)$ is the maximum tangle crossing number of a tanglegram with $n$ leaves, we prove $\frac{1}{2}\binom{n}{2}(1-o(1))\le\gamma(n)<\frac{1}{2}\binom{n}{2}$. Further, we provide an algorithm for computing non-trivial lower bounds on the tangle crossing number in $O(n^4)$ time. This lower bound may be tight, even for tanglegrams with tangle crossing number $\Theta(n^2)$.Comment: 13 pages, 6 figure

Counting Power Domination Sets in Complete m-ary Trees

Motivated by the question of computing the probability of successful power domination by placing k monitors uniformly at random, in this paper we give a recursive formula to count the number of power domination sets of size k in a labeled complete m-ary tree. As a corollary we show that the desired probability can be computed in exponential with linear exponent time

A positivity phenomenon in Elser\u27s Gaussian-cluster percolation model

Veit Elser proposed a random graph model for percolation in which physical dimension appears as a parameter. Studying this model combinatorially leads naturally to the consideration of numerical graph invariants which we call Elser numbers elsk(G), where G is a connected graph and k a nonnegative integer. Elser had proven that els1(G)=0 for all G. By interpreting the Elser numbers as reduced Euler characteristics of appropriate simplicial complexes called nucleus complexes, we prove that for all graphs G, they are nonpositive when k=0 and nonnegative for k⩾2. The last result confirms a conjecture of Elser. Furthermore, we give necessary and sufficient conditions, in terms of the 2-connected structure of G, for the nonvanishing of the Elser numbers

A positivity phenomenon in Elser's Gaussian-cluster percolation model

Veit Elser proposed a random graph model for percolation in which physical dimension appears as a parameter. Studying this model combinatorially leads naturally to the consideration of numerical graph invariants which we call Elser numbers els_k(G), where G is a connected graph and k a nonnegative integer. Elser had proven that els_1(G) = 0 for all G. By interpreting the Elser numbers as reduced Euler characteristics of appropriate simplicial complexes called nucleus complexes, we prove that for all graphs G, they are nonpositive when k = 0 and nonnegative for k ≥ 2. The last result confirms a conjecture of Elser. Furthermore, we give necessary and sufficient conditions, in terms of the 2-connected structure of G, for the nonvanishing of the Elser numbers