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# Graph Algo Assign

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Kuratowski's theorem

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### Graph Algo Assign

1. 1. Kuratowski’s Theorem, Statement and Implication of Robertson-Seymour Theorem
2. 2. Outline <ul><li>Kuratowski’s theorem </li></ul><ul><li>Robertson Seymour Theorem </li></ul><ul><li>Consequences </li></ul>
3. 3. Kuratowski’s theorem <ul><li>Why important? </li></ul>
4. 4. Kuratowski’s theorem <ul><li>Theorem Statement : A graph G is embeddable in the plane iff it does not contain a subgraph homeomorphic to the complete graph K5 or complete bipartite graph K3,3 </li></ul>
5. 5. Kuratowski’s theorem <ul><li>Natural question--- </li></ul>
6. 6. Kuratowski’s theorem <ul><li>Wagner’s conjecture </li></ul>
7. 7. Robertson Seymour Theorem <ul><li>Theorem Statement : Every downwardly closed set of (isomorphism classes of) finite graphs is precisely the set of all (isomorphism classes of) graphs that lack a certain set of finitely many forbidden minors . </li></ul>
8. 8. Robertson Seymour Theorem <ul><li>A graph H is a minor of a graph G if H can be made from G by deleting or contracting edges. </li></ul>
9. 9. Robertson Seymour Theorem <ul><li>A graph H is a minor of a graph G if H can be made from G by deleting or contracting edges. </li></ul>
10. 10. Robertson Seymour Theorem <ul><li>A graph H is a minor of a graph G if H can be made from G by deleting or contracting edges. </li></ul>
11. 11. Robertson Seymour Theorem <ul><li>A graph H is a minor of a graph G if H can be made from G by deleting or contracting edges. </li></ul>
12. 12. Robertson Seymour Theorem <ul><li>A downwardly closed set S of isomorphism classes of graphs is a set such that if G ∈ S and H is a minor of G , then H ∈ S . </li></ul>
13. 13. Robertson Seymour Theorem <ul><li>A downwardly closed set S of isomorphism classes of graphs is a set such that if G ∈ S and H is a minor of G , then H ∈ S . </li></ul><ul><li>Examples:- the set of all forests </li></ul>
14. 14. Robertson Seymour Theorem <ul><li>Theorem Statement : Every downwardly closed set of (isomorphism classes of) finite graphs is precisely the set of all (isomorphism classes of) graphs that lack a certain set of finitely many forbidden minors . </li></ul>