1. Perfect Matching for Biconnected Cubic Graphs in O(nlog 2 n) Time Krzysztof Diks & Piotr Stańczyk

2. Petersen theorem (1891) Cubic graph without perfect matching contains at least three bridges Conclusion: every biconnected cubic graph has a perfect matching

3. Frink’s algorithm - O(n 2 ) time (1926) Observation: at least one of two local reductions of n+2-vertex biconnected graph leads to n- vertex biconnected graph Idea: keep reducing a graph (preserving biconnectivity) until it consists of 2 vertices.

4. Frink’s algorithm - O(n 2 ) time (1926) Reduce graph to 2-vertex (preserving biconnectivity) – O(n) phases, O(n) time each Construct arbitrary matching – O(1) Reverse all reductions (4 cases possible) – O(n) phases, O(n) time each (worst case)

5. Biedl’s et al. algorithm – O(nlog 4 n) time (SODA,1999) Valid reduction selection  Instead of computing biconnected components from scratch use dynamic graph biconnectivity algorithm – O(log 4 n) per edge removal/addition/query Elimination of case (d) from the revertion process  1) Select an edge e which is excluded from the matching  2) Perform reductions against edges incident to e (edge e „travels” along the graph)  3) Construct initial matching excluding edge e  4) Reverse all reductions – case d) does not occur, as e is never matched Complexity: O(nlog 4 n)

6. New algorithm – O(nlog 2 n) time Replace dynamic graph biconnectivity algorithm with:  Dynamic graph connectivity algorithm – O(log 2 n) deterministic / O(lognloglog 3 n) randomized time  Tarjan’s dynamic trees – O(logn) (addition, removal of edges, change of tree’s root, LCA computation) How to perform reduction phase with a new approach?

7. New algorithm – O(nlog 2 n) time Remove reduction edge along with all incident vertices / edges. Two cases are possible:  Resulting graph is not connected If so, graph has a structure presented in figure (a) One reduction leads to not connected graph (can be verified with dynamic connectivity algorithm) The other reduction leads to biconnected graph (Frink’s theorem)

8. New algorithm – O(nlog 2 n) time  Resulting graph is connected Analyze changes in spanning tree maintained by dynamic connectivity data-structure, update Tarjan’s dynamic trees Analyze edges of a spanning tree T spanned by vertices A, B, E and F. Select reduction which covers all spanned edges with cycles (use LCA operation from Tarjan’s trees) Selected reduction maintains biconnectivity…

9. New algorithm – O(nlog 2 n) time Selected reduction maintains biconnectivity:  Added edges and edges spanned by A, B, C and D are covered by a cycle  Any other edge e → before reduction there was a cycle c covering e. If c is a cycle in reduced graph → OK Otherwise, c contains some removed edges E. Let f be a cycle in T+E containing all edges of E. Symmetric difference of f and c is a cycle in reduced graph  Each edge belongs to some cycle → graph is biconnected Overall complexity: O(nlog 2 n) deterministic, O(nlognloglog 3 n) randomized

10. Possible improvement Replacement of dynamic connectivity algorithm with decremental counterpart – how to handle edge insertion? New vertex and edge representation: Allowed Not allowed

11. Possible improvement It is not allowed to perform reductions against not allowed edges – how to bypass it…  Introduce support for reverse points in decremental connectivity data-structure  In case of performing reduction against not allowed edge, it is required to execute reverse point first, three cases are possible…

12. Possible improvement Even cycle case:

13. Possible improvement Odd cycle case: not allowed – introduces bridge

14. Possible improvement No cycle case:

15. Thank you Any questions?