Geometric Spanning Cycles in Bichromatic Point Sets

Given a set of points in the plane each colored either red or blue, we find non-self-intersecting geometric spanning cycles of the red points and of the blue points such that each edge of the red spanning cycle is crossed at most three times by the blue spanning cycle and vice-versa

Simple DFS on the Complement of a Graph and on Partially Complemented Digraphs

A complementation operation on a vertex of a digraph changes all outgoing arcs into non-arcs, and outgoing non-arcs into arcs. A partially complemented digraph $\widetilde{G}$ is a digraph obtained from a sequence of vertex complement operations on $G$. Dahlhaus et al. showed that, given an adjacency-list representation of $\widetilde{G}$, depth-first search (DFS) on $G$ can be performed in $O(n + \widetilde{m})$ time, where $n$ is the number of vertices and $\widetilde{m}$ is the number of edges in $\widetilde{G}$. To achieve this bound, their algorithm makes use of a somewhat complicated stack-like data structure to simulate the recursion stack, instead of implementing it directly as a recursive algorithm. We give a recursive $O(n+\widetilde{m})$ algorithm that uses no complicated data-structures

Connectivity, tree-decompositions and unavoidable-minors

The results in this thesis are steps toward bridging the gap between the handful of exact structure theorems known for minor-closed classes of graphs, and the very general, yet wildly qualitative, Graph Minors Structure Theorem. This thesis introduces a refinement of the notion of tree-width. Tree-width is a measure of how “tree-like” a graph is. Essentially, a graph is tree-like if it can be decomposed across a collection of non-crossing vertex-separations into small pieces. In our variant, which we call k-tree-width, we require that the vertex-separations each have order at most k. Tree-width and branch-width are related parameters in a graph, and we introduce a branch-width-like variant for k-tree-width. We find a dual notion, in terms of tangles, for our branch-width parameter, and we prove a generalization of Robertson and Seymour’s Grid Theorem

Intestinal Dendritic Cells in Health and Gut Inflammation

Broad Medical Research Programme, Crohn’s in Childhood Research Association and a MRC Doctoral Training Award

An implicit representation of chordal comparability graphs in linear time

Abstract. Ma and Spinrad have shown that every transitive orientation of a chordal comparability graph is the intersection of four linear orders. That is, chordal comparability graphs are comparability graphs of posets of dimension four. Among other uses, this gives an implicit representation of a chordal comparability graph using O(n) integers so that, given two vertices, it can be determined in O(1) time whether they are adjacent, no matter how dense the graph is. We give a linear-time algorithm for finding the four linear orders, improving on their bound of O(n 2).