Weihrauch Reducibility and Finite-Dimensional Subspaces

In this thesis we study several principles involving subspaces and decompositions of vector spaces, matroids, and graphs from the perspective of Weihrauch reducibility. We study the problem of decomposing a countable vector space or countable matroid into 1-dimensional subspaces. We also study the problem of producing a finite-dimensional or 1-dimensional subspace of a countable vector space, and related problems for producing finite-dimensional subspaces of a countable matroid. This extends work in the reverse mathematics setting by Downey, Hirschfeldt, Kach, Lempp, Mileti, and Montalb´an (2007) and recent work of Hirst and Mummert (2017). Finally, we study the problem of producing a nonempty subset of a countable graph that is equal to a finite union of connected components and the problem of producing a nonempty subset of a countable graph that is equal to a union of connected components that omits at least one connected component. This extends work of Gura, Hirst, and Mummert (2015). We briefly investigate some of these problems in the reverse mathematics setting

Equivalences on Acyclic Orientations

The cyclic and dihedral groups can be made to act on the set Acyc(Y) of acyclic orientations of an undirected graph Y, and this gives rise to the equivalence relations ~kappa and ~delta, respectively. These two actions and their corresponding equivalence classes are closely related to combinatorial problems arising in the context of Coxeter groups, sequential dynamical systems, the chip-firing game, and representations of quivers. In this paper we construct the graphs C(Y) and D(Y) with vertex sets Acyc(Y) and whose connected components encode the equivalence classes. The number of connected components of these graphs are denoted kappa(Y) and delta(Y), respectively. We characterize the structure of C(Y) and D(Y), show how delta(Y) can be derived from kappa(Y), and give enumeration results for kappa(Y). Moreover, we show how to associate a poset structure to each kappa-equivalence class, and we characterize these posets. This allows us to create a bijection from Acyc(Y)/~kappa to the union of Acyc(Y')/~kappa and Acyc(Y'')/~kappa, Y' and Y'' denote edge deletion and edge contraction for a cycle-edge in Y, respectively, which in turn shows that kappa(Y) may be obtained by an evaluation of the Tutte polynomial at (1,0).Comment: The original paper was extended, reorganized, and split into two papers (see also arXiv:0802.4412

Orderly Spanning Trees with Applications

We introduce and study the {\em orderly spanning trees} of plane graphs. This algorithmic tool generalizes {\em canonical orderings}, which exist only for triconnected plane graphs. Although not every plane graph admits an orderly spanning tree, we provide an algorithm to compute an {\em orderly pair} for any connected planar graph $G$, consisting of a plane graph $H$ of $G$, and an orderly spanning tree of $H$. We also present several applications of orderly spanning trees: (1) a new constructive proof for Schnyder's Realizer Theorem, (2) the first area-optimal 2-visibility drawing of $G$, and (3) the best known encodings of $G$ with O(1)-time query support. All algorithms in this paper run in linear time.Comment: 25 pages, 7 figures, A preliminary version appeared in Proceedings of the 12th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2001), Washington D.C., USA, January 7-9, 2001, pp. 506-51

Binomial edge ideals and rational normal scrolls

Let $X$ be the Hankel matrix of size $2\times n$ and let $G$ be a closed graph on the vertex set $[n].$ We study the binomial ideal $I_G\subset K[x_1,\ldots,x_{n+1}]$ which is generated by all the $2$-minors of $X$ which correspond to the edges of $G.$ We show that $I_G$ is Cohen-Macaulay. We find the minimal primes of $I_G$ and show that $I_G$ is a set theoretical complete intersection. Moreover, a sharp upper bound for the regularity of $I_G$ is given

Genus Ranges of 4-Regular Rigid Vertex Graphs

We introduce a notion of genus range as a set of values of genera over all surfaces into which a graph is embedded cellularly, and we study the genus ranges of a special family of four-regular graphs with rigid vertices that has been used in modeling homologous DNA recombination. We show that the genus ranges are sets of consecutive integers. For any positive integer $n$, there are graphs with $2n$ vertices that have genus range ${m,m+1,...,m'}$ for all $0\le m<m'\le n$, and there are graphs with $2n-1$ vertices with genus range ${m,m+1,...,m'}$ for all $0\le m<m' <n$ or $0<m<m'\le n$. Further, we show that for every $n$ there is $k<n$ such that ${h}$ is a genus range for graphs with $2n-1$ and $2n$ vertices for all $h\le k$. It is also shown that for every $n$, there is a graph with $2n$ vertices with genus range ${0,1,...,n}$, but there is no such a graph with $2n-1$ vertices