Graphs on Surfaces

Graphs on Surfaces provides an accessible and fairly comprehensive treatment of recent developments on generalized duality for graphs on surfaces, and its applications. The authors illustrate the interdependency between duality, medial graphs and knots; how this interdependency is reflected in algebraic invariants of graphs and knots; and how this interdependency can be exploited to solve problems in graph theory and knot theory. Taking a constructive approach, the authors emphasize how the ideas and constructions arise from considering classical constructions such as geometric duals and Tait graphs, by removing artificial restrictions in these constructions, by localizing global operations, or by broadening the setting to embedded graphs. The authors include a description on how these adaptations may be accomplished and what the benefits of doing so are.§The authors present a self-contained introduction to graphs on surfaces, twisted duality, and topological graph and knot polynomials that is accessible both to graph theorists and to knot theorists. Aimed at those with basic graph theory and knot theory this book is accessible to graduate students and researchers in either area. Because the area is advancing so rapidly, the authors give a comprehensive overview of the field while including a robust bibliography, hoping to provide the reader with the necessary knowledge and background to read research papers on these topics as they appear. The reader will come away from the text convinced of advantages of considering these higher genus analogues of constructions of plane and abstract graphs, and with a good understanding of how they arise.