1. B. Li F. Moataz N. Nisse K. Suchan 1 Minimum Size Tree Decompositions 1, Inria, France 2, University Nice Sophia Antipolis, CNRS, I3S, France 3, Institute of Applied Mathematics, CAS, Beijing, China 4, FIC, Universidad Adolfo Ibanez, Santiago, Chile 5, WMS, AGH, Univ. of Science and Technology, Krakow, Poland 1,2,3 2,11,2 4,5 Thank you to B. Li for her slides LAGOS, Praia das Fontes, May 12th 2015

2. Motivations to Study Tree Decompositions 2 Dynamic programming based on tree decomposition [B. Courcelle, Information and Computation, 1990] Many NP-hard problems in general graphs linear in graphs of bounded treewidth All problems expressible in monadic second order logic are linear solvable in graphs of bounded treewidth

3. 3 Characterizations for small treewidth is a forest [Folklore] no -minor in no following minors in [Wald and Colbourn, Networks, 1983] [Arnborg and Proskurowski, Discrete Math. 1990] Graphs of treewidth at most 4 can be reduced to empty by at most 6 reduction rules. [Sanders, Discrete Math., 1995] e.g. outerplanar graphs, series parallel graphs Our goal: Explore other parameters to understand better and compute tree decomposition for algorithmic applications

4. 4 Tree-Decomposition of a pair - For is a tree: - there iss.t. - induces a subtree in For called bag, It’s called path decomposition if T is a path.

5. 5 Tree-Decomposition of Size of a tree decomposition, the number of bags Width 3 Size 6 Minimum size tree decomposition of width at most k Width 3 Size 4 Minimum length path decomposition of width at most k Treewidth, is the minimum width. Width of a tree decomposition, the size of the largest bag Width 3 Length 4

6. 6 Related Work [D. Dereniowski et al., CoRR, 2013] NP-hard to compute for any fixed - Polynomial algorithms for computing - in general graphs NP-hard to compute for any fixed in connected graphs - Minimum length path decompositionof width at most k

7. Our contributions NP-hard to computefor any fixed - General approach for computing - Polynomial algorithm for computing - Polynomial algorithms for computing - 7 in general graphs (resp. connected graphs) for any fixed in the class of forests and 2-connected outerplanar graphs (resp. ) Minimum size tree decompositionof width at most k NP-hard in planar graphs with -

8. Reduction from 3-partition Instance: a list of positive integers, NP-hardness of computing Question: is there a partition of into sets s.t. for each Construct a graph s.t. Yes for 3-partition of width at most 4 and size at most has a tree decomposition based on [D. Dereniowski et al., CoRR, 2013]

9. Construction of Graph andcopies of andcopies of edges disjoint union of and [D. Dereniowski et al., CoRR, 2013] Yes for 3-partition of width at most 4 and length at most s has a path decomposition copies of

10. 10 Any tree decomposition of is a path decomposition of width 4 and size s Path-Decomposition & Tree-Decomposition of is NP-hard

11. 11 NP-hardness of computing If computing is NP-hard in general graphs, then computing is NP-hard in connected graphs. has a tree decomposition of width and size NP-hard to computefor any fixed in general graphs (resp. connected graphs) (resp. )

12. 12 General Approach for Computing find a set of vertices s.t. containing as a leaf bag Givenan d there is a tree decomposition of width and size is called a k-potential leaf

13. 13 General Approach for Computing If there is a-time algorithm that, computes a k-potential leaf of thencan be computed in for any be the the class of graphs of treewidth Let in Prove by induction on Find a k-potential leaf

14. 14 Polynomial Algorithm for computing Characterization of all 2-potential leaves in the class of graphs of treewidth 2:

15. 15 Polynomial in Trees Characterization of 3-potential leaves in trees

16. 16 Polynomial in 2-connected outerplanar graphs 3-potential leaves in 2-connected outerplanar graphs are

17. Future Works 17 - Compute in the class of graphs of treewidth 2 or 3 is a 3-potential leaf of but not 3-potential leaf of It seems more tricky.

18. Future Works 18 - Compute in trees There are 5-potential leaf that are not connected It is different when

19. Future Works 19 Obrigado !