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A linear time algorithm for recognizing a K 5 -minor

This algorithm focuses on recognizing K5-minors in highly connected graphs. It includes constructions of 1-cut, 2-cut, and (3,3)-cut decompositions as part of the process. The key properties of the algorithm involve unique (not K3,3) structures and linear-sized (3,3)-cut block trees. The approach is recursive, involving brute force for small graphs and building smaller graphs for larger ones. The running time is O(|V(G)|), where G is a 3-connected graph with certain limitations. The algorithm involves building smaller graphs through various methods such as removing degree-3 vertices, contracting matchings, and ensuring the resulting graph remains 3-connected.

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A linear time algorithm for recognizing a K 5 -minor

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  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

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