Connectivity for Matroids and Graphs.

This dissertation studies connectivity for matroids and graphs. The main results generalize Tutte\u27s Wheels and Whirls Theorem and have numerous applications. In Chapter 2, we prove two structural theorems for 3-connected matroids. An element e of a 3-connected matroid M is essential if neither the deletion M$\\$e nor the contraction M/e is 3-connected. Tutte\u27s Wheels and Whirls Theorem proves that the only 3-connected matroids in which every element is essential are the wheels and whirls. If M is not a wheel or a whirl, we prove that the essential elements of M can be partitioned into classes where two elements are in the same class if M has a fan containing both. In particular, M must have at least two non-essential elements. In the second structural theorem, we show that if M has a fan with 2k or 2k + 1 elements for some $k \geq \ 2$, then M can be obtained by sticking together a (k + 1)-spoked wheel and a certain 3-connected minor of M. In Chapters 3 and 4, we characterize all 3-connected matroids whose set of non-essential elements has rank two. In particular, we completely determine all 3-connected matroids with exactly two non-essential elements. In Chapter 5, we derive some consequences of these results for the 3-connected binary matroids and graphs. We prove that there are exactly six classes of 3-connected binary matroids whose set of non-essential elements has rank two and we prove that there are exactly two classes of graphs, multi-dimensional wheels and twisted wheels, with exactly two non-essential edges. In Chapter 6, we use our first structural theorem to investigate the set of elements e in a 3-connected matroid M such that the simplification of M/e is 3-connected. We get best-possible lower bounds on the number of such elements thereby improving a result which was derived by Cunningham and Seymour independently. We also give some generalizations of the Wheels and Whirls Theorem and the Wheels Theorem

Roots in 3-manifold topology

Let C be some class of objects equipped with a set of simplifying moves. When we apply these to a given object M in C as long as possible, we get a root of M. Our main result is that under certain conditions the root of any object exists and is unique. We apply this result to different situations and get several new results and new proofs of known results. Among them there are a new proof of the Kneser-Milnor prime decomposition theorem for 3-manifolds and different versions of this theorem for cobordisms, knotted graphs, and orbifolds.Comment: This is the version published by Geometry & Topology Monographs on 29 April 200

Contraction Obstructions for Connected Graph Searching

We consider the connected variant of the classic mixed search game where, in each search step, cleaned edges form a connected subgraph. We consider graph classes with bounded connected (and monotone) mixed search number and we deal with the question whether the obstruction set, with respect of the contraction partial ordering, for those classes is finite. In general, there is no guarantee that those sets are finite, as graphs are not well quasi ordered under the contraction partial ordering relation. In this paper we provide the obstruction set for $k=2$, where $k$ is the number of searchers we are allowed to use. This set is finite, it consists of 177 graphs and completely characterises the graphs with connected (and monotone) mixed search number at most 2. Our proof reveals that the "sense of direction" of an optimal search searching is important for connected search which is in contrast to the unconnected original case. We also give a double exponential lower bound on the size of the obstruction set for the classes where this set is finite