Minimal Connectivity

A k-connected graph such that deleting any edge / deleting any vertex / contracting any edge results in a graph which is not k-connected is called minimally / critically / contraction-critically k-connected. These three classes play a prominent role in graph connectivity theory, and we give a brief introduction with a light emphasis on reduction- and construction theorems for classes of k-connected graphs.Comment: IMADA-preprint-math, 33 page

A successful concept for measuring non-planarity of graphs: the crossing number

AbstractThis paper surveys how the concept of crossing number, which used to be familiar only to a limited group of specialists, emerges as a significant graph parameter. This paper has dual purposes: first, it reviews foundational, historical, and philosophical issues of crossing numbers, second, it shows a new lower bound for crossing numbers. This new lower bound may be helpful in estimating crossing numbers

Constant mean curvature surfaces

In this article we survey recent developments in the theory of constant mean curvature surfaces in homogeneous 3-manifolds, as well as some related aspects on existence and descriptive results for $H$-laminations and CMC foliations of Riemannian $n$-manifolds.Comment: 102 pages, 17 figure

Bridging the Gap Between Tree and Connectivity Augmentation: Unified and Stronger Approaches

We consider the Connectivity Augmentation Problem (CAP), a classical problem in the area of Survivable Network Design. It is about increasing the edge-connectivity of a graph by one unit in the cheapest possible way. More precisely, given a $k$-edge-connected graph $G=(V,E)$ and a set of extra edges, the task is to find a minimum cardinality subset of extra edges whose addition to $G$ makes the graph $(k+1)$-edge-connected. If $k$ is odd, the problem is known to reduce to the Tree Augmentation Problem (TAP) -- i.e., $G$ is a spanning tree -- for which significant progress has been achieved recently, leading to approximation factors below $1.5$ (the currently best factor is $1.458$). However, advances on TAP did not carry over to CAP so far. Indeed, only very recently, Byrka, Grandoni, and Ameli (STOC 2020) managed to obtain the first approximation factor below $2$ for CAP by presenting a $1.91$-approximation algorithm based on a method that is disjoint from recent advances for TAP. We first bridge the gap between TAP and CAP, by presenting techniques that allow for leveraging insights and methods from TAP to approach CAP. We then introduce a new way to get approximation factors below $1.5$, based on a new analysis technique. Through these ingredients, we obtain a $1.393$-approximation algorithm for CAP, and therefore also TAP. This leads to the currently best approximation result for both problems in a unified way, by significantly improving on the above-mentioned $1.91$-approximation for CAP and also the previously best approximation factor of $1.458$ for TAP by Grandoni, Kalaitzis, and Zenklusen (STOC 2018). Additionally, a feature we inherit from recent TAP advances is that our approach can deal with the weighted setting when the ratio between the largest to smallest cost on extra links is bounded, in which case we obtain approximation factors below $1.5$

Aspects of matroid connectivity and uniformity.

In approaching a combinatorial problem, it is often desirable to be armed with a notion asserting that some objects are more highly structured than others. In particular, focusing on highly structured objects may avoid certain degeneracies and allow for the core of the problem to be addressed. In matroid theory, the principle notion fulfilling this role of “structure” is that of connectivity. This thesis proves a number of results furthering the knowledge of matroid connectivity and also introduces a new structural notion, that of generalised uniformity. The first part of this thesis considers 3-connected matroids and the presence of elements which may be deleted or contracted without the introduction of any non-minimal 2-separations. Principally, a Wheels-and-Whirls Theorem and then a Splitter Theorem is established, guaranteeing the existence of such elements, provided certain well-behaved structures are not present. The second part of this thesis generalises the notion of a uniform matroid by way of a 2-parameter property capturing “how uniform” a given matroid is. Initially, attention is focused on matroids representable over some field. In particular, a finiteness result is established and a specific class of binary matroids is completely determined. The concept of generalised uniformity is then considered more broadly by an analysis of its relevance to a number of established matroid notions and settings. Within that analysis, a number of equivalent characterisations of generalised uniformity are obtained. Lastly, the third part of the thesis considers a highly structured class of matroids whose members are defined by the nature of their circuits. A characterisation is achieved for the regular members of this class and, in general, the infinitely many excluded series minors are determined