Topological Theory of Graphs / / Yanpei Liu.

• Superior document: Title is part of eBook package: De Gruyter DG Plus eBook-Package 2017
• VerfasserIn: Liu, Yanpei,
• TeilnehmendeR: University of Science & Technology,.
• Place / Publishing House: Berlin ;, Boston : : De Gruyter, , [2017]; ©2017
• Year of Publication: 2017
• Language: English
• Physical Description: 1 online resource (XII, 357 p.)

This book introduces polyhedra as a tool for graph theory and discusses their properties and applications in solving the Gauss crossing problem. The discussion is extended to embeddings on manifolds, particularly to surfaces of genus zero and non-zero via the joint tree model, along with solution algorithms. Given its rigorous approach, this book would be of interest to researchers in graph theory and discrete mathematics.

This book presents a topological approach to combinatorial configurations, in particular graphs, by introducing a new pair of homology and cohomology via polyhedra. On this basis, a number of problems are solved using a new approach, such as the embeddability of a graph on a surface (orientable and nonorientable) with given genus, the Gauss crossing conjecture, the graphicness and cographicness of a matroid, and so forth. Notably, the specific case of embeddability on a surface of genus zero leads to a number of corollaries, including the theorems of Lefschetz (on double coverings), of MacLane (on cycle bases), and of Whitney (on duality) for planarity. Relevant problems include the Jordan axiom in polyhedral forms, efficient methods for extremality and for recognizing a variety of embeddings (including rectilinear layouts in VLSI), and pan-polynomials, including those of Jones, Kauffman (on knots), and Tutte (on graphs), among others. Contents Preliminaries Polyhedra Surfaces Homology on Polyhedra Polyhedra on the Sphere Automorphisms of a Polyhedron Gauss Crossing Sequences Cohomology on Graphs Embeddability on Surfaces Embeddings on Sphere Orthogonality on Surfaces Net Embeddings Extremality on Surfaces Matroidal Graphicness Knot Polynomials

Table of Contents

Frontmatter -- Preface to DG Edition -- Preface to USTC Edition -- Contents -- 1. Preliminaries -- 2. Polyhedra -- 3. Surfaces -- 4. Homology on Polyhedra -- 5. Polyhedra on the Sphere -- 6. Automorphisms of a Polyhedron -- 7. Gauss Crossing Sequences -- 8. Cohomology on Graphs -- 9. Embeddability on Surfaces -- 10. Embeddings on Sphere -- 11. Orthogonality on Surfaces -- 12. Net Embeddings -- 13. Extremality on Surfaces -- 14. Matroidal Graphicness -- 15. Knot Polynomials -- Bibliography -- Subject Index -- Author Index