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.

A planarizing set of a graph is a set of edges or vertices whose removal leaves a planar graph. It is shown that, if G is an n-vertex graph of maximum degree d and orientable genus g, then there exists a planarizing set of O( p dgn) edges. This result is tight within a constant factor. Similar results are obtained for planarizing vertex sets and for graphs embedded on nonorientable surfaces. Planarizing edge and vertex sets can be found in O(n + g) time, if an embedding of G on a surface of genus g is given. We also construct an approximation algorithm that finds an O( p gn log g) planarizing vertex set of G in O(n log g) time if no genus-g embedding is given as an input. 1 Introduction A graph G is planar if G can be drawn in the plane so that no two edges intersect. Planar graphs arise naturally in many applications of graph theory, e.g. in VLSI and circuit design, in network design and analysis, in computer graphics, and is one of the most intensively studied class of graphs [2...

All rights reserved. This dissertation may not be reproduced in whole or in part, by mimeograph or other means, without the permission of the author. Supervisor: M. R. Fellows A substantial part of the history of graph theory deals with the study and classi-cation of sets of graphs that share common properties. One predominant trend is to characterize graph families by sets of minimal forbidden graphs (within some partial ordering on the graphs). For example, the famous Kuratowski Theorem classi es the planar graph family by two forbidden graphs (in the topological partial order). Most, if not all, of the current approaches for nding these forbidden substructure characterizations use extensive and specialized case analysis. Thus, until now, for a xed graph family,thistype of mathematical theorem proving often required months or even years of human e ort. The main focus of this dissertation is to develop a practical theory for automating (with distributed computer programming) this clas-sic part of graph theory. We extend and (more importantly) implement avariation

Abstract: We study the growth rate on the number obstructions (forbidden minors) for families of graphs that are based on parameterized graph problems. Our main result shows that if the de ning graph problem is NP-complete then the growth rate on the number of obstructions must be super-polynomial or else the polynomial-time P hierarchy must collapse to 3.We illustrate the rapid growth rate of parameterized lower ideals by computing (and counting) the obstructions for the graph families with independence plus size at most k, k 12. Key Words: graph minors, obstruction sets, polynomial hierarchy Category: F.4, F.m, G.2

In 1890 Heawood [10] established an upper bound for the chromatic number of a graph embedded on a surface of Euler genus g 1.

We derive an O(n2)-time algorithm for calculating the sequence of numbers g0(G), g1(G), g2(G),... of distinct ways to embed a 3-regular Halin graph G on the respective orientable surfaces S0, S1, S2,.... Key topological features are a quadrangular decomposition of plane Halin graphs and a new recombinant-strands reassembly process that fits pieces together three-at-a-vertex. Key algorithmic features are reassembly along a post-order traversal, with just-in-time dynamic assignment of roots for quadrangular pieces encountered along the tour. 1.

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

Abstract. This paper investigates properties of the zero-divisor graph of a commutative ring and its genus. In particular, we determine all isomorphism classes of finite commutative rings with identity whose zero-divisor graph has genus 1. 1.

This is a survey of studies on topological graph theory developed by Japanese people in the recent two decades and presents a big bibliography including almost all papers written by Japanese topological graph theorists.

We obtain asymptotic formulas for the number of rooted 2-connected and 3-connected surface maps on an orientable surface of genus g with respect to vertices and edges simultaneously. We also derive the bivariate version of the large facewidth result for random 3-connected maps. These results are then used to derive asymptotic formulas for the number of labelled k-connected graphs of orientable genus g for k ≤ 3. 1