1. Planar Graphs prepared and Instructed by Arie Girshson Semester B, 2014 June 2014Planar Graphs1

2. Planar Graphs - Background June 2014Planar Graphs2 Planar Embedding Planar Graph (G) (vertices, edges)

3. June 2014Planar Graphs3 A relevant topology in studies of planar graphs deal with Jordan curves – Simple closed curve (continuous non- self-intersecting curve whose origin and terminus coincide). The union of the edges in a cycle of a plane graph constitutes a Jordan curve. Jordan curve J partitions the plane into two disjoint open sets, interior of J and exterior of J. Clearly J=int J ∩ ext J. Jordan curve theorem states that any line joining a point in int J to a point in ext J must meet J in some point. int J ext J The Jordan Curve Theorem J

4. June 2014Planar Graphs4 Theorem: K 5 is nonplanar int C ext C int C 1 int C 2 int C 3 Hence, contradicts the assumption that G is a plane graph.

5. Embedding on a Surface June 2014Planar Graphs5 A Graph G is said to be embeddable on a surface S, if it can be drawn in S so that its edges intersect only at their ends. Such drawing is called embedding of G on S. Embedding of K 5 on a torus: PP RR PP Q Q Representation of the torus as a rectangle in which opposite sides are identified.

6. Duality / Dual Graphs June 2014Planar Graphs6 A plane graph G partitions the plane into connected regions. The closures of these regions called faces of G. Each plane graph has exactly one unbounded face, called the exterior/outer face.

7. June 2014Planar Graphs7 Example:

8. June 2014Planar Graphs8 Subdivision

9. June 2014Planar Graphs9 Dual Graph Graph G Graph G *

10. June 2014Planar Graphs10

11. June 2014Planar Graphs11 Conclusion: Isomorphic plane graphs may have non- isomorphic duals. Example: The following plane graphs are isomorphic. Are the dual graphs isomorphic ? NO The plane graph on the left has a face of degree five, whereas the right one has a face of degree four.

12. June 2014Planar Graphs12 A few relations, which are direct consequences of the definition of G * :

13. June 2014Planar Graphs13

14. June 2014Planar Graphs14 G and G * G\e and G*\e*

15. Directed Dual Graph June 2014Planar Graphs15

16. June 2014Connectivity16 Example: Directed dual graph.

17. Euler’s Formula June 2014Planar Graphs17 Simple formula relating the number of vertices, edges & faces in a connected plane graph (established by Euler).

18. June 2014Planar Graphs18 The theorem follows by the principle of induction.

19. June 2014Planar Graphs19 Corollary: All planar embeddings of a connected planar graph have the same number of faces.

20. June 2014Planar Graphs20