1. Graph Theory Ch6 Planar Graphs

2. Basic Definitions  curve, polygon curve, drawing  crossing, planar, planar embedding, and plane graph  open set  region, face

4. Proposition: K 5 and K 3,3 is not planar

5. Restricted Jordan Curve Theorem Theorem. A simple closed polygonal curve C consisting of finite number of segments partitions the plane into exactly 2 faces, each have C as boundary

6. Dual Graphs Definition: let function F on a graph F(G) = { faces of G } if there exits f: V(G * ) → F(G) so that f is 1-1 & onto and for all x, y V(G * ), there is an edge connects x, y iff there is an edge e in G that f(x) and f(y) are on the different side of e.

7.  A cut-edge in G becomes a loop in G *  For all x V(G * ) and X = f(x), x is in the interior of X  Each edge e in G there is exactly one e * in G * that e and e * crosses.  (G * ) * =G iff G is connected pf: a) for all G, G * is connected b) each face in G * contains exactly one vertex of G  Two embeddings of a planar graph may have non-isomorphic duals.

8. Length of a face length of a face is defined as total length of the boundary of the face. 2e(G) = ∑L (F i ) Theorem. edges in G form a cycle in G iff the corresponding edges in G * form a bond in G * Theorem. the follows are equivalent A) G is bipartite B) every face of G has even length C) G * is Eulerian

9. Outerplane graph Def: outerplanar, outerplane graph a graph is outerplanar if it has an embedding that every vertex is on the boundary of the unbounded face. The boundary of the outer face of a 2-connected outerplane graph is a spanning cycle K 4 and K 2,3 are not outerplanar.

10. Every simple outerplane graph has 2 non-adjacent vertex of degree at most 2 pf: 1. n(G) < 3, every vertex has degree ≤2 2. n(G) = 4 holds. (think about K 4 – {any edges}) 3. n(G) ≥ 4

11. Euler ’ s Formula n – e + f = 2 All planar embeddings of connected graph G have the same number of faces A graph with k components, n – e + f = k+1

12. For simple planar graphs, e(G) ≤ 3n(G) – 6, if G is triangle free, e(G) ≤ 2n(G) – 4 pf: 2e = ∑L (F i ) ≥ 3f-----(*) f = e – n + 2 => e ≤ 3n – 6 for triangle free case, 3f in (*) -> 4f K 5 and K 3,3 are not planar

13. Maximal Planar Graph Def. Maximal planar graph: a simple planar graph that is not a spanning subgraph of any other planar graph. Proposition. The follows are equivalent A)G has 3n-6 edges B)G is a triangulation C)G is a maximal plane graph

14. Regular Polyhedra A graph embeds in the plane iff it embeds on a sphere For a regular polyhedra of degree k and all faces ’ length are l e( 2/k + 2/l -1 ) = 2 => (2/k) + (2/l) > 1 => (k – 2)(l – 2) < 4 hence k and l can only be klf 334 346 438 3512 5320

15. Graph Theory Ch6 Planar Graphs (continued)

16. Kuratowski ’ s Theorem Theorem. A graph is planar iff it does not contain a subdivision of K 5 or K 3,3. Kuratowski subgraph: a subgraph contains a subdivision of K 5 or K 3,3. minimal nonplanar graph: a nonplanar graph that every proper subgraph is planar

17. Lemma 1 if F is the edge set of a face in a planar embedding of G, then G has an embedding with F being the edge set of the unbounded face.

18. Lemma 2 every minimal planar graph is 2-connected. Lemma 3 let S = {x, y} be a separating set of G, if G is nonplanar, there Exist some S-lobe adding (x, y) is nonplanar.

19. Lemma 4 if G is a graph with Fewest Edges among all nonplanar graphs without Kuratowski subgraphs G is 3-connected

20. Convex embedding: planar embedding that each face boundary is a convex polygon Theorem. Every 3-connected planar graph has a convex embedding Theorem. Every 3-connected graph G with at least 5 vertices has an edge e such that G˙e is 3-connected.

21. Lemma 5. if G has no Kuratowski subgraph, G˙e has no Kuratowski subgraph

22. Theorem. (Tutte1960) if G is a 3-connected graph without subdivision of K 5 or K 3,3, then G has a convex embedding in the plane with no three vertices on a line Pf: induction on n(G) K 4 :

23. n(G) > 4: exist e that G˙e is 3-connected. G˙e has no Kuratowski subgraph. e z, H = G˙e H-z is 2-connected.

24. Definition: H is a minor of G if a copy of H can be obtained by deleting or contracting edges of G. G is planar iff neither K 5 nor K 3,3 is a minor of G. Nonseparating Let G be a subdivisions of a 3-connected graph.G is planar iff every edge e exactly lies in 2 nonseparating cycles.

25. H-fragment Conflict Planarity testing If a planar embedding of H can be extended to a planar embedding of G, then in that extension every H-fragment of G appears inside a single face of H.

26. Planarity testing 1. find a cycle G 0 2. for each G i -fragment B, determine all faces of G i that contain all vertices of attachment of B. call it F(B) 3. if F(B) is empty for some B, stop (FAIL). Else, choose one. 4. choose a path P between 2 vertices of attachment of B. embed P across F(B). Result in G i+1.