1 Introduction. - 1. 1 Sets. - 1. 2 Graphs. - 1. 3 Subgraphs. - 1. 4 Cocyeles. - 1. 5 Forests trees circuits and paths. - 1. 6 Some elementary results. - 1. 7 Theorems about trees. - 1. 8 Spanning trees. - 1. 9 Eulerian graphs. - 1. 10 Bipartite graphs. - 1. 11 Contractions. - 1. 12 Menger's theorem and n-connected graphs. - 1. 13 2-connected graphs. - 1. 14 Blocks. - 1. 15 The cycle and cocycle spaces of a graph. - 1. 16 Double covers. - 2 Maps. - 2. 1 Permutations. - 2. 2 Maps. - 2. 3 Imbeddings of maps. - 2. 4 3-graphs. - 2. 5 From maps to gems and back again. - 2. 6 Premaps. - 3 Classification of Surfaces. - 3. 1 Dipoles. - 3. 2 Reduced and unitary 3-graphs. - 3. 3 Canonical gems. - 3. 4 Planar graphs. - 4 Consistent and Coherent Orientations. - 4. 1 Orientations. - 4. 2 Pairwise coherently orientable nets. - 4. 3 Families of circuits. - 4. 4 Rings. - 5 Non-separating Curves in Surfaces. - 5. 1 The main results and their topological implications. - 5. 2 Permutation pairs. - 5. 3 A condition for a b-cycle to separate. - 5. 4 Fundamental sets of semicycles. - 6 Mac Lane's Theorem for 3-Graphs. - 6. 1 Congruence. - 6. 2 Semicycle covers. - 6. 3 Boundary covers. - 6. 4 Partial congruence. - 6. 5 Mac Lane's theorem. - 6. 6 Whitney's characterisation. - 7 Kuratowski's Theorem. - 7. 1 Corollaries of Mac Lane's theorem. - 7. 2 Kuratowski's theorem. - 7. 3 Wagner's theorem. - 8 Duality. - 8. 1 Duals. - 8. 2 Constructing orthogonal graphs. - 8. 3 Duality for planar graphs. - 8. 4 The zigzag space. - 8. 5 The principal edge tripartition for planar graphs. - 8. 6 Walks. - 8. 7 Principal cycles and principal cocycles. - 8. 8 Diagonals. - 8. 9 Every planar graph has a diagonal. - 8. 10 No non-planar graph has a diagonal. - 9 Rings of Bonds. - 9. 1 Chordal graphs. - 9. 2 Rings of bonds. - 10 Bridges. - 10. 1 Residues and bridges. - 10. 2 Tutte's characterisation. - List of Symbols. Language: English