1. Turing Kernelization for Finding Long Paths and Cycles in Restricted Graph Classes Bart M. P. Jansen June 3rd, WORKER 2015, Nordfjordeid, Norway

2. Finding long paths and cycles 2

3. YearAuthors 1985Monien 1993Bodlaender 1995Alon et al. 1995Alon et al. 2006Kneis et al. 2007Chen et al. 2007Chen et al. 2008Koutis 2009Williams 2010Björklund et al. 2013Fomin et et al. 3

4. Preprocessing for path and cycle problems 4

5. Relaxed notions of preprocessing 5

6. Turing kernelization 6

7. Our results 7 The difficult part of finding long paths and cycles in these graph classes can be confined to small subtasks

8. Adaptivity 8

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13. Decompose-Query-Reduce 13

14. Circumference of triconnected graphs 14

15. Decomposition into triconnected components 15 Every graph can be decomposed into triconnected components [Tutte 1966]

16. Decomposition into triconnected components 16

17. Exact statement for graph decomposition 17

18. Kernelizing using the decomposition 18

19. Finding a 2-separation to reduce (I) 19

20. Finding a 2-separation to reduce (II) 20

21. Summary of the kernelization 21

22. Extensions 22

23. Conclusion 23 Is there a non-adaptive Turing kernel?

24. Lower bounds for Turing kernels 24

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26. Colored paths are harder to find 26