, Observe that is an equivalence relation. For a set A ? ?(H), we define reduce H (A) as the operation which returns a set containing one element of each equivalence class of A/ . The main idea of our algorithm is to call reduce H at each step of our dynamic programming algorithm in order to keep the size of a set of partial solutions manipulated small

, For every A ? ?(H), we have |reduce H (A)| ? n k · 2 k(log 2 (k)+1) and we can moreover compute reduce H (A)

. Proof, To prove that reduce H (A) ? n k · 2 k(log 2 (k)+1) , it is enough to bound the number of equivalence classes of

, First observe that i?[k] deg aux H (P) (v i ) = 2|V (H)| when P = ?, since each isolated vertex in H |P gives a loop in aux H (P). Moreover, when P contains an edge, removing an edge from a partial solution P of H increases i?[k] deg aux H (P) (v i ) by two; indeed, this edge removal splits a maximal path of H |P into two maximal paths. Therefore, any partial solution P satisfies that i?[k] deg aux H (P) (v i ) ? 2|V (H)|; in particular each vertex of aux H (P) has degree at most 2|V (H)|. As aux H (P) contains k vertices, We claim that, for every P ? ?(H), we have i?[k] deg aux H (P) (v i ) ? 2|V (H)|

.. .. Is, We conclude that partitions ?(H) into at most n k · 2 k(log 2 k+1) equivalences classes. It remains to prove that we can compute reduce H (A) in time O(|A| · nk log 2 (nk)). First observe that, for every P ? ?(H), we can compute aux H (P) in time O(nk). Moreover, we can also compute the degree sequence of aux H (P) and the connected components of aux H (P) in time O(nk). Thus, by using the right data structures, we can decide whether P 1 P 2 in time O(nk). Furthermore, by using a self-balancing binary search tree, we can compute reduce H (A) in time O(|A| · nk log 2 (|reduce H (A)|)). Since log 2 (|reduce H (A)|) ? k log 2 (2nk), Since the number of partitions of {v 1

, The rest of this section is dedicated to prove that, for a set of partial solutions A of H, the set reduce H (A) is equivalent to A, i.e., if A contains a partial solution that forms a Hamiltonian cycle with a complement solution, then reduce H (A) also. Our results are based on a kind of equivalence between Hamiltonian cycles and red-blue Eulerian trails

, If P ? ?(H) and Q ? ?(H) form a Hamiltonian cycle, then the multigraph aux H (P) aux H (Q) admits a red-blue Eulerian trail

, From the definitions of a partial solution and of a complement solution, Suppose that P ? ?(H) and Q ? ?(H) form a Hamiltonian cycle C. Let M := aux H (P) aux H (Q)

, Q are all the H-paths in G |Q

,. .. ?-p-1, P , Q appear in C in this order, ? for each x ? [ ], the first end-vertices of P x is the last end-vertex of Q x?1 and the last end-vertex of P x is the first end-vertex of Q x

, Since aux H (P) has the same set of connected components as aux H (P ), we know that aux H (P ) M is also connected. Moreover, for every i ? [k], we have deg aux H (P) (v i ) = deg aux H (P ) (v i ) = deg M

, ) M admits a red-blue Eulerian trail. Thus, for every P ? A and multigraph M with blue edges such that aux H (P) M admits a red-blue Eulerian trail, there exists P ? reduce H (A) such that aux H (P ) M admits a red-blue Eulerian trail. Hence

, One easily checks that H is a transitive relation. Now, assuming that A H B, we have reduce H (A) B because reduce H (A) H A

, Our algorithm computes recursively, for every k-labeled graph H arising in the k-expression of G, a set A H such that A H H ?(H) and |A H | ? n k · 2 k(log 2 (k)+1) . In order to prove the correctness of our algorithm

. Proof and . First, observe that H has the same set of vertices and edges as D. Thus, we have ?(H) = ?(D) and ?(H) = ?(D)

P. Let, To prove the lemma, it is sufficient to prove that there exists P ? A D such that aux H (P ) M contains a red-blue Eulerian trail. Let f be a bijective function such that ? for every edge e of aux D (P) with endpoints v and v i , for some , f (e) is an edge of aux H (P) with endpoints v and v j , and ? for every loop e with endpoint v i , f (e) is a loop of aux H (P) with endpoint v j . By construction of aux D (P) and aux H (P), such a function exists, We construct the multigraph M from M and T by successively doing the following: ? For every edge e of the multigraph aux D (P) with endpoints v and v i , take the subwalk W = (v , f (e)

, ? For every loop e with endpoint v i in the multigraph aux D (P), take the subwalk W =

, there exists P ? A D such that aux D (P ) M contains a red-blue Eulerian trail. Observe that aux H (P) (respectively M) is obtained from aux D (P ) (resp. M ) by replacing each edge associated with {v i , v k } or {v i } in aux D (P ) (resp. M ) with an edge associated with {v j , v k } or {v j } respectively, By construction, one can construct from T a red-blue Eulerian trail of aux D (P) M . Since A D D ?(D)

. Respectively, So, we can obtain a red-blue Eulerian trail of aux H (P ) M from a red-blue Eulerian trail of aux H (P) M by replacing (v i , e, v j ) with the sequence (v i , e 1 , v i , f, v j , e 2 , v j ) where f is the blue edge we add to M to obtain M . It implies the claim. Now, since A H B, there exists P ? A such that aux H (P ) M admits a red-blue Eulerian trail T . Let W be the subwalk of T such that W =

, Since every partial solution of H is obtained from the union of a partial solution of D and a subset of E H i,j of size at most n

. D-=-a-d-h-?, D) + 0(i, j)

, disjoint union of two graphs or relabeling cannot create a Hamiltonian cycle. Thus, by minimality, we have H = ? i,j (D) such that ? D is a k-labeled graph arising in ? and i

. ?-e(c)-?-e,

J. ?-e-h-i, Q. Let-p-:=-e(c)-?-e(d), and Q. :=-e(c)-?-e-h-i,j-.-observe-that-p-?-?(d), By Lemma 4.69, the multigraph aux D (P) aux D (Q) contains a red-blue Eulerian trail. Since A D D ?(D), there exists P ? A D such that aux D (P ) aux D (Q) contains a red-blue Eulerian trail. As Q ? E H i,j , we deduce that, for every ? [k] \ {i, j}, we have deg aux D (P ) (v ) = 0 and deg aux H (P ) (v i ) = deg aux H (P ) (v j ). For the other direction, suppose that the latter condition holds. Let Q be the graph on the vertex set V (G) such that it contains exactly deg aux H (P) (v i ) many edges between the set of vertices labeled i and the set of vertices labeled j. Observe that aux H (Q) consists of deg aux H (P) (v i ) many edges between v i and v j . Therefore

, By Lemma 4.68, for every A ? ?(H), we can compute reduce H (A) in time O(|A| · nk 2 log 2 (nk)). Observe that, for every k-labeled graph D arising in ? and such that A D is computed before A H , we have |A D | ? n k · 2 k(log 2 (k)+1) . It follows that: ? If H = D ? F , then we have |A D ? A F | ? n 2k · 2 2k(log 2 (k)+1) . Thus, we can compute A H := reduce H (A D ? A F ) in time O, Running time. Let H be a k-labeled graph arising in ?. Observe that if H = i(v) or H = ? i?j (D), then we compute A H in time O(1)

?. If and H. =-?-i,

, First observe that, for every partial solution P of H, we have |P + (i, j)| ? n 2 and we can compute the set P + (i, j) in time O(n 2 ). Moreover, by Lemma 4.68, for every ? {0, . . . , n ? 1}, we have |A D | ? n k · 2 k(log 2 (k)+1) and thus, we deduce that |A D + (i, j)| ? n k+2 ·2 k(log 2 (k)+1) and that A +1 D can be

. Thus,

. D-?-·-·-·-?-a-n-d-|-?,

, Since ? uses at most O(n) disjoint union operations and O(nk 2 ) unary operations

, M has a red-blue Eulerian trail

, M * has an Eulerian trail

, The underlying undirected graph of M * has at most one connected component containing an edge, and, for each vertex v of M * , deg + M * (v) = deg ? M * (v)

, Even though G is strongly connected, it does not have a red-blue Eulerian trail, and one can check that M * has two connected components containing an edge. To decide whether the underlying undirected graph of M * has at most one connected component containing an edge, multiple arcs are useless. So, it is enough to keep one partial solution P for each degree sequence in aux H (P) and for each set, the condition that "for each vertex v of M * , deg + M * (v) = deg ? M * (v)" can be translated to that, for each vertex v of M , the number of blue incoming arcs is the same as the number of red outgoing arcs, and the number of red incoming arcs is the same as the number of blue outgoing arcs

, v k }, deg + aux H (P 1 ) (v) = deg + aux H (P 2 ) (v) and deg ? aux H (P 1 ) (v) = deg ? aux

, If P 1 P 2 , then P 1 is part of a directed Hamiltonian cycle in G if and only if P 2 is part of a directed Hamiltonian cycle in G. From the definition of

. |-?-n-2k, Thus we can follow the lines of the proof for undirected graphs, and easily deduce that one can solve Directed Hamiltonian Cycle in time n O(k), vol.2

, The vertices of out-degree zero are called leaves of D. The Min Leaf Out-Branching problem asks for a given digraph D and an integer , whether there is a spanning out-tree of D with at most leaves. This problem generalizes Hamiltonian Path (and also Hamiltonian Cycle) by taking = 1. Ganian, Hlin?ný, and Obdr?álek  showed that there is an n 2 O(k) -time algorithm for solving Min Leaf Out-Branching problem

, In Section 5.2, we prove that one can compute in polynomial time the number of minimal transversals of ?-acyclic hypergraphs. In Subsection 5.2.3, we show that, as a corollary, we can count in polynomial time the minimal dominating sets of strongly chordal graphs and we discuss about the possible extensions of this corollary. In Subsection 5.2.4, we conclude this chapter by some open questions concerning the counting of minimal transversals and in particular, we discuss about how some existing parameters on hypergraphs could be used to extend our result. , some results are known concerning width measures. Arnborg, Lagergren, and Seese  extended Courcelle's theorem by showing that every counting problem expressible in MSO 2 are FPT parameterized by the tree-width of the input graph. Makowsky  proved that evaluating the Tutte polynomial is FPT when parameterized by tree-width. Moreover, Courcelle, Makowsky, and Rotics  generalized Theorem 2.52 by showing that every counting problem expressible in MSO 1 is FPT parameterized by the clique-width of the input graph. Efficient algorithms are known for the counting variants of NP-hard problems parameterized by tree-width. For example, Bodlaender et al.  designed a framework called determinant approach which provides 2 O(tw(G)) ·n time algorithms for the counting variants of many connectivity problems such as Hamiltonian Cycle and Steiner Tree. Recently, Golovach et al. [73, Theorem 36] proved the following theorem about counting the 1-minimal and 1-maximal (?, ?)-dominating sets with the d-neighbor-width as parameter. For a graph G, a (?, ?)-dominating set D ? V (G) of G is 1-maximal (resp. 1-minimal )

, The minimal transversals of H containing x are {a, x, c}, {a, x, d} and {b, x, d}. Observe that removing x from these sets directly yields a minimal transversal of H \ H(x) = {{a, b}, {c, d}}. However, adding x to a minimal transversal of H \ H(x) does not give necessarily a minimal transversal of H. For example

, In fact, we can show in general that T ? x ? mtr(H) if and only if T is a minimal transversal of H \ H(x) and T is not a transversal of H. Consequently, the number of minimal transversals of H containing x is #mtr(H \ H(x)) ? #(tr(H) ? mtr

H. Hypergraph and B. Sub-hypergraph-h-?-h, we define the B-blocked transversals of H to be the transversals T of H such that each vertex x of T has a private in H \ H (B). In particular, if y is a vertex of B, then y cannot be in a B-blocked transversal of H . Observe also that if y ? V (H) \ V (H ), then y cannot be in a B-blocked transversal of H . Fact 5.5. Given a sub-hypergraph H of a hypergraph H and B ? V (H), T is a B-blocked transversal of H if and only if T

, As an example, let H be the hypergraph depicted in Figure 5.1, H = H and B = {x}. The only B-blocked transversal of H is {b, c}. While {b, d} is a minimal transversal of H \ H(x), it is not a B-blocked transversal as the hyperedge {x, c} does not intersect {b, d}. We call the set H (B) the blocked hyperedges. Intuitively, H (B) is the set of hyperedges that cannot be used as privates in a transversal of H . We denote by btr B (H ) the set of B-blocked transversals of H

, Observe that by definition, mtr(H) = btr ? (H). Moreover, if H(B) = H = ?, then btr H (H) = ? as mtr(?) = {?} and ? / ? tr(H). When B = {x}, we denote btr B (H) by btr x (H)

S. Given, We extend this notation to mtr and btr as well. The following summarizes observations about blocked transversals, we denote by tr(H, S) := {T ? tr(H) : T ? S}

, Fact 5.6. Let H be a sub-hypergraph of a hypergraph H and B, S ? V (H). Then, 1. btr B (H ) = btr B?V (H ) (H )

S. Btr-b-(h-,-s)-=-btr-b-(h,

, One checks easily that #mtr(H) = #mtr(H, V (H) \ {x}) + #mtr(H)(x), Therefore, we have #mtr(H) = #btr ? (H) = #btr ? (H, V (H) \ {x}) + #btr ? (H \ H(x)) ? #btr x (H)

H. , ,. .. Vertices-x-1, and .. , x q such that, for all 1 ? i ? q, #btr x i (H i ) and #btr ? (H i ) can be computed in polynomial time if #btr x j (H j ) and #btr ? (H j ) are known for all j < i. As a consequence, one can compute #mtr(H) by classical dynamic programming for any ?-acyclic hypergraph. The end of this subsection is dedicated to the proof of several crucial lemmas concerning recursive formulas for computing the number of blocked transversals, and that will be useful in our algorithm

S. , B. , and ,. , Lemma 5.7. Let H be a hypergraph

, it means that there exists a hyperedge e ? H such that e?S = ?. In this case, btr B (H, S) = ?. Moreover, there exists i ? [1, k] such that C i = {?} and e ? H i . Thus, btr B (H i , S) = ? and the equality holds in this case

S. Let-t-?-btr-b-(h, We show that for all i ? k, T i =

S. ). T-?-v-(h-i-)-?-btr-b-(h-i and . Let-e-?-h-i-.-since-e-?-h,-we-have-e-?-t-=-?, Thus e ? T i = ?, that is, T i ? tr(H i , S). Moreover, let y ? T i . By definition of T , there exists e ? H \ H(B) such that e is private for y w.r.t. T . Observe that we have T i ? S ? V (H i ) = V (C i ) since T ? S. Thus, y ? V (C i ) and e ? S ? C i , because C i is a connected component

, It remains to show that T ? mtr(H \ H(B)). Let y ? T . By definition of T , there exists i such that y ? T i . Thus, there exists e ? H i \ H i (B) that is private for y w.r.t. T i . Moreover, since H i (B) = H i ? H(B), we know that e / ? H(B). As C 1 , . . . , C k are the connected component of H[S], we have that, for every j = i, V (C i ) ? V (C j ) = ?. Moreover, for all ? k, we have S ? V (H ) = V (C ), As H = k i=1 H i , there exists i such that e ? H i . Thus e ?T i = ? and thus e ?T = ?, that is, T ? tr(H)

E. Thus and . ?-t-=-e-?-t-i-=-{y}, other words, e is private for y w.r.t. T and H \ H(B). That is T ? btr B (H)

, We recall that btr B (H, S)(x) is the set of B-blocked transversals T of H such that T ? S and x ? T . The following lemma shows that for any B-blocked transversal T ? S of H containing x

, By definition, x ? T and T ? S, thus we only have to show that T = T \ {x} ? btr B (H 1 ), Proof. Let H, vol.1

. Let-y-?-t, In other words, T is a minimal transversal of H 1 \ H 1 (B) which concludes the proof. To complete the previous lemma, we show that for each B-blocked transversal T ? S of H \ H(x), we have T ? {x} is a B-blocked transversal of H if and only if T is not a (B ? {x}, Since T is a minimal transversal of H \ H(B), there exists e ? H \ H(B) such that , otherwise e would not be private for y w.r.t. T . Thus e ? H \ (H(B) ? H(x)), that is, e is private to y w.r.t. T in H 1 \ H 1 (B)

S. and B. ?-v-(h)-and-x-?-s, We have {{x}} btr B (H 1 , S \ {x}) \ btr B (H, S)(x) = {{x}} btr B?{x}

, We prove the lemma by proving first the left-to-right inclusion (Claim 5.9.1) and then the right-to-left inclusion (Claim 5.9.2)

, Claim 5.9.1. For every T ? {{x}} btr B (H 1 , S \ {x}) \ btr B (H, S)(x), we have T \ {x} ? btr B?{x}

, Assume towards a contradiction that T / ? tr(H 2 ), i.e., there exists e ? H 2 such that e ? T = ?. We prove that it implies T ? btr B (H, S)(x). First, observe that T ? tr(H), since T ? tr(H 1 ) = tr(H \ H(x)) and T = T ? {x}. Thus, we have e ? T = {x} and e ? H(x). As e ? H 2 = H \ (H(B) ? H(x)), we have e ? H \ H(B) and then e is a private hyperedge for x w.r.t. T and H \ H(B). Furthermore

. T-?-mtr, H 1 \ H 1 (B)). Thus, every vertex in T has a private hyperedge w.r.t. T and H \ H(B)

T. As and . Tr, we can conclude that T ? mtr(H \ H(B)). Finally, we have T ? S by assumption

, Let y ? T . Since T ? btr B (H 1 ), there exists f ? H 1 \ H 1 (B) such that f ? T = {y}. Since H 1 \ H 1 (B) = H 2 \ H 2 (B ? {x}), every y ? T have a private hyperedge in H 2 \ H 2 (B ? {x}), that is T ? mtr, )(x) which is a contradiction. Thus, T ? tr

, Claim 5.9.2. For every T ? {x} btr B?{x} (H 2 , S \ {x}), we have T ? {x} btr B (H 1 , S \ {x}) \ btr B (H, S)(x)

, For each 1 ? i ? n and each 1 ? j ? m, let tab[i, j, 0] be #btr ? (H x i e j , [? x i ]), and for each > i, let tab[i, j, ] be #btr x (H x i e j

H. Precompute and . Ej,

, ] from the recursive formula of #btr ? (H xi ej

, Compute tab[i, j, ] from the recursive formula of #btr x (H xi ej

A. Abboud, A. Backurs, and V. Williams, Tight hardness results for LCS and other sequence similarity measures, IEEE 56th Annual Symposium on Foundations of Computer Science, FOCS 2015, pp.59-78, 2015.

N. Alon, R. Yuster, and U. Zwick, Color-coding, J. ACM, vol.42, issue.4, pp.844-856, 1995.

S. Arnborg, J. Lagergren, and D. Seese, Easy problems for tree-decomposable graphs, J. Algorithms, vol.12, issue.2, pp.308-340, 1991.

A. Backurs and P. Indyk, Edit distance cannot be computed in strongly subquadratic time (unless SETH is false), SIAM J. Comput, vol.47, issue.3, pp.1087-1097, 2018.

R. Belmonte and M. Vatshelle, Graph classes with structured neighborhoods and algorithmic applications, Theoret. Comput. Sci, vol.511, pp.54-65, 2013.

B. Bergougnoux and . Mamadou-moustapha-kanté, More applications of the dneighbor equivalence: acyclic and connectivity constraints, 2018.

L. Hans and . Bodlaender, A linear time algorithm for finding tree-decompositions of small treewidth, Proceedings of the Twenty-Fifth Annual ACM Symposium on Theory of Computing, pp.226-234, 1993.

L. Hans and . Bodlaender, Treewidth: Characterizations, applications, and computations, Graph-Theoretic Concepts in Computer Science, 32nd International Workshop, pp.1-14, 2006.

L. Hans, M. Bodlaender, S. Cygan, J. Kratsch, and . Nederlof, Deterministic single exponential time algorithms for connectivity problems parameterized by treewidth, Inform. and Comput, vol.243, pp.86-111, 2015.

L. Hans, R. Bodlaender, F. V. Downey, D. Fomin, and . Marx, The Multivariate Algorithmic Revolution and Beyond -Essays Dedicated to Michael R. Fellows on the Occasion of His 60th Birthday, Lecture Notes in Computer Science, vol.7370, 2012.

L. Hans, P. Bodlaender, M. S. Grønås-drange, F. V. Dregi, D. Fomin et al., A c k n 5-approximation algorithm for treewidth, SIAM J. Comput, vol.45, issue.2, pp.317-378, 2016.

S. Kellogg, J. Booth, and . Johnson, Dominating sets in chordal graphs, SIAM J. Comput, vol.11, issue.1, pp.191-199, 1982.

R. B. Borie, R. G. Parker, and C. A. Tovey, Automatic generation of lineartime algorithms from predicate calculus descriptions of problems on recursively constructed graph families, Algorithmica, vol.7, issue.5&6, pp.555-581, 1992.

A. Brandstädt, J. P. Van-bang-le, and . Spinrad, Graph classes: a survey, SIAM Monographs on Discrete Mathematics and Applications. Society for Industrial and Applied Mathematics (SIAM), 1999.

J. Brault-baron, F. Capelli, and S. Mengel, Understanding model counting for beta-acyclic cnf-formulas, 32nd International Symposium on Theoretical Aspects of Computer Science, STACS 2015, pp.143-156, 2015.

B. Bui-xuan, O. Suchý, J. A. Telle, and M. Vatshelle, Feedback vertex set on graphs of low clique-width, European J. Combin, vol.34, issue.3, pp.666-679, 2013.

B. Bui-xuan, J. A. Telle, and M. Vatshelle, Boolean-width of graphs, IWPEC, pp.61-74, 2009.

B. Bui-xuan, J. A. Telle, and M. Vatshelle, Fast dynamic programming for locally checkable vertex subset and vertex partitioning problems, Theoret. Comput. Sci, vol.511, pp.66-76, 2013.

K. Cameron, Induced matchings, Discrete Applied Mathematics, vol.24, issue.1-3, pp.97-102, 1989.

F. Capelli, Structural restrictions of CNF formulas: application to model counting and knowledge compilation, 2016.

F. Capelli, Understanding the complexity of #sat using knowledge compilation, 32nd Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2017, pp.1-10, 2017.

L. Chandran, D. Issac, and A. Karrenbauer, On the parameterized complexity of biclique cover and partition, 11th International Symposium on Parameterized and Exact Computation, IPEC 2016, vol.11, pp.1-11, 2016.

S. Chechik, D. H. Larkin, L. Roditty, and G. Schoenebeck, Robert Endre Tarjan, and Virginia Vassilevska Williams. Better approximation algorithms for the graph diameter, Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014, pp.1041-1052, 2014.

J. Cheetham, K. H. Frank, A. Dehne, U. Rau-chaplin, P. J. Stege et al., Solving large FPT problems on coarse-grained parallel machines, J. Comput. Syst. Sci, vol.67, issue.4, pp.691-706, 2003.

A. Stephen and . Cook, The complexity of theorem-proving procedures, Proceedings of the 3rd Annual ACM Symposium on Theory of Computing, pp.151-158, 1971.

D. G. Corneil, M. Habib, J. Lanlignel, B. Reed, and U. Rotics, Polynomial-time recognition of clique-width ? 3 graphs, Discrete Appl. Math, vol.160, issue.6, pp.834-865, 2012.

D. G. Corneil and U. Rotics, On the relationship between clique-width and treewidth, SIAM J. Comput, vol.34, issue.4, pp.825-847, 2005.

B. Courcelle, J. A. Makowsky, and U. Rotics, Linear time solvable optimization problems on graphs of bounded clique-width, Theory Comput. Syst, vol.33, issue.2, pp.125-150, 2000.

B. Courcelle, The monadic second-order logic of graphs III: tree-decompositions, minor and complexity issues, ITA, vol.26, pp.257-286, 1992.

B. Courcelle, The monadic second-order logic of graphs XVI : Canonical graph decompositions, Logical Methods in Computer Science, vol.2, issue.2, 2006.

B. Courcelle and J. Engelfriet, Graph structure and monadic second-order logic, of Encyclopedia of Mathematics and its Applications, vol.138, 2012.

B. Courcelle, J. Engelfriet, and G. Rozenberg, Handle-rewriting hypergraph grammars, J. Comput. Syst. Sci, vol.46, issue.2, pp.218-270, 1993.

B. Courcelle, J. A. Makowsky, and U. Rotics, On the fixed parameter complexity of graph enumeration problems definable in monadic second-order logic, Discrete Applied Mathematics, vol.108, issue.1-2, pp.23-52, 2001.

B. Courcelle and M. Mosbah, Monadic second-order evaluations on treedecomposable graphs, Theor. Comput. Sci, vol.109, issue.1&2, pp.49-82, 1993.

B. Courcelle and S. Olariu, Upper bounds to the clique width of graphs, Discrete Applied Mathematics, vol.101, issue.1-3, pp.77-114, 2000.

R. Curticapean, The simple, little and slow things count: on parameterized counting complexity, 2015.

M. Cygan, V. Fedor, L. Fomin, D. Kowalik, D. Lokshtanov et al., Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms, 2015.

M. Cygan, J. Nederlof, M. Pilipczuk, M. Pilipczuk, J. M. Van-rooij et al., Solving connectivity problems parameterized by treewidth in single exponential time (extended abstract), 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science-FOCS 2011, pp.150-159, 2011.

G. Damiand, M. Habib, and C. Paul, A simple paradigm for graph recognition: application to cographs and distance hereditary graphs, Theor. Comput. Sci, vol.263, issue.1-2, pp.99-111, 2001.

, of Graduate texts in mathematics, Graph Theory, vol.173, 2012.

G. Rodney, M. R. Downey, and . Fellows, Fixed-parameter intractability, Proceedings of the Seventh Annual Structure in Complexity Theory Conference, pp.36-49, 1992.

G. Rodney, M. R. Downey, and . Fellows, Parameterized Complexity. Monographs in Computer Science, 1999.

G. Rodney, M. R. Downey, and . Fellows, Fundamentals of parameterized complexity. Texts in Computer Science, 2013.

A. Durand and M. Hermann, On the counting complexity of propositional circumscription, Inform. Process. Lett, vol.106, issue.4, pp.164-170, 2008.

T. Eiter and G. Gottlob, Identifying the minimal transversals of a hypergraph and related problems, SIAM J. Comput, vol.24, issue.6, pp.1278-1304, 1995.

T. Eiter and G. Gottlob, Hypergraph transversal computation and related problems in logic and AI, European Workshop on Logics in Artificial Intelligence, pp.549-564, 2002.

T. Eiter, K. Makino, and G. Gottlob, Computational aspects of monotone dualization: a brief survey, Discrete Appl. Math, vol.156, issue.11, pp.2035-2049, 2008.

M. Khaled, R. Elbassioni, S. Raman, R. Ray, and . Sitters, On the approximability of the maximum feasible subsystem problem with 0/1-coefficients, Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms, pp.1210-1219, 2009.

W. Espelage, F. Gurski, and E. Wanke, How to solve NP-hard graph problems on clique-width bounded graphs in polynomial time, Graph-theoretic concepts in computer science, vol.2204, pp.117-128, 2001.

S. Fafianie, H. L. Bodlaender, and J. Nederlof, Speeding up dynamic programming with representative sets: An experimental evaluation of algorithms for steiner tree on tree decompositions, Algorithmica, vol.71, issue.3, pp.636-660, 2015.

M. Farber, Domination, independent domination, and duality in strongly chordal graphs, Discrete Applied Mathematics, vol.7, issue.2, pp.115-130, 1984.

M. R. Fellows, Parameterized complexity: The main ideas and some research frontiers, Algorithms and Computation, 12th International Symposium, pp.291-307, 2001.

M. R. Fellows, Parameterized complexity: The main ideas and connections to practical computing, Electr. Notes Theor. Comput. Sci, vol.61, pp.1-19, 2002.

M. R. Fellows, F. A. Rosamond, U. Rotics, and S. Szeider, Clique-width minimization is np-hard, Proceedings of the 38th Annual ACM Symposium on Theory of Computing, pp.354-362, 2006.

P. Festa, P. M. Pardalos, and M. G. Resende, Feedback set problems, Christodoulos A. Floudas and Panos M. Pardalos, pp.1005-1016, 2009.

H. Fleischner, Eulerian graphs and related topics, vol.1, 1990.

J. Flum and M. Grohe, The parameterized complexity of counting problems, SIAM J. Comput, vol.33, issue.4, pp.892-922, 2004.

V. Fedor, P. A. Fomin, D. Golovach, S. Lokshtanov, and . Saurabh, Intractability of clique-width parameterizations, SIAM J. Comput, vol.39, issue.5, pp.1941-1956, 2010.

V. Fedor, P. A. Fomin, D. Golovach, S. Lokshtanov, and . Saurabh, Almost optimal lower bounds for problems parameterized by clique-width, SIAM J. Comput, vol.43, issue.5, pp.1541-1563, 2014.

V. Fedor, P. A. Fomin, D. Golovach, S. Lokshtanov, M. Saurabh et al., Clique-width III: hamiltonian cycle and the odd case of graph coloring, ACM Trans. Algorithms, vol.15, issue.1, p.27, 2019.

V. Fedor, P. A. Fomin, J. Golovach, and . Raymond, On the tractability of optimization problems on h-graphs, 26th Annual European Symposium on Algorithms, vol.30, 2018.

V. Fedor, D. Fomin, F. Lokshtanov, S. Panolan, and . Saurabh, Efficient computation of representative families with applications in parameterized and exact algorithms, J. ACM, vol.63, issue.4, 2016.

L. Fortnow, The status of the P versus NP problem, Commun. ACM, vol.52, issue.9, pp.78-86, 2009.

V. Froidure, Rangs des relations binaires et semigroupes de relations non ambigus, 1995.

R. Ganian and P. Hlin?ný, On parse trees and Myhill-Nerode-type tools for handling graphs of bounded rank-width, Discrete Appl. Math, vol.158, issue.7, pp.851-867, 2010.

R. Ganian, P. Hlin?ný, and J. Obdr?álek, Clique-width: when hard does not mean impossible, 28th International Symposium on Theoretical Aspects of Computer Science, vol.9, pp.404-415, 2011.

R. Ganian, P. Hlin?ný, and J. Obdr?álek, A unified approach to polynomial algorithms on graphs of bounded (bi-)rank-width, European J. Combin, vol.34, issue.3, pp.680-701, 2013.

M. R. Garey and D. S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness, 1979.

M. R. Garey, D. S. Johnson, G. L. Miller, and C. H. Papadimitriou, The complexity of coloring circular arcs and chords, SIAM J. Matrix Analysis Applications, vol.1, issue.2, pp.216-227, 1980.

J. F. Geelen, A. M. Gerards, and G. Whittle, Branch-width and well-quasiordering in matroids and graphs, J. Comb. Theory, Ser. B, vol.84, issue.2, pp.270-290, 2002.

B. Godlin, T. Kotek, and J. A. Makowsky, Evaluations of graph polynomials, Graph-Theoretic Concepts in Computer Science, 34th International Workshop, pp.183-194, 2008.

R. Leslie-ann-goldberg, J. Gysel, and . Lapinskas, Approximately counting locallyoptimal structures, J. Comput. Syst. Sci, vol.82, issue.6, pp.1144-1160, 2016.

P. A. Golovach, P. Heggernes, D. Mamadou-moustapha-kanté, . Kratsch, H. Sigve et al., Output-polynomial enumeration on graphs of bounded (local) linear mim-width, Algorithmica, pp.1-28, 2017.

P. A. Golovach, D. Lokshtanov, S. Saurabh, and M. Zehavi, Cliquewidth III: the odd case of graph coloring parameterized by cliquewidth, Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, pp.262-273, 2018.

M. Charles and G. , Algorithmic graph theory and perfect graphs, Elsevier, vol.57, 2004.

M. Charles-golumbic and U. Rotics, On the clique-width of some perfect graph classes, Int. J. Found. Comput. Sci, vol.11, issue.3, pp.423-443, 2000.

G. Gottlob, N. Leone, and F. Scarcello, Hypertree decompositions and tractable queries, J. Comput. Syst. Sci, vol.64, issue.3, pp.579-627, 2002.

G. Gottlob and R. Pichler, Hypergraphs in model checking: Acyclicity and hypertree-width versus clique-width, SIAM J. Comput, vol.33, issue.2, pp.351-378, 2004.

G. Gottlob and S. Szeider, Fixed-parameter algorithms for artificial intelligence, constraint satisfaction and database problems, Comput. J, vol.51, issue.3, pp.303-325, 2008.

P. Hlinený, A parametrized algorithm for matroid branch-width, SIAM J. Comput, vol.35, issue.2, pp.259-277, 2005.

, Petr Hlin?ný and Sang-il Oum. Finding branch-decompositions and rank-decompositions, SIAM J. Comput, vol.38, issue.3, pp.1012-1032, 2008.

R. Impagliazzo and R. Paturi, On the complexity of k-sat, J. Comput. Syst. Sci, vol.62, issue.2, pp.367-375, 2001.

R. Impagliazzo, R. Paturi, and F. Zane, Which problems have strongly exponential complexity?, 39th Annual Symposium on Foundations of Computer Science, FOCS '98, pp.653-663, 1998.

L. Jaffke, J. A. O-joung-kwon, and . Telle, A note on the complexity of feedback vertex set parameterized by mim-width, 2017.

L. Jaffke, J. A. O-joung-kwon, and . Telle, Polynomial-time algorithms for the longest induced path and induced disjoint paths problems on graphs of bounded mim-width, 12th International Symposium on Parameterized and Exact Computation, vol.21, pp.1-21, 2017.

L. Jaffke, J. A. O-joung-kwon, and . Telle, A unified polynomial-time algorithm for feedback vertex set on graphs of bounded mim-width, 35th Symposium on Theoretical Aspects of Computer Science, STACS 2018, vol.42, pp.1-42, 2018.

M. Järvisalo, D. L. Berre, O. Roussel, and L. Simon, The international SAT solver competitions, vol.33, 2012.

V. Jelínek, The rank-width of the square grid, Discrete Applied Mathematics, vol.158, issue.7, pp.841-850, 2010.

J. Jeong, E. J. Kim, and S. Oum, Finding branch-decompositions of matroids, hypergraphs, and more, 45th International Colloquium on Automata, Languages, and Programming, vol.80, 2018.

T. Jiang and B. Ravikumar, Minimal NFA problems are hard, SIAM J. Comput, vol.22, issue.6, pp.1117-1141, 1993.

O. Dong-yeap-kang, . Kwon, J. F. Torstein, J. A. Strømme, and . Telle, A width parameter useful for chordal and co-comparability graphs, Theor. Comput. Sci, vol.704, pp.1-17, 2017.

K. Mamadou-moustapha, The rank-width of directed graphs, 2007.

V. Mamadou-moustapha-kanté, A. Limouzy, L. Mary, and . Nourine, On the enumeration of minimal dominating sets and related notions, SIAM J. Discrete Math, vol.28, issue.4, pp.1916-1929, 2014.

V. Mamadou-moustapha-kanté, A. Limouzy, L. Mary, T. Nourine, and . Uno, On the enumeration and counting of minimal dominating sets in interval and permutation graphs, Algorithms and computation, vol.8283, pp.339-349, 2013.

M. Mamadou, M. Kanté, and . Rao, The rank-width of edge-coloured graphs, Theory Comput. Syst, vol.52, issue.4, pp.599-644, 2013.

M. Mamadou, T. Kanté, and . Uno, Counting minimal dominating sets, Theory and Applications of Models of Computation -14th Annual Conference, pp.333-347, 2017.

R. M. Karp, Reducibility among combinatorial problems, Proceedings of a symposium on the Complexity of Computer Computations, pp.85-103, 1972.

K. H. Kim, Boolean matrix theory and applications, vol.70, 1982.

D. Kobler and U. Rotics, Polynomial algorithms for partitioning problems on graphs with fixed clique-width (extended abstract), Proceedings of the Twelfth Annual Symposium on Discrete Algorithms, pp.468-476, 2001.

D. Kobler and U. Rotics, Edge dominating set and colorings on graphs with fixed clique-width, Discrete Applied Mathematics, vol.126, issue.2-3, pp.197-221, 2003.

A. Kotzig, Moves without forbidden transitions in a graph. Matematick? ?asopis, vol.18, pp.76-80, 1968.

M. A. Langston, Practical FPT implementations and applications (invited talk), First International Workshop, pp.291-292, 2004.

M. A. Langston, A. D. Perkins, A. M. Saxton, J. A. Scharff, and B. H. Voy, Innovative computational methods for transcriptomic data analysis: A case study in the use of FPT for practical algorithm design and implementation, Comput. J, vol.51, issue.1, pp.26-38, 2008.

L. Levin, Universal sequential search problems, Problemy Peredachi Informatsii, vol.9, issue.3, pp.115-116, 1973.

J. A. Makowsky, Colored tutte polynomials and kaufman brackets for graphs of bounded tree width, Proceedings of the Twelfth Annual Symposium on Discrete Algorithms, pp.487-495, 2001.

J. A. Makowsky, U. Rotics, I. Averbouch, and B. Godlin, Computing graph polynomials on graphs of bounded clique-width, Graph-Theoretic Concepts in Computer Science, 32nd International Workshop, pp.191-204, 2006.

R. Matai and S. Singh, Traveling salesman problem: an overview of applications, formulations, and solution approaches, Traveling Salesman Problem, 2010.

S. Mengel, Lower bounds on the mim-width of some graph classes, Discrete Applied Mathematics, 2017.

P. Montealegre and I. Todinca, On distance-d independent set and other problems in graphs with "few" minimal separators, Graph-Theoretic Concepts in Computer Science -42nd International Workshop, pp.183-194, 2016.

H. Moser and S. Sikdar, The parameterized complexity of the induced matching problem, Discrete Applied Mathematics, vol.157, issue.4, pp.715-727, 2009.

K. Mulmuley, V. Umesh, V. V. Vazirani, and . Vazirani, Matching is as easy as matrix inversion, Combinatorica, vol.7, issue.1, pp.105-113, 1987.

Y. Okamoto, T. Uno, and R. Uehara, Counting the number of independent sets in chordal graphs, J. Discrete Algorithms, vol.6, issue.2, pp.229-242, 2008.

S. Ordyniak, D. Paulusma, and S. Szeider, Satisfiability of acyclic and almost acyclic CNF formulas, Theor. Comput. Sci, vol.481, pp.85-99, 2013.

J. Orlin, Contentment in graph theory: Covering graphs with cliques. Indagationes Mathematicae (Proceedings), vol.80, pp.406-424, 1977.

S. Oum, Graphs of Bounded Rank Width, 2005.

S. Oum, Rank-width and vertex-minors, J. Combin. Theory Ser. B, vol.95, issue.1, pp.79-100, 2005.

S. Oum, Rank-width is less than or equal to branch-width, Journal of Graph Theory, vol.57, issue.3, pp.239-244, 2008.

S. Oum, Approximating rank-width and clique-width quickly, ACM Trans. Algorithms, vol.5, issue.1, 2009.

S. Oum, Rank-width: Algorithmic and structural results, Discrete Applied Mathematics, vol.231, pp.15-24, 2017.

S. Sang-il-oum, M. Hortemo-saether, and . Vatshelle, Faster algorithms for vertex partitioning problems parameterized by clique-width, Theoret. Comput. Sci, vol.535, pp.16-24, 2014.

P. Sang-il-oum and . Seymour, Approximating clique-width and branch-width, J. Combin. Theory Ser. B, vol.96, issue.4, pp.514-528, 2006.

R. Paige and . Robert-e-tarjan, Three partition refinement algorithms, SIAM Journal on Computing, vol.16, issue.6, pp.973-989, 1987.

M. Pilipczuk, Problems parameterized by treewidth tractable in single exponential time: A logical approach, Mathematical Foundations of Computer Science 2011 -36th International Symposium, MFCS 2011, pp.520-531, 2011.

J. , S. Provan, and M. O. Ball, The complexity of counting cuts and of computing the probability that a graph is connected, SIAM J. Comput, vol.12, issue.4, pp.777-788, 1983.

M. Rao, Décompositions de Graphes et Algorithmes Efficaces, 2006.

N. Robertson and P. D. Seymour, Graph minors. II. algorithmic aspects of tree-width, J. Algorithms, vol.7, issue.3, pp.309-322, 1986.

N. Robertson and P. D. Seymour, Graph minors. x. obstructions to tree-decomposition, J. Comb. Theory, Ser. B, vol.52, issue.2, pp.153-190, 1991.

H. Sigve, M. Saether, and . Vatshelle, Hardness of computing width parameters based on branch decompositions over the vertex set, Electronic Notes in Discrete Mathematics, vol.49, pp.301-308, 2015.

J. , A. Telle, and A. Proskurowski, Algorithms for vertex partitioning problems on partial k-trees, SIAM J. Discrete Math, vol.10, issue.4, pp.529-550, 1997.

L. G. Valiant, The complexity of computing the permanent, Theoret. Comput. Sci, vol.8, issue.2, pp.189-201, 1979.

L. G. Valiant, The complexity of enumeration and reliability problems, SIAM J. Comput, vol.8, issue.3, pp.410-421, 1979.

M. Vatshelle, New width parameters of graphs, 2012.

V. Vassilevska and W. , Multiplying matrices faster than coppersmith-winograd, Proceedings of the 44th Symposium on Theory of Computing Conference, STOC 2012, pp.887-898, 2012.

V. Vassilevska and W. , Hardness of easy problems: Basing hardness on popular conjectures such as the strong exponential time hypothesis, 10th International Symposium on Parameterized and Exact Computation, IPEC 2015, pp.17-29, 2015.

. Chain-chin-yen, C. T. Richard, and . Lee, The weighted perfect domination problem and its variants, Discrete Applied Mathematics, vol.66, issue.2, pp.147-160, 1996.