Math 2.0

Combinatorics Definition

Graph embedding

Builds on GraphTopological embeddingOrientable surface

Definition

Let G=(V,E)G = (V, E) be a finite graph and let Σ\Sigma be a surface. An embedding of GG in Σ\Sigma consists of a point pv∈Σp_v \in \Sigma for each vertex vv and a subset γe⊆Σ\gamma_e \subseteq \Sigma for each edge ee such that:

  1. The points pvp_v are distinct.
  2. For each edge e=uve = uv, the set γe\gamma_e is the image of a topological embedding of [0,1][0, 1] into Σ\Sigma with ends pup_u and pvp_v.
  3. An arc γe\gamma_e contains no point pwp_w other than its two ends.
  4. Two distinct arcs meet at most in common ends.

Remarks

Example: drawing K4K_4 in the plane as a triangle with a central vertex joined to its three corners satisfies all four clauses. Non-example: drawing K4K_4 as a square with both diagonals is not an embedding, because the diagonals cross at an inner point, which violates clause 4. A graph is planar exactly when it has an embedding in the plane, or equivalently in the sphere. The genus of GG is the least gg such that GG embeds in the orientable surface of genus gg; for example K5K_5 and K3,3K_{3,3} have genus 11.

Reading path

To understand this page from the ground up, read these 64 pages in order. The foundations come first.

  1. Element definition
  2. Euclidean space object
  3. Function concept
  4. Graph definition
  5. Infimum concept
  6. Proposition concept
  7. Real numbers object
  8. Supremum concept
  9. Set definition
  10. Fréchet derivative definition
  11. Function of several variables concept
  12. Vector-valued function concept
  13. Jordan measure definition
  14. Conjunction definition
  15. Disjunction definition
  16. Implication definition
  17. Predicate definition
  18. Empty set object
  19. Family of sets definition
  20. Ordered pair definition
  21. Limit of a function of several variables concept
  22. Intersection definition
  23. Partial derivative concept
  24. Union definition
  25. Existential quantifier definition
  26. Universal quantifier definition
  27. Cartesian product definition
  28. Continuity in R^n definition
  29. Differentiable function of several variables definition
  30. Partial derivative definition
  31. Subset definition
  32. Union of a family of sets definition
  33. Uniqueness quantifier definition
  34. Double integral definition
  35. Binary relation definition
  36. Differentiable on an open set definition
  37. Gradient definition
  38. Higher partial derivatives definition
  39. Topological space definition
  40. Iterated derivative definition
  41. Function definition
  42. Subspace topology definition
  43. Codomain definition
  44. Injective function definition
  45. C^k function characterised by continuous derivatives statement
  46. Metric space definition
  47. Image of a set definition
  48. Preimage of a set definition
  49. Surjective function definition
  50. Parametrised curve definition
  51. Open ball definition
  52. Bijective function definition
  53. Parametrised surface definition
  54. Metric topology definition
  55. Inverse function definition
  56. Tangent plane and normal vector of a surface definition
  57. Open set definition
  58. Surface area definition
  59. Continuous map of topological spaces definition
  60. Homeomorphism definition
  61. Surface integral definition
  62. Orientable surface definition
  63. Topological embedding definition
  64. Graph embedding definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Graph_embedding
About this page
TagVK1XXMR2WW
URIm2://VK1XXMR2WW
Kinddefinition
WritingW0 stub
MSC05C10, 57M15
DefinesGraph embedding
UsesGraph, Topological embedding, Orientable surface
Changed2 November 2026

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