1. Planar Graphs (part 2) prepared and Instructed by Gideon Blocq Semester B, 2014 June 2014Planar Graphs (part 2)1

2. Goal of the presentation June 2014Planar Graphs2 Two main theorems: 1.Unicity in embedding of planar graphs. Specifically in 3-connected planar graphs. 2.Kuratowski’s theorem.

3. Bridges (Fragments) June 2014Planar Graphs3

4. June 2014Planar Graphs4 Figure 1: Cycle C

5. Bridges of Cycles June 2014Planar Graphs5

6. Bridges of Cycles June 2014Planar Graphs6 We now consider plane graphs. C is a simple closed curve in the plane. Each bridge is either in Int(C) (inner bridge) or in Ext(C) (outer bridge). Theorem 2: Inner (outer) bridges avoid one an other. Proof by contradiction for inner bridges: Suppose by contradiction that two inner bridges B, B’ overlap. Two options: (1) B and B’ are skew (2) Equivalent 3-bridges.

7. Bridges of Cycles June 20147 u v v' u' P P’ u v P u' v' H K

8. June 201488 KH

9. 99 K

10. 10 Unique plane embeddings

11. June 201411 Unique plane embeddings Theorem 5: Every simple 3-connected planar graph has a unique planar embedding. By Theorem 4, the facial cycles are its non-separating cycles. The latter are defined in terms of the abstract structure of the graph, hence they are the same for every embedding.

12. June 201412 Part 2: Kuratowski’s theorem A minor of a graph G is any graph obtainable by means of a sequence of vertex and edge deletions and edge contractions. By an F-minor of G, we mean a minor of G which is isomorphic to F. Every F-subdivision also has an F-minor. Why? Minor

13. June 201413 Part 2: Kuratowski’s theorem

14. June 201414

15. June 201415 e

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17. Appendix June 2014Planar Graphs17 Lemma 3 (intermediate): Let G be a planar graph and f a face in some embedding. Then G admits a planar embedding whose outer face has the same boundry as f.