Planar Graphs (part 2)

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

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

3. Bridges (Fragments) Let H be a proper subgraph of connected G. The set E(G)\E(H) may be split up into two classes: An edge e, which has both ends in V(H). For each component F of G-V(H), the edges of F together with the edges linking F to H. The elements of are denoted as the vertices of attachment (vot). The subgraphs induces by these classes are denoted as “bridges”. A bridge is trivial if it has no internal vertices. A bridge with k vertices of attachment to H is a k-bridge. Two bridges with same vot are equivalent. Planar Graphs

4. Figure 1: Cycle C Planar Graphs

5. Bridges of Cycles A k-bridge with , partitions C into k segments. Two bridges avoid each other if all vot of one bridge lie in a single segment of the other; otherwise they overlap. Two bridges B and B’ are skew if there exist distinct vot {u,v} of B and {u’,v’} of B’, which occur in cyclic order u,u’,v,v’. Look at Figure 1 and check which bridges overlap/avoid and which are skew. Theorem 1:Overlapping bridges are either skew or equivalent 3-bridges. Planar Graphs

6. Bridges of Cycles 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. Planar Graphs

7. Bridges of Cycles Case 1: Let uPv be a path in B and u’P’v’ be a path in B’. Consider the subgraph, which is plane, since G is plane. Now let K be the plane graph by adding a vertex, w, in Ext(C) and connecting all vertices of H to w. However, K is a subdivision of , hence this is a contradiction. u' u' K H P P v u u v P’ P’ v' v'

8. Lemma 2 (intermediate from old book of Bondy): If a bridge B has three vot, then there exists in and three paths in B joining to and respectively, such that for , and only have in common. (holds also for non-plane graphs). Case 2: B and B’ are equivalent 3-bridges. According to Lemma 2, there exists an as depicted in the figure. We add a vertex w as done in Case 1. H K 8

9. Case (ii): B and B’ are equivalent 3-bridges. The new graph K has to be planar since and we only added an edge that does not intersect. However, K is a subdivision of , hence this is a contradiction. K 9

10. Unique plane embeddings A non-seperating cycle has no chords and at most one non-trivial bridge. Theorem 4:A cycle in a simple 3-connected plane graph is a facial cycle iff it is nonseparating. Suppose that C is not a facial cycle. Then there must be an inner and outer bridge of C. G is simple, so no loops. The bridges are non-trivial or chords. Suppose C is a facial cycle. We may assume (from Lemma 3 in the appendix) that all bridges are inner bridges. From Theorem 2, they all avoid each other. If C has a chord xy, then {x,y} is a vertex cut. If C has two nontrivial bridges, they lie on a single segment of the other bridge and {x,y} is a vertex cut. Both contradict the 3-connectivity. 10

11. 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. 11

12. 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 12

13. Part 2: Kuratowski’s theorem Main Theorem: A graph is planar iff it contains no minor isomorphic to or Proposition 1: Minors of planar graphs are planar. Proposition 2 (Thomassen): Let G be a 3-connected graph with . Then G contains an edge e, such that, when contracting e (G\e), G is still 3-connected. Proposition 3: In a loopless 3-connected plane graph, the neighbors of any vertex lie on a common cycle. 13

14. Theorem 6: Every 3-connected nonplanar graph has a K-minor. Proof: We assume that G is simple and non-planar. For , all graphs are planar. Thus consider We proceed by induction on n. By Proposition 2, G contains an edge e, such that G\e is 3-connected. We assume by induction that if H is non-planar it as a K-minor. Any minor of H is a minor of G, hence G is non-planar. Thus, we assume H is planar. Consider an embedding of H. Denote by z the vertex formed by contracting e. By proposition 3, the neighbors of z lie on a cycle C, the boundary of a face f of . 14

15. Denote by and the bridges of C in G\e that contain x and y respectively. Case 1: and avoid each other. Then and can be embedded in the face f, such that x and y belong to the same face of the resulting plane graph. We can now add e and get a planar embedding of G, which is a contradiction. e 15

16. Case 2: and overlap. Thus according to Theorem 1, they are either skew or 3-equivalent. In both cases, by adding e, it is easy to see that G has a K-minor, which proves the main theorem. G has a minor G has a minor 16

17. Appendix 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. Planar Graphs