---

This book studies algebraic representations of graphs in order to investigate combinatorial structures via local symmetries. Topological, combinatorial and algebraic classifications are distinguished by invariants of polynomial type and algorithms are designed to determine all such classifications with complexity analysis. Being a summary of the author´s original work on graph embeddings, this book is an essential reference for researchers in graph theory. ContentsAbstract GraphsAbstract MapsDualityOrientabilityOrientable MapsNonorientable MapsIsomorphisms of MapsAsymmetrizationAsymmetrized Petal BundlesAsymmetrized MapsMaps within SymmetryGenus PolynomialsCensus with PartitionsEquations with PartitionsUpper Maps of a GraphGenera of a GraphIsogemial GraphsSurface Embeddability

---

Table of Content:PrefaceChapter 1 Abstract Graphs1.1 Graphs and Networks1.2 Surfaces1.3 Embeddings1.4 Abstract Representation1.5 NotesChapter 2 Abstract Maps2.1 Ground Sets2.2 Basic Permutations2.3 Conjugate Axiom2.4 Transitive Axiom2.5 Included Angles2.6 NotesChapter 3 Duality3.1 Dual Maps3.2 Deletion of an Edge3.3 Addition of an Edge3.4 Basic Transformation3.5 NotesChapter 4 Orientability4.1 Orientation4.2 Basic Equivalence4.3 Euler Characteristic4.4 Pattern Examples4.5 NotesChapter 5 Orientable Maps5.1 Butterflies5.2 Simplified Butterflies5.3 Reduced Rules5.4 Orientable Principles5.5 Orientable Genus5.6 NotesChapter 6 Nonorientable Maps6.1 Barflies6.2 Simplified Barflies6.3 Nonorientable Rules6.4 Nonorientable Principles6.5 Nonorientable Genus6.6 NotesChapter 7 Isomorphisms of Maps7.1 Commutativity7.2 Isomorphism Theorem7.3 Recognition7.4 Justification7.5 Pattern Examples7.6 NotesChapter 8 Asymmetrization8.1 Automorphisms8.2 Upper Bounds of Group Order8.3 Determination of the Group8.4 Rootings8.5 NotesChapter 9 Asymmetrized Petal Bundles9.1 Orientable Petal Bundles9.2 Planar Pedal Bundles9.3 Nonorientable Pedal Bundles9.4 The Number of Pedal Bundles9.5 NotesChapter 10 Asymmetrized Maps10.1 Orientable Equation10.2 Planar Rooted Maps10.3 Nonorientable Equation10.4 Gross Equation10.5 The Number of Rooted Maps10.6 NotesChapter 11 Maps Within Symmetry11.1 Symmetric Relation11.2 An Application11.3 Symmetric Principle11.4 General Examples11.5 NotesChapter 12 Genus Polynomials12.1 Associate Surfaces12.2 Layer Division of a Surface12.3 Handle Polynomials12.4 Crosscap Polynomials12.5 NotesChapter 13 Census with Partitions13.1 Planted Trees13.2 Hamiltonian Cubic Maps13.3 Halin Maps13.4 Biboundary Inner Rooted Maps13.5 General Maps13.6 Pan-Flowers13.7 NotesChapter 14 Equations with Partitions14.1 The Meson Functional14.2 General Maps on the Sphere14.3 Nonseparable Maps on the Sphere14.4 Maps Without Cut-Edge on Surfaces14.5 Eulerian Maps on the Sphere14.6 Eulerian Maps on Surfaces14.7 NotesChapter 15 Upper Maps of a Graph15.1 Semi-Automorphisms on a Graph15.2 Automorphisms on a Graph15.3 Relationships15.4 Upper Maps with Symmetry15.5 Via Asymmetrized Upper Maps15.6 NotesChapter 16 Genera of Graphs16.1 A Recursion Theorem16.2 Maximum Genus16.3 Minimum Genus16.4 Average Genus16.5 Thickness16.6 Interlacedness16.7 NotesChapter 17 Isogemial Graphs17.1 Basic Concepts17.2 Two Operations17.3 Isogemial Theorem17.4 Nonisomorphic Isogemial Graphs17.5 NotesChapter 18 Surface Embeddability18.1 Via Tree-Travels18.2 Via Homology18.3 Via Joint Trees18.4 Via Configurations18.5 NotesAppendix 1 Concepts of Polyhedra, Surfaces, Embeddings and MapsAppendix 2 Table of Genus Polynomials for Embeddings and Maps of Small SizeAppendix 3 Atlas of Rooted and Unrooted Maps for Small GraphsBibliography

---

Yanpei Liu, Beijing Jiaotong University, Beijing, China

---