Drawing a graph http://mathworld.wolfram.com/GraphEmbedding.html.

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Drawing a graph http://mathworld.wolfram.com/GraphEmbedding.html

https://reference.wolfram.com/language/ref/GraphPlot.html Graph Theory and Complex Networks by Maarten van Steen

Graph Theory and Complex Networks by Maarten van Steen

What is a planar embedding? drp.math.umd.edu/Project-Slides/Characteristics of Planar Graphs.pptx What is a planar embedding? K4 http://www.boost.org/doc/libs/1_49_0/libs/graph/doc/figs/planar_plane_straight_line.png

Kuratowski’s Theorem (1930) A graph is planar if and only if it does not contain a subdivision of K5 or K3,3. http://www.math.ucla.edu/~mwilliams/pdf/petersen.pdf

Kuratowski Subgraphs What is a subdivision? K5 K3,3 http://www.boost.org/doc/libs/1_49_0/libs/graph/doc/figs/k_5_and_k_3_3.png What is a subdivision? Kuratowski Subgraphs http://www.personal.kent.edu/~rmuhamma/GraphTheory/MyGraphTheory/Diagrams/g83.gif http://www.personal.kent.edu/~rmuhamma/GraphTheory/MyGraphTheory/Diagrams/g82.gif

For a planar connected graph |V| - |E| + |F| = 2 Euler characteristic (simple form): = number of vertices – number of edges + number of faces Or in short-hand, = |V| - |E| + |F| where V = set of vertices E = set of edges F = set of faces = set of regions & the notation |X| = the number of elements in the set X. For a planar connected graph |V| - |E| + |F| = 2

Defn: A tree is a connected graph that does not contain a cycle. A forest is a graph whose components are trees. Lemma 2.1: Any tree with n vertices has n-1 edges. χ = 8 – 7 + 1 = 2 χ = 8 – 8 + 2 = 2 χ = 8 – 9 + 3= 2

= |V| – |E| + |F| = 1 – 0 + 1 = 2

= |V| – |E| + |F| = 2 – 1 + 1 = 2

= |V| – |E| + |F| = 3 – 2 + 1 = 2

= |V| – |E| + |F| = 4 – 3 + 1 = 2

= |V| – |E| + |F| = 5 – 4 + 1 = 2

= |V| – |E| + |F| = 8 – 7 + 1 = 2

= |V| – |E| + |F| = 8 – 8 + 2 = 2 Not a tree.

= 8 – 9 + 3 = 2 Not a tree. = |V| – |E| + |F| = 8 – 9 + 3 = 2 For the brave of heart, consider graphs drawn on other surfaces such as a torus or Klein bottle. For fun, see http://youtu.be/Q6DLWJX5tbs or www.geometrygames.org. Not a tree.

Euler’s fomula: For a planar connected graph |V| - |E| + |F| = 2 where V = set of vertices, E = set of edges, F = set of faces = set of regions Defn: A tree (or acyclic graph) is a connected graph that does not contain a cycle. A forest is a graph whose components are trees. Lemma 2.1: Any tree with n vertices has n-1 edges. Thm 2.9: For any connected planar graph with |V| ≥ 2, |E| ≤ 3|V| - 6 Cor 2.4: K5 is nonplanar. Thm 2.10: K3,3 is nonplanar. Cor: A graph is planar if and only if it does not contain a subdivision of K5 or K3,3.