Minor-Obstructions for Apex-Pseudoforests

A graph is called a pseudoforest if none of its connected components contains more than one cycle. A graph is an apex-pseudoforest if it can become a pseudoforest by removing one of its vertices. We identify 33 graphs that form the minor-obstruction set of the class of apex-pseudoforests, i.e., the set of all minor-minimal graphs that are not apex-pseudoforests

The PC-Tree algorithm, Kuratowski subdivisions, and the torus.

The PC-Tree algorithm of Shih and Hsu (1999) is a practical linear-time planarity algorithm that provides a plane embedding of the given graph if it is planar and a Kuratowski subdivision otherwise. Remarkably, there is no known linear-time algorithm for embedding graphs on the torus. We extend the PC-Tree algorithm to a practical, linear-time toroidality test for K3;3-free graphs called the PCK-Tree algorithm. We also prove that it is NP-complete to decide whether the edges of a graph can be covered with two Kuratowski subdivisions. This greatly reduces the possibility of a polynomial-time toroidality testing algorithm based solely on edge-coverings by subdivisions of Kuratowski subgraphs

Classification of finite groups with toroidal or projective-planar permutability graphs

Let $G$ be a group. The permutability graph of subgroups of $G$, denoted by $\Gamma(G)$, is a graph having all the proper subgroups of $G$ as its vertices, and two subgroups are adjacent in $\Gamma(G)$ if and only if they permute. In this paper, we classify the finite groups whose permutability graphs are toroidal or projective-planar. In addition, we classify the finite groups whose permutability graph does not contain one of $K_{3,3}$, $K_{1,5}$, $C_6$, $P_5$, or $P_6$ as a subgraph.Comment: 30 pages, 8 figure

On the purity of minor-closed classes of graphs

Given a graph H with at least one edge, let gapH(n) denote the maximum difference between the numbers of edges in two n-vertex edge-maximal graphs with no minor H. We show that for exactly four connected graphs H (with at least two vertices), the class of graphs with no minor H is pure, that is, gapH(n) = 0 for all n ≥ 1; and for each connected graph H (with at least two vertices) we have the dichotomy that either gapH(n) = O(1) or gapH(n) = ⊝(n). Further, if H is 2-connected and does not yield a pure class, then there is a constant c > 0 such that gapH(n) ∼ cn. We also give some partial results when H is not connected or when there are two or more excluded minors

Local Structure for Vertex-Minors

This thesis is about a conjecture of Geelen on the structure of graphs with a forbidden vertex-minor; the conjecture is like the Graph Minors Structure Theorem of Robertson and Seymour but for vertex-minors instead of minors. We take a step towards proving the conjecture by determining the "local structure''. Our first main theorem is a grid theorem for vertex-minors, and our second main theorem is more like the Flat Wall Theorem of Robertson and Seymour. We believe that the results presented in this thesis provide a path towards proving the full conjecture. To make this area more accessible, we have organized the first chapter as a survey on "structure for vertex-minors''

Gonality of metric graphs and Catalan-many tropical morphisms to trees

This thesis consists of two points of view to regard degree-(g′+1) tropical morphisms Φ : (Γ,w) → Δ from a genus-(2g′) weighted metric graph (Γ,w) to a metric tree Δ, where g′ is a positive integer. The first point of view, developed in Part I, is purely combinatorial and constructive. It culminates with an application to bound the gonality of (Γ,w). The second point of view, developed in Part II, incorporates category theory to construct a unified framework under which both Φ and higher dimensional analogues can be understood. These higher dimensional analogues appear in the construction of a moduli space Gtrop/g→0,d parametrizing the tropical morphisms Φ, and a moduli spaceMtrop/g parametrizing the (Γ,w). There is a natural projection map Π : Gtrop/g→0,d →Mtrop/g that sends Φ : (Γ,w) → Δ to (Γ,w). The strikingly beautiful result is that when g = 2g′ and d = g′+1, the projection Π itself is an indexed branched cover, thus having the same nature as the maps Φ that are being parametrized. Moreover, fibres of Π have Catalan-many points. Each part has its own introduction that motivates and describes the problem from its own perspective. Part I and its introduction are based on two articles which are joint work with Jan Draisma. Part II contains material intended to be published as two articles. There is also a layman summary available at the beginning

Proceedings of the 3rd International Workshop on Optimal Networks Topologies IWONT 2010

Peer Reviewe

List Coloring Some Classes of 1-Planar Graphs

In list coloring we are given a graph G and a list assignment for G which assigns to each vertex of G a list of possible colors. We wish to find a coloring of the vertices of G such that each vertex uses a color from its list and adjacent vertices are given different colors. In this thesis we study the problem of list coloring 1-planar graphs, i.e., graphs that can be drawn in the plane such that any edge intersects at most one other edge. We also study the closely related problem of simultaneously list coloring the vertices and faces of a planar graph, known as coupled list coloring. We show that 1-planar bipartite graphs are list colorable whenever all lists are of size at least four, and further show that this coloring can be found in linear time. In pursuit of this result, we show that the previously known edge partition of a 1-planar graph into a planar graph and a forest can be found in linear time. A wheel graph consists of a cycle of vertices, all of which are adjacent to an additional center vertex. We show that wheel graphs are coupled list colorable when all lists are of size five or more and show that this coloring can be found in linear time. Possible extensions of this result to planar partial 3-trees are discussed. Finally, we discuss the complexity of list coloring 1-planar graphs, both in parameterized and unparameterized settings