For an arbitrary fixed surface S, a linear time algorithm is presented that for a given graph G either finds an embedding of G in S or identifies a subgraph of G that is homeomorphic to a minimal forbidden subgraph for embeddability in S. A side result of the proof of the algorithm is that minimal forbidden subgraphs for embeddability in S cannot be arbitrarily large. This yields a constructive proof of the result of Robertson and Seymour that for each closed surface there are only finitely many minimal forbidden subgraphs. The results and methods of this paper can be used to solve more general embedding extension problems.

This chapter gives a general overview of two emerging techniques for discrete optimization that have footholds in mathematics, computer science, and operations research: branch decompositions and tree decompositions. Branch decompositions and tree decompositions along with their respective connectivity invariants, branchwidth and treewidth, were first introduced to aid in proving the Graph Minors Theorem, a wellknown conjecture (Wagner’s conjecture) in graph theory. The algorithmic importance of branch decompositions and tree decompositions for solving NP-hard problems modelled on graphs was first realized by computer scientists in relation to formulating graph problems in monadic second order logic. The dynamic programming techniques utilizing branch decompositions and tree decompositions, called branch decomposition and tree decomposition based algorithms, fall into a class of algorithms known as fixed-parameter tractable algorithms and have been shown to be effective in a practical setting for NP-hard problems such as minimum domination, the travelling salesman problem, general minor containment, and frequency assignment problems.

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A monumental project in graph theory was recently completed. The project, started by Robertson and Seymour, and later joined by Thomas, led to entirely new concepts and a new way of looking at graph theory. The motivating problem was Kuratowski’s characterization of planar graphs, and a far-reaching generalization of this, conjectured by Wagner: If a class of graphs is minor-closed (i.e., it is closed under deleting and contracting edges), then it can be characterized by a finite number of excluded minors. The proof of this conjecture is based on a very general theorem about the structure of large graphs: If a minor-closed class of graphs does not contain all graphs, then every graph in it is glued together in a tree-like fashion from graphs that can almost be embedded in a fixed surface. We describe the precise formulation of the main results and survey some of its applications to algorithmic and structural problems in graph theory.

Graph minors and the theory of graphs embedded in surfaces are

Let K be an induced non-separating subgraph of a graph G, andletB be the bridge of K in G. Obstructions for extending a given 2-cell embedding of K to an embedding of G in the same surface are considered. It is shown that it is possible to find a nice obstruction which means that it has bounded branch size up to a bounded number of “almost disjoint ” millipedes. Moreover, B contains a nice subgraph ˜ B with the following properties. If K is 2-cell embedded in some surface and F is a face of K, then ˜ B admits exactly the same types of embeddings in F as B. A linear time algorithm to construct such a universal obstruction ˜ B is presented. At the same time, for every type of embeddings of ˜ B, an embedding of B ofthesametypeis determined. 1

In the core of the seminal Graph Minor Theory of Robertson and Seymour lies a powerful theorem capturing the “rough ” structure of graphs excluding a fixed minor. This result was used to prove Wagner’s Conjecture that finite graphs are well-quasi-ordered under the graph minor relation. Recently, a number of beautiful results that use this structural result have appeared. Some of these along with some other recent advances on graph minors are surveyed. Keywords: Graph minor theory, Tree-width, Tree-decomposition, Path-decomposition, Complete

A projective plane is equivalent to a disk with antipodal points identified. A graph is projective planar if it can be drawn on the projective plane with no crossing edges. A linear time algorithm for projective planar embedding has been described by Mohar. We provide a new approach that takes O(n 2 ) time but is much easier to implement. We programmed a variant of this algorithm and used it to computationally verify the known list of all the projective plane obstructions. Key words: graph algorithms, surface embedding, graph embedding, projective plane, forbidden minor, obstruction 1 Background A graph G consists of a set V of vertices and a set E of edges, each of which is associated with an unordered pair of vertices from V . Throughout this paper, n denotes the number of vertices of a graph, and m is the number of edges. A graph is embeddable on a surface M if it can be drawn on M without crossing edges. Archdeacon's survey [2] provides an excellent introduction to topologica...

The spindle surface S is the pinched surface formed by identifying two points on the sphere. In this paper we examine cubic graphs that minimally do not embed on the spindle surface. We give the complete list of 21 cubic graphs that form the topological obstruction set in the cubic order for graphs that embed on S.

this paper we give a survey of the topics and results in topological graph theory. We offer neither breadth, as there are numerous areas left unexamined, nor depth, as no area is completely explored. Nevertheless, we do offer some of the favorite topics of the author and attempt to place them 1