JT/TR Version 1.1 26/02/19
Planarity
Jeff Trim
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The concept of planarity in graph theory, which refers to the ability to draw a graph on a plane without any edges crossing. It covers topics such as euler's relation, bipartite graphs, hamiltonian cycles, and the kuratowski's theorem for determining the planarity of a graph. The document also provides examples and step-by-step illustrations of how to transform a non-planar graph into a planar one using techniques like edge contraction. This resource would be valuable for students studying discrete mathematics, decision mathematics, or graph theory at the advanced level, as it delves into the theoretical foundations and practical applications of planarity in these fields.
Typology: Cheat Sheet
2023/2024
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Planarity
Jeff Trim
Discrete/Decision Mathematics Topics AQA Edexcel MEI OCR A Nodes/vertices, degree/order AS AS D1 MwA AS Arcs/edges, simple, connected, related vocabulary AS AS D1 MwA AS Trees AS MwA AS Euler’s relation: V - E + F = 2 AS A Level Bipartite graphs, Km,n AS MwA A Level Walk, trail, path, cycle AS AS Eulerian, semi-Eulerian graphs AS AS D1 AS Hamiltonian cycles AS A Level D1 A Level Complete graphs, Kn AS AS D1 AS Isomorphic Equivalence A Level AS D1 AS Planar graphs AS AS D1 A Level Subdivision and contraction A Level A Level Planarity Algorithm A Level D Kuratowski’s Theorem A Level A Level Thickness A Level Complement of a graph A Level Ore’s theorem A Level
Graph Theory Topic Mapping
Imagineering
What did you see?
Can you do the same for a triangular prism?
Victor Schlegel (1843-1905)
“It is always possible by suitable choice of the centre of projection to make the projection of one face completely contain the projections of all the other faces. This is called a Schlegel diagram of the polyhedron.” Duncan Sommerville, writing in 1929, described this approach, which was introduced by Victor Schlegel in 1886.
Name Faces Vertices Edges
- Euler’s Relation: F + V = E +
- Cube
- Tetrahedron
- Octahedron
- Dodecahedron
- Icosahedron
Two graphs are isomorphically equivalent if one can be stretched, distorted, or by repositioning the vertices transformed into the other. (An isomorphism is a one-to-one matching.)
Isomorphic Equivalence
Isomorphic Equivalence
Isomorphic Equivalence
Isomorphic Equivalence
K
n denotes the complete graph on n vertices; in this graph every pair of vertices is joined directly by one edge. K 3 K 4 K 5
Complete graphs, K
n
E G W
(This graph is now K 2,
Bipartite graphs, K
3,
E G W
(This is now the finished graph of K 3,
Bipartite graphs, K
3,
Eulerian graph Semi-Eulerian graph contains an Eulerian cycle contains an Eulerian path Are either of these graphs traversable? (Is there an Eulerian path which traverses each edge once?)
Cycles
Is there a path which visits every vertex once? Hamiltonian graph contains a Hamiltonian cycle