We set this number to 0 if there is no (T, L)-out-branching. For an out-tree T in a digraph D, Leaf(T ) denotes the set of leaves in the set of internal vertices of T . Given a vertex x in an out-tree T let T x denote the subtree of T rooted at x. [1] F. Abu-Khzam Kernelization algorithms for d-hitting set problems Parameterized algorithms for directed maximum leaf problems Spanning directed trees with many leaves Color-coding: a new method for finding simple paths, cycles and other small subgraphs within large graphs Bipartite Graphs and Their Applications, Algorithms and Data Structures Proc. ICALP 2007 Proceedings of the twenty-sixth annual ACM symposium on Theory of computing, STOC '94 Calvo, G. Durán, and F. Protti. New advances about a conjecture on Helly circle graphs. Electronic Notes in Discrete Mathematics Latin- American Conference on Combinatorics, Graphs and Applications, pp.434-4451, 1990. ,
Treewidth: Structure and Algorithms, Structural Information and Communication Complexity, pp.11-25, 2007. ,
DOI : 10.1007/978-3-540-72951-8_3
Open problems in parameterized and exact computation, Proceedings of the 2nd International Workshop on Parameterized and Exact Computation, pp.6-31, 2006. ,
On problems without polynomial kernels, Journal of Computer and System Sciences, vol.75, issue.8, pp.423-434, 2009. ,
DOI : 10.1016/j.jcss.2009.04.001
URL : http://doi.org/10.1016/j.jcss.2009.04.001
Clustering with partial information, Proceedings of the 33rd international symposium on Mathematical Foundations of Computer Science, MFCS '08, pp.144-155, 2008. ,
DOI : 10.1016/j.tcs.2009.12.016
URL : http://doi.org/10.1016/j.tcs.2009.12.016
(Meta) kernelization, Proceedings of the 2009 50th Annual IEEE Symposium on Foundations of Computer Science, FOCS '09, pp.629-638, 2009. ,
DOI : 10.1109/focs.2009.46
URL : https://hal.archives-ouvertes.fr/lirmm-00904532
Cross-composition: A new technique for kernelization lower bounds, STACS, pp.165-176, 2011. ,
URL : https://hal.archives-ouvertes.fr/hal-00573603
Kernel bounds for disjoint cycles and disjoint paths, ESA'09, pp.635-646, 2009. ,
URL : https://hal.archives-ouvertes.fr/lirmm-00806805
A Faster FPT Algorithm for Finding Spanning Trees with Many Leaves, Mathematical Foundations of Computer Science 2003, pp.259-268, 2003. ,
DOI : 10.1007/978-3-540-45138-9_20
An FPT algorithm for directed spanning k-leaf. Manuscript ,
A 3/2-approximation algorithm for finding spanning trees with many leaves in cubic graphs, Graph-Theoretic Concepts in Computer Science, pp.66-77, 2008. ,
Reducing prime graphs and recognizing circle graphs, Combinatorica, vol.54, issue.3, pp.243-254, 1987. ,
DOI : 10.1007/BF02579301
Circle Graph Obstructions, Journal of Combinatorial Theory, Series B, vol.60, issue.1, pp.107-144, 1994. ,
DOI : 10.1006/jctb.1994.1008
URL : http://doi.org/10.1006/jctb.1994.1008
A polynomial kernel for multicut in trees, STACS, pp.183-194, 2009. ,
URL : https://hal.archives-ouvertes.fr/inria-00359171
A polyhedral approach to an integer multicommodity flow problem, Discrete Applied Mathematics, vol.101, issue.1-3, pp.13-36, 2000. ,
DOI : 10.1016/S0166-218X(99)00187-0
Nondeterminism within $P^ * $, SIAM Journal on Computing, vol.22, issue.3, pp.560-572, 1993. ,
DOI : 10.1137/0222038
ON THE HARDNESS OF APPROXIMATING MULTICUT AND SPARSEST-CUT, computational complexity, vol.15, issue.2, pp.94-114, 2006. ,
DOI : 10.1007/s00037-006-0210-9
Improved Parameterized Upper Bounds for Vertex Cover, In Mathematical Foundations of Computer Science Lecture Notes in Computer Science, vol.4162, pp.238-249, 2006. ,
DOI : 10.1007/11821069_21
On the parameterized max-leaf problems: digraphs and undirected graphs, 2008. ,
An Improved Parameterized Algorithm for??the??Minimum Node Multiway Cut Problem, Algorithmica, vol.3, issue.1, pp.1-13, 2009. ,
DOI : 10.1007/s00453-007-9130-6
A fixed-parameter algorithm for the directed feedback vertex set problem, Proceedings of the 40th annual ACM symposium on Theory of computing, STOC '08, pp.177-186, 2008. ,
Lower Bounds for Kernelizations and??Other??Preprocessing??Procedures, Mathematical Theory and Computational Practice, pp.118-128, 2009. ,
DOI : 10.1007/s00224-010-9270-y
Directeds-t numberings, Rubber bands, and testing digraphk-vertex connectivity, Combinatorica, vol.13, issue.115, pp.435-451, 1994. ,
DOI : 10.1007/BF01302965
URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.65.1422
Minimal multicut and maximal integer multiflow: A survey, European Journal of Operational Research, vol.162, issue.1, pp.55-69, 2005. ,
DOI : 10.1016/j.ejor.2003.10.037
URL : https://hal.archives-ouvertes.fr/hal-01124670
The monadic second-order logic of graphs. I. Recognizable sets of finite graphs, Information and Computation, vol.85, issue.1, pp.12-75, 1990. ,
DOI : 10.1016/0890-5401(90)90043-H
URL : https://hal.archives-ouvertes.fr/hal-00353765
Circle graphs and monadic second-order logic, Journal of Applied Logic, vol.6, issue.3, pp.416-442, 2008. ,
DOI : 10.1016/j.jal.2007.05.001
URL : https://hal.archives-ouvertes.fr/hal-00333840
Handle-rewriting hypergraph grammars, Journal of Computer and System Sciences, vol.46, issue.2, pp.218-270, 1993. ,
DOI : 10.1016/0022-0000(93)90004-G
URL : http://doi.org/10.1016/0022-0000(93)90004-g
Decomposition of Directed Graphs, SIAM Journal on Algebraic Discrete Methods, vol.3, issue.2, pp.214-228, 1982. ,
DOI : 10.1137/0603021
Multicut is FPT, Proceedings of the 43rd annual ACM symposium on Theory of computing, pp.459-468, 2011. ,
URL : https://hal.archives-ouvertes.fr/lirmm-00741933
Diamond-free circle graphs are Helly circle, Discrete Mathematics, vol.310, issue.4, pp.845-849, 2010. ,
DOI : 10.1016/j.disc.2009.09.022
URL : https://hal.archives-ouvertes.fr/lirmm-00432897
FPT algorithms and kernels for the Directed k-Leaf problem, Journal of Computer and System Sciences, vol.76, issue.2, pp.144-152, 2010. ,
DOI : 10.1016/j.jcss.2009.06.005
URL : https://hal.archives-ouvertes.fr/lirmm-00432901
Reducing multicut to bounded treewidth, 2010. ,
Well-Quasi-Order of Relabel Functions, Order, vol.54, issue.12, pp.301-315, 2010. ,
DOI : 10.1007/s11083-010-9174-0
URL : https://hal.archives-ouvertes.fr/lirmm-00806804
On Finding Directed Trees with Many Leaves, Parameterized and Exact Computation, pp.86-97, 2009. ,
DOI : 10.1007/978-3-642-11269-0_7
URL : https://hal.archives-ouvertes.fr/lirmm-00432919
Induced subgraphs and well-quasi-ordering, Journal of Graph Theory, vol.59, issue.4, pp.427-435, 1990. ,
DOI : 10.1002/jgt.3190140406
A Characterization of Circle Graphs, European Journal of Combinatorics, vol.5, issue.3, pp.223-238, 1984. ,
DOI : 10.1016/S0195-6698(84)80005-0
Satisfiability allows no nontrivial sparsification unless the polynomial-time hierarchy collapses, Proceedings of the 42nd ACM symposium on Theory of computing, STOC '10, pp.251-260, 2010. ,
Open problems ? parameterized complexity and approximation algorithms, Number 09511 in Dagstuhl Seminar Proceedings, pp.26-90, 2010. ,
Self-stabilizing systems in spite of distributed control, Communications of the ACM, vol.17, issue.11, pp.643-644, 1974. ,
DOI : 10.1145/361179.361202
Spanning trees with many leaves, Journal of Graph Theory, vol.13, issue.4, pp.189-197, 2001. ,
DOI : 10.1002/jgt.1013
Parameterized Complexity, pp.19-25, 1999. ,
DOI : 10.1007/978-1-4612-0515-9
URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.21.3797
An approximation algorithm for the maximum leaf spanning arborescence problem, ACM Transactions on Algorithms, vol.6, issue.3, pp.1-46, 2010. ,
DOI : 10.1145/1798596.1798599
On Intersection Graphs of Arcs and Chords in a Circle, p.12, 2000. ,
On a conjecture concerning helly circle graphs, Pesquisa Operacional, vol.23, issue.1, pp.221-229, 2003. ,
DOI : 10.1590/S0101-74382003000100016
Practical split decomposition via graphlabelled trees, 2010. ,
URL : https://hal.archives-ouvertes.fr/lirmm-00805193
Quasi linear-time circle graph recognition. Manuscript, p.12, 2010. ,
Intersection theorems for systems of sets, Journal of the London Mathematical Society, vol.35, pp.85-90, 1960. ,
Fixed-parameter tractability is polynomial-time extremal structure theory I: The case of max leaf, Proc. of ACiD, 2005. ,
FPT is P-time extremal structure I, Algorithms and Complexity in Durham 2005, Proceedings of the first ACiD Workshop, pp.1-41, 2005. ,
Improved Approximation Algorithms for Minimum Weight Vertex Separators, SIAM Journal on Computing, vol.38, issue.2, pp.629-657, 2008. ,
DOI : 10.1137/05064299X
URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.597.5634
A generalization of Nemhauser and Trotter's local optimization theorem, 26th International Symposium on Theoretical Aspects of Computer Science, pp.409-420, 2009. ,
Coordinatized kernels and catalytic reductions: An improved FPT algorithm for max leaf spanning tree and other problems, FSTTCS, pp.240-251, 2000. ,
Kernel(s) for problems with no kernel: On out-trees with many leaves, STACS, pp.421-432, 2009. ,
URL : https://hal.archives-ouvertes.fr/inria-00358112
Parameterized complexity theory, pp.19-25, 2006. ,
A Linear Vertex Kernel for Maximum Internal Spanning Tree, Algorithms and Computation, pp.275-282, 2009. ,
DOI : 10.1007/978-3-642-10631-6_29
URL : https://hal.archives-ouvertes.fr/lirmm-00432674
Solving Connected Dominating Set Faster than 2 n, Algorithmica, vol.40, issue.4, pp.153-166, 2008. ,
DOI : 10.1007/s00453-007-9145-z
URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.103.5347
Sharp separation and applications to exact and parameterized algorithms, pp.72-83, 2010. ,
DOI : 10.1007/978-3-642-12200-2_8
URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.157.3390
Bidimensionality and Kernels, SODA, pp.503-510, 2010. ,
DOI : 10.1137/1.9781611973075.43
URL : http://arxiv.org/abs/1606.05689
Infeasibility of instance compression and succinct PCPs for NP, Proceedings of the fourtieth annual ACM symposium on Theory of computing, STOC 08, pp.133-142, 2008. ,
DOI : 10.1145/1374376.1374398
Theory of Relations, volume 118 of Studies in Logic and the Foundations of Mathematics, 1986. ,
Recognizing circle graphs in polynomial time, Journal of the ACM, vol.36, issue.3, pp.435-473, 1989. ,
DOI : 10.1145/65950.65951
A short note on the approximability of the maximum leaves spanning tree problem, Information Processing Letters, vol.52, issue.1, pp.45-49, 1994. ,
DOI : 10.1016/0020-0190(94)90139-2
Approximate max-flow min-(multi)cut theorems and their applications, STOC, pp.698-707, 1993. ,
DOI : 10.1137/s0097539793243016
URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.51.3217
Primal-dual approximation algorithms for integral flow and multicut in trees, Algorithmica, vol.11, issue.5, pp.3-20, 1997. ,
DOI : 10.1007/BF02523685
Circle graph obstructions under pivoting, Journal of Graph Theory, vol.85, issue.8, pp.1-11, 2009. ,
DOI : 10.1002/jgt.20363
URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.100.8993
A logical approach to multicut problems, Information Processing Letters, vol.103, issue.4, pp.136-141, 2007. ,
DOI : 10.1016/j.ipl.2007.03.005
Spanning trees in graphs of minimum degree 4 or 5, Discrete Mathematics, vol.104, issue.2, pp.167-183, 1992. ,
DOI : 10.1016/0012-365X(92)90331-9
Finding topological subgraphs is fixed-parameter tractable, Proceedings of the 43rd annual ACM symposium on Theory of computing, STOC '11, pp.479-488, 2011. ,
DOI : 10.1145/1993636.1993700
URL : http://arxiv.org/abs/1011.1827
FPT Algorithms for Path-Transversals and Cycle-Transversals Problems in Graphs, IWPEC, pp.129-140, 2008. ,
DOI : 10.1007/978-3-540-79723-4_13
URL : https://hal.archives-ouvertes.fr/hal-00824933
Algorithm design techniques for parameterized graph modification problems, pp.28-33, 2006. ,
Complexity and Exact Algorithms for Multicut, In SOFSEM, pp.303-312, 2006. ,
DOI : 10.1007/11611257_28
URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.72.8925
Fixed-parameter tractability and data reduction for multicut in trees, Networks, vol.47, issue.3, pp.124-135, 2005. ,
DOI : 10.1002/net.20081
Invitation to data reduction and problem kernelization, ACM SIGACT News, vol.38, issue.1, pp.31-45, 2007. ,
DOI : 10.1145/1233481.1233493
Minimizing NLC-Width is NP-Complete, Graph-Theoretic Concepts in Computer Science, pp.69-80, 2005. ,
DOI : 10.1007/11604686_7
Ordering by Divisibility in Abstract Algebras, Proceedings of the London Mathematical Society, pp.3-2326, 1952. ,
DOI : 10.1112/plms/s3-2.1.326
URL : http://plms.oxfordjournals.org/cgi/content/short/s3-2/1/326
NLC2-Decomposition in Polynomial Time, Lecture Notes in Computer Science, vol.1665, pp.110-121, 1999. ,
DOI : 10.1007/3-540-46784-X_12
URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.139.8074
Recent developments on graphs of bounded clique-width, Second Workshop on Graph Classes, Optimization, and Width Parameters, pp.2747-2761, 2009. ,
DOI : 10.1016/j.dam.2008.08.022
Approximation through multicommodity flow, Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science, pp.726-737, 1990. ,
DOI : 10.1109/FSCS.1990.89595
Spanning Trees with Many Leaves, SIAM Journal on Discrete Mathematics, vol.4, issue.1, pp.99-106, 1991. ,
DOI : 10.1137/0404010
A New Algorithm for Finding Trees with Many Leaves, Algorithms and Computation, pp.270-281, 2008. ,
DOI : 10.1007/978-3-540-92182-0_26
Limits and Applications of Group Algebras for Parameterized Problems, Automata, Languages and Programming, pp.653-664, 2009. ,
DOI : 10.1007/978-3-642-02927-1_54
Well-quasiordering finite trees with gap-condition. Proof of Harvey Friedman's conjecture, Annals of Mathematics, vol.130, pp.215-226, 1989. ,
Well quasi ordering depends on the labels, Acta Scientiarum Mathematicarum, vol.55, pp.59-65, 1991. ,
An algorithm for planarity testing of graphs, Theory of Graphs: International Symposium: Rome, pp.215-232, 1966. ,
Rubber bands, convex embeddings and graph connectivity, Combinatorica, vol.13, issue.1, pp.91-102, 1988. ,
DOI : 10.1007/BF02122557
Parameterized graph separation problems, Theoretical Computer Science, vol.351, issue.3, pp.394-406, 2006. ,
DOI : 10.1016/j.tcs.2005.10.007
URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.102.9011
Chordal Deletion is Fixed-Parameter Tractable, Algorithmica, vol.2, issue.1, pp.747-768, 2010. ,
DOI : 10.1007/s00453-008-9233-8
URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.108.7774
Treewidth reduction for constrained separation and bipartization problems, STACS, pp.561-572, 2010. ,
URL : https://hal.archives-ouvertes.fr/inria-00455767
Constant Ratio Fixed-Parameter Approximation of the Edge Multicut Problem, ESA, pp.647-658, 2009. ,
DOI : 10.1007/978-3-642-04128-0_58
Fixed-parameter tractability of multicut parameterized by the size of the cutset, Proceedings of the 43rd annual ACM symposium on Theory of computing, pp.469-478, 2011. ,
Zur allgemeinen kurventheorie, Fundamenta Mathematicae, vol.10, pp.96-115, 1927. ,
DOI : 10.1007/bf02307626
Algorithmic Aspects of Graph Connectivity, 2008. ,
DOI : 10.1017/CBO9780511721649
Reconnaissance des graphes de cordes, Discrete Mathematics, vol.54, issue.3, pp.329-337, 1985. ,
DOI : 10.1016/0012-365X(85)90117-7
URL : http://doi.org/10.1016/0012-365x(85)90117-7
On well-quasi-ordering infinite trees, Mathematical Proceedings of the Cambridge Philosophical Society, vol.60, issue.03, pp.61697-720, 1965. ,
DOI : 10.2307/1993287
Invitation to Fixed Parameter Algorithms (Oxford Lecture Series in Mathematics and Its Applications), 2006. ,
Excluding a bipartite circle graph from line graphs, Journal of Graph Theory, vol.62, issue.2, pp.183-203, 2009. ,
DOI : 10.1002/jgt.20353
Un bel ordre d'abritement et ses rapports avec les bornes d'une multirelation . Comptes Rendus de l'Académie des Sciences, pp.1677-1680, 1972. ,
An Amortized Search Tree Analysis for k-Leaf Spanning Tree, SOFSEM, pp.672-684, 2010. ,
DOI : 10.1007/978-3-642-11266-9_56
Almost 2-SAT is fixed-parameter tractable, Journal of Computer and System Sciences, vol.75, issue.8, pp.435-450, 2009. ,
DOI : 10.1016/j.jcss.2009.04.002
Finding odd cycle transversals, Operations Research Letters, vol.32, issue.4, pp.299-301, 2004. ,
DOI : 10.1016/j.orl.2003.10.009
Graph minors. V. Excluding a planar graph, Journal of Combinatorial Theory, Series B, vol.41, issue.1, pp.92-114, 1986. ,
DOI : 10.1016/0095-8956(86)90030-4
URL : http://doi.org/10.1006/jctb.1999.1919
Graph minors. IV. Tree-width and well-quasi-ordering, Journal of Combinatorial Theory, Series B, vol.48, issue.2, pp.227-254, 1990. ,
DOI : 10.1016/0095-8956(90)90120-O
URL : http://doi.org/10.1006/jctb.1999.1919
Graph Minors .XIII. The Disjoint Paths Problem, Journal of Combinatorial Theory, Series B, vol.63, issue.1, pp.65-110, 1995. ,
DOI : 10.1006/jctb.1995.1006
URL : http://doi.org/10.1006/jctb.1999.1919
Graph Minors. XX. Wagner's conjecture, Journal of Combinatorial Theory, Series B, vol.92, issue.2, pp.325-357, 2004. ,
DOI : 10.1016/j.jctb.2004.08.001
Quickly Excluding a Planar Graph, Journal of Combinatorial Theory, Series B, vol.62, issue.2, pp.323-348, 1994. ,
DOI : 10.1006/jctb.1994.1073
URL : http://doi.org/10.1006/jctb.1994.1073
Graph Searching and a Min-Max Theorem for Tree-Width, Journal of Combinatorial Theory, Series B, vol.58, issue.1, pp.22-33, 1993. ,
DOI : 10.1006/jctb.1993.1027
Cut problems and their application to divide-and-conquer. In Approximation algorithms for NP-hard problems, 1996. ,
2-approximation for finding trees with many leaves, Proc. of ESA 1998, pp.441-452 ,
Recognition of Circle Graphs, Journal of Algorithms, vol.16, issue.2, pp.264-282, 1994. ,
DOI : 10.1006/jagm.1994.1012
Constructing full spanning trees for cubic graphs, Information Processing Letters, vol.13, issue.1, pp.8-11, 1981. ,
DOI : 10.1016/0020-0190(81)90141-1
On better-quasi-ordering countable series-parallel orders. Transactions of the, pp.2491-2505, 2000. ,
A quadratic kernel for feedback vertex set, Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '09, pp.115-119, 2009. ,
DOI : 10.1137/1.9781611973068.13
k-NLC graphs and polynomial algorithms, Discrete Applied Mathematics, vol.54, issue.2-3, pp.251-266, 1994. ,
DOI : 10.1016/0166-218X(94)90026-4
On calculating connected dominating set for efficient routing in ad hoc wireless networks, Proceedings of the 3rd international workshop on Discrete algorithms and methods for mobile computing and communications , DIALM '99, pp.7-14, 1999. ,
DOI : 10.1145/313239.313261