A linear time algorithm for recognizing a K 5 -minor

1. A linear time algorithm for recognizing a K5-minor Bruce Reed Zhentao Li

2. Definitions K5 K5-model

3. Connectivity G2 G1 G3

4. Wagner’s theorem for K5 A 3-connected graph without a K5-model or a cut of size 3 which splits it into at least 3 components is either planar or L L

5. K5 minor containment Construction of 1-cut, 2-cut, and (3,3)-cut decomposition K5 minor containment in “highly” connected graphs

6. K5 minor containment Construction of 1-cut, 2-cut, and (3,3)-cut decomposition K5 minor containment in “highly” connected graphs 1 and 2-cuts (3,3)-cut decomp Test planarity Test if the graph is L (HT73) (HT74)

7. Finding a (3,3)-block tree Some assumptions • G is 3-connected • G has no K5-minor • |E(G)|<64|V(G)| (RS95) Properties • Unique (not K3,3) • Linear size (3,3)-cut (3,3)-block tree

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

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

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

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

12. Building a smaller graph by: Contracting a matching 1 2 3 4 5 6 7 8 9 (2,4) (1,3) 1 (1,7) 1 (7,9) (2,8) 2 2 (7,9) 3 (6,9) 3 4 7 (4,5) 4 7 (1,6) 5 8 5 8 6 9 6 9 G H