1. A linear time algorithm for recognizing a K 5 -minor Bruce Reed Zhentao Li

2. Definitions K5K5 K 5 -model

3. Connectivity G3G3 G2G2 G1G1

4. Wagner’s theorem for K 5 A 3-connected graph without a K 5 -model or a cut of size 3 which splits it into at least 3 components is either planar or L L

5. K 5 minor containment Construction of 1-cut, 2-cut, and (3,3)-cut decomposition K 5 minor containment in “highly” connected graphs

6. K 5 minor containment Construction of 1-cut, 2-cut, and (3,3)-cut decomposition K 5 minor containment in “highly” connected graphs Test planarity Test if the graph is L 1 and 2-cuts (3,3)-cut decomp (HT73)(HT74)

7. Finding a (3,3)-block tree Some assumptions G is 3-connected G has no K 5 -minor |E(G)|<64|V(G)| (RS95) Properties Unique (not K 3,3 ) Linear size (3,3)-block tree (3,3)-cut

8. A recursive algorithm Use brute force if the graph is small. Otherwise, build a smaller graph to recurse on. G H >  |V(G)| vertices Running time: |V(G)|[1+(1-  )+(1-  ) 2 +(1-  ) 3 +…]=O(|V(G)|)

9. Building a smaller graph by: Removing degree 3 vertices Common neighbours Rest of the graph Common neighbours Common neighbours

10. Building a smaller graph by: Contracting a matching GH Induced Low degree vertices Size >  |V(G)| Resulting graph is 3-connected

11. (3,3)-block tree for H(3,3)-block tree for G Building a smaller graph by: Contracting a matching

12. G H 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 (1,3) (7,9) (2,4) (1,7) (2,8) (7,9) (6,9) (4,5) (1,6) 123456789123456789