K. R. Abrahamson and M. R. Fellows. Finite automata, bounded treewidth and well-quasiordering. In N. Robertson and P. Seymour, editors, Proceedings of the AMS Summer Workshop on Graph Minors, Graph Structure Theory, Contemporary Mathematics vol. 147, pages 539--564. American Mathematical Society, 1993.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....on the maximum degree of vertices in the graph. Finite index corresponds to finite state : there exists a linear time algorithm that decides the property on graphs, given with a tree decomposition of bounded treewidth. Moreover, this algorithm is of a special, well described structure. See e.g. [1]. Note that a reduction rule (H 1 ; H 2 ) 2 R is safe for a property P if and only if H 1 P;l H 2 (if H 1 and H 2 are l terminal graphs) Below, we give a lemma on the existence of subgraphs of a certain size and type in graphs with bounded treewidth. This lemma will be used to show that there ....

K. R. Abrahamson and M. R. Fellows. Finite automata, bounded treewidth and wellquasiordering. In Proceedings of the AMS Summer Workshop on Graph Minors, Graph Structure Theory, Contemporary Mathematics vol. 147, pages 539--564. American Mathematical Society, 1993.

....improved by Fellows and Langston in [25] where, among others, they prove that, for any fixed k, an O(n 3) algorithm can be constructed checking whether a graph has cutwidth at most k. Furthermore, a technique introduced in [24] see also [6] further reduced the bound to O(n2) while in [1] a general method is given to construct a linear time algorithm that decides whether a given graph has cutwidth at most k, for k constant. Finally, in [22] an explicit constructive linear time algorithm was presented able to output the optimal vertex ordering in case of a positive answer. This ....

....B(j 1) 3(j 1) 3(j) 1, and 3.B(j) B(j 1) 3l(j 1) Let A : a, a] and B : b, b, be two sequences in S. We say that A B if r = r2 and V i i ai hi. In general, we say that A B if there exist extensions 2i C (A) and C (B) such that A . For example if A = [1, 7, 2, 6, 4] 9 and B = 5, 7, 3, 8] then A B because = 5,7,7,7,4,8,8,8,8] is an extension of B, 21 = 1,7,2,6,4,4,4,4,4] is an extension of A, and 21 . The following lemma corresponds to Corollary 3.11 of [12] LEMMA 2.8. If A and B are two sequences then A B if and only if (A) 4 (B) Suppose ....

[Article contains additional citation context not shown here]

K. R. Abrahamson and M. R. Fellows. Finite automata, bounded treewidth and well-quasiordering. In IN. Robertson and P. Seymour, editors, Proceedings of the AMS Summer Workshop on Graph Minors, Graph Structure Theory, Contemporary Mathematics vol. 147 , pages 539 564. American 34 Mathematical Society, 1993.

.... paper as an important intermediate step to obtain explicit and constructive algorithms that solve the treewidth k and pathwidth k problems in linear time (k fixed) Results of a similar nature as ours were independently obtained by Lagergren and Arnborg [31] and by Abrahamson and Fellows [1]. It should be noted, that for k = 1; 2; 3; 4, linear time and space algorithms based on graph rewriting exist for the treewidth k problem [6, 33, 42] We also solve a different, related problem, with basically the same algorithms: for each constant k, we have a polynomial time algorithm, ....

....Use the partial ordering OE on characterizations in full sets, or stronger forms of such partial orderings, and try to remove many characteristics from full sets that are dominated by other better characteristics in that full set. ffl Use (Myhill Nerode) state reduction techniques. See e.g. [7, 1]. ffl Use memoization ; i.e. do not compute full sets at once, but always try to find characteristics that can be included in a (not yet full) set of a node i that is as close to the root as possible. As soon as we find a characteristic in the set of the root of T we are done. This approach may ....

K. R. Abrahamson and M. R. Fellows. Finite automata, bounded treewidth and well-quasiordering. In Graph Structure Theory, Contemporary Mathematics vol. 147, pages 539--564. American Mathematical Society, 1993.

.... k 2 ) O(k 2 n k 3 ) steps (take in mind that if G is a partial k tree then E(G) O(kjV (G)j) 2 We mention that the property of being a core is an EMS property (i.e. involves counting or summing evaluations over sets de nable on monadic second order logic) and therefore, the results in [CMR,ALS91,AF93] imply the existence of a polynomial time algorithm for deciding it. So far, no explicit algorithm has been reported for this problem. We consider the isomorphism problem on cores. Notice that graphs G and H are isomorphic i jE(G; H)j jE(H; G)j 1. This check can be done in polynomial time ....

K.R. Abrahamson and M.R. Fellows. Finite automata, bounded treewidth and well-quasiordering. In N. Robertson and P. Seymour, editors, Proceedings of the AMS Summer Workshop on Graph Minors, Graph Structure Theory, Contemporary Mathematics vol. 147, pages 539{ 564. American Mathematical Society, 1993.

....at most w. # # Because of the large hidden constant in the complexity of linear time algorithm of Theorem 2.5, this theorem does not provide practical algorithms. It still provides a simple way to determine if a property is linear time decidable on partial k trees. Abrahamson and Fellows [AF93] gave a more straightforward automata theoretic proof of Courcelle s theorem. Unfortunately, the analogue of Courcelle s theorem does not hold for NP complete problems which have a monadic second order definition on graphs of locally bounded treewidth. Instead, there is a similar theorem for a ....

Karl Abrahamson and Michael R. Fellows. Finite automata, bounded treewidth and well-quasiordering. In Graph structure theory (Seattle, WA,

.... k 2 ) O(k 2 n k 3 ) steps (take in mind that if G is a partial k tree then E(G) O(kjV (G)j) We mention that the property of being a core is an EMS property (i.e. involves counting or summing evaluations over sets de nable on monadic second order logic) and therefore, the results in [Cou90b, ALS91, AF93] imply the existence of a polynomial time algorithm for deciding it. So far, no explicit algorithm has been reported for this problem. We consider the isomorphism problem on cores. Notice that graphs G and H are isomorphic i jE(G; H)j jE(H; G)j 1. This check can be done in polynomial time ....

Karl R. Abrahamson and Michael R. Fellows. Finite automata, bounded treewidth and well-quasiordering. In N. Robertson and P. Seymour, editors, Proceedings of the AMS Summer Workshop on Graph Minors, Graph Structure Theory, Contemporary Mathematics vol. 147, pages 539-564. American Mathematical Society, 1993.

....improved by Fellows and Langston in [23] where, among others, they prove that, for any xed k, an O(n 3 ) algorithm can be constructed checking whether a graph has cutwidth at most k. Furthermore, a technique introduced in [22] see also [6] further reduced the bound to O(n 2 ) while in [1] a general method is given to construct a linear time algorithm that decides whether a given graph has cutwidth at most k, for k constant. Finally, in [47] an explicit constructive linear time algorithm was presented able to output the optimal vertex ordering in case of a positive answer. This ....

....Proof. Let r i = jl i j; i = 1; 2 and = j j = jSj. As in the proof of Lemma 3.1, we set G i = V (G) E(G i ) i = 1; 2 and we observe that l is a vertex ordering for both G i ; i = 1; 2 where jlj = r 1 r 2 . We use the notations Q i = QG i ;l (0) 24 (Q h 1 ) [ 6 9 1 ] Q h 1 = Q h 1 [ Q h 1 (1) Q h 1 (2) z 6 8 7 6 6 9 7 9 Q h 1 [ Q h 1 (2) Q h 1 (2) z 6 7 4 7 3 8 1 ] B h 1 = 6 6 9 9 9 1 ] l = m h 2 z m h 5 z ] Q h 1 ( Q h 1 (1) Q h 1 ( Q h 1 (2) Q ....

[Article contains additional citation context not shown here]

K. R. Abrahamson and M. R. Fellows. Finite automata, bounded treewidth and well-quasiordering. In N. Robertson and P. Seymour, editors, Proceedings of the AMS Summer Workshop on Graph 34 Minors, Graph Structure Theory, Contemporary Mathematics vol. 147, pages 539-564. American Mathematical Society, 1993.

....improved by Fellows and Langston in [11] where, among others, they prove that for any xed k, a O(n 3 ) algorithm can be constructed checking whether a graph has cutwidth at most k. Furthermore, a technique introduced in [10] see also [3] further reduced the bound to O(n 2 ) while in [1] it is given a general method to construct a linear time algorithm that decides whether a given graph has cutwidth at most k, for k constant. However the methodology in [1] gives only a decision algorithm: it does not give any way to construct, in the case of a positive answer, the corresponding ....

....at most k. Furthermore, a technique introduced in [10] see also [3] further reduced the bound to O(n 2 ) while in [1] it is given a general method to construct a linear time algorithm that decides whether a given graph has cutwidth at most k, for k constant. However the methodology in [1] gives only a decision algorithm: it does not give any way to construct, in the case of a positive answer, the corresponding vertex ordering. In this paper, we give an explicit description, for any k 1, of a linear time algorithm that checks whether an input graph G has cutwidth k and, if this ....

K. R. Abrahamson and M. R. Fellows. Finite automata, bounded treewidth and well-quasiordering. In N. Robertson and P. Seymour, editors, Proceedings of the AMS Summer Workshop on Graph Minors, Graph Structure Theory, Contemporary Mathematics vol. 147, pages 539-564. American Mathematical Society, 1993.

....improved by Fellows and Langston in [11] where, among others, they prove that for any fixed k, a O(n 3 ) algorithm can be constructed checking whether a graph has cutwidth at most k. Furthermore, a technique introduced in [10] see also [3] further reduced the bound to O(n 2 ) while in [1] it is given a general method to construct a linear time algorithm that decides whether a given graph has cutwidth at most k, for k constant. However the methodology in [1] gives only a decision algorithm: it does not give any way to construct, in the case of a positive answer, the corresponding ....

....at most k. Furthermore, a technique introduced in [10] see also [3] further reduced the bound to O(n 2 ) while in [1] it is given a general method to construct a linear time algorithm that decides whether a given graph has cutwidth at most k, for k constant. However the methodology in [1] gives only a decision algorithm: it does not give any way to construct, in the case of a positive answer, the corresponding vertex ordering. In this paper, we give an explicit description, for any k 1, of a linear time algorithm that checks whether an input graph G has cutwidth k and, if this is ....

K. R. Abrahamson and M. R. Fellows. Finite automata, bounded treewidth and well-quasiordering. In N. Robertson and P. Seymour, editors, Proceedings of the AMS Summer Workshop on Graph Minors, Graph Structure Theory, Contemporary Mathematics vol. 147, pages 539--564. American Mathematical Society, 1993.

.... k 2 ) O(k 2 n k 3 ) steps (take in mind that if G is a partial k tree then E(G) O(kjV (G)j) We mention that the property of being a core is an EMS property (i.e. involves counting or summing evaluations over sets de nable on monadic second order logic) and therefore, the results in [Cou90b, ALS91, AF93] imply the existence of a polynomial time algorithm for deciding it. So far, no explicit algorithm has been reported for this problem. We consider the isomorphism problem on cores. Notice that graphs G and H are isomorphic i jE(G; H)j jE(H; G)j 1. This check can be done in polynomial time ....

Karl R. Abrahamson and Michael R. Fellows. Finite automata, bounded treewidth and well-quasiordering. In N. Robertson and P. Seymour, editors, Proceedings of the AMS Summer Workshop on Graph Minors, Graph Structure Theory, Contemporary Mathematics vol. 147, pages 539-564. American Mathematical Society, 1993.

....is NP Complete. Furthermore, the H decomposition problem is not expressible in Extended Monadic Second Order logic. This follows from the even stronger fact, pointed to us by M. Fellows, that the H decomposition problem is not finite state even when H is a tree by using the methods shown in [1] and in [10] We can, however, show the following: Theorem 1.1 Let k be a fixed integer, and let H be a star. There exists a polynomial time algorithm that, given a k slim graph G, finds an H decomposition of G if one exists. We can also solve the H decomposition problem for a much wider class ....

K. Abrahamson and M. Fellows, Finite automata, bounded treewidth and well-quasiordering, In: Graph Structure Theory, N. Robertson and P. Seymour eds. AMS Contemporary Math. 147 (1993), 539-564.

....improved by Fellows and Langston in [11] where, among others, they prove that for any fixed k, a O(n 3 ) algorithm can be constructed checking whether a graph has cutwidth at most k. Furthermore, a technique introduced in [10] see also [2] further reduced the bound to O(n 2 ) while in [1] a general method 2 is given to construct a linear time algorithm that decides whether a given graph has cutwidth at most k, for k constant. However the methodology in [1] gives only a decision algorithm: it does not give any method to construct the corresponding layout in the case of a positive ....

....at most k. Furthermore, a technique introduced in [10] see also [2] further reduced the bound to O(n 2 ) while in [1] a general method 2 is given to construct a linear time algorithm that decides whether a given graph has cutwidth at most k, for k constant. However the methodology in [1] gives only a decision algorithm: it does not give any method to construct the corresponding layout in the case of a positive answer, the corresponding layout. In this paper, we give an explicit description, for any k 1, of a linear time algorithm that checks whether an input graph G has cutwidth ....

K. R. Abrahamson and M. R. Fellows. Finite automata, bounded treewidth and well-quasiordering. In N. Robertson and P. Seymour, editors, Proceedings of the AMS Summer Workshop on Graph Minors, Graph Structure Theory, Contemporary Mathematics vol. 147, pages 539--564. American Mathematical Society, 1993.

....improved by Fellows and Langston in [11] where, among others, they prove that for any fixed k, a O(n 3 ) algorithm can be constructed checking whether a graph has cutwidth at most k. Furthermore, a technique introduced in [10] see also [3] further reduced the bound to O(n 2 ) while in [1] it is given a general method to construct a linear time algorithm that decides whether a given graph has cutwidth at most k, for k constant. However the methodology in [1] gives only a decision algorithm: it does not give any way to construct, in the case of a positive answer, the corresponding ....

....at most k. Furthermore, a technique introduced in [10] see also [3] further reduced the bound to O(n 2 ) while in [1] it is given a general method to construct a linear time algorithm that decides whether a given graph has cutwidth at most k, for k constant. However the methodology in [1] gives only a decision algorithm: it does not give any way to construct, in the case of a positive answer, the corresponding vertex ordering. In this paper, we give an explicit description, for any k 1, of a linear time algorithm that checks whether an input graph G has cutwidth k and, if this is ....

K. R. Abrahamson and M. R. Fellows. Finite automata, bounded treewidth and well-quasiordering. In N. Robertson and P. Seymour, editors, Proceedings of the AMS Summer Workshop on Graph Minors, Graph Structure Theory, Contemporary Mathematics vol. 147, pages 539--564. American Mathematical Society, 1993.

....improved by Fellows and Langston in [11] where, among others, they prove that for any fixed k, a O(n 3 ) algorithm can be constructed checking whether a graph has cutwidth at most k. Furthermore, a technique introduced in [10] see also [2] further reduced the bound to O(n 2 ) while in [1] a general method 2 is given to construct a linear time algorithm that decides whether a given graph has cutwidth at most k, for k constant. However the methodology in [1] gives only a decision algorithm: it does not give any method to construct the corresponding layout in the case of a positive ....

....at most k. Furthermore, a technique introduced in [10] see also [2] further reduced the bound to O(n 2 ) while in [1] a general method 2 is given to construct a linear time algorithm that decides whether a given graph has cutwidth at most k, for k constant. However the methodology in [1] gives only a decision algorithm: it does not give any method to construct the corresponding layout in the case of a positive answer, the corresponding layout. In this paper, we give an explicit description, for any k # 1, of a linear time algorithm that checks whether an input graph G has ....

[Article contains additional citation context not shown here]

K. R. Abrahamson and M. R. Fellows. Finite automata, bounded treewidth and well-quasiordering. In N. Robertson and P. Seymour, editors, Proceedings of the AMS Summer Workshop on Graph Minors, Graph Structure Theory, Contemporary Mathematics vol. 147, pages 539--564. American Mathematical Society, 1993.

....decomposition of G 1 Phi H of width at most k, if and only if there is an extension of (T 2 ; 2 ) that is a branch decomposition of G 2 Phi H of width at most k. The two lemmas above imply together that the property that a graph has branchwidth at most k is finite state, or regular (see e.g. [1]) A class of graphs G is finite state, if the equivalence relation on l terminal graphs G H , 8K : G Phi K 2 G , H Phi K 2 G) has a finite number of equivalence classes, for each fixed l. Combining the above results, and results and techniques from [1, 3, 6, 7, 13] we obtain the following ....

....is finite state, or regular (see e.g. 1] A class of graphs G is finite state, if the equivalence relation on l terminal graphs G H , 8K : G Phi K 2 G , H Phi K 2 G) has a finite number of equivalence classes, for each fixed l. Combining the above results, and results and techniques from [1, 3, 6, 7, 13], we obtain the following result. The results show that each of these is computable, although real practical efficiency and doabily is not guaranteed. 11 Theorem 10 One can construct, for each k, Delta linear time algorithms that decide whether a graph has branchwidth at most k (these ....

K. R. Abrahamson and M. R. Fellows. Finite automata, bounded treewidth and well-quasiordering. In N. Robertson and P. Seymour, editors, Proceedings of the AMS Summer Workshop on Graph Minors, Graph Structure Theory, Contemporary Mathematics vol. 147, pages 539--564. American Mathematical Society, 1993.

....be used to compute the obstruction set of the class of graphs with treewidth k. Bodlaender and Kloks also show how if existing, a tree decomposition with treewidth at most k can be computed in the same time bounds. Results of a similar flavor were obtained independently by Abrahamson and Fellows [1]. Recognition algorithms for graphs with treewidth k (k constant) have been designed by Arnborg et al. 4] These algorithms use linear time, but polynomial, not linear memory (it is allowed that the algorithm consults the contents of memory that is never written to) A disadvantage of this ....

K. R. Abrahamson and M. R. Fellows. Finite automata, bounded treewidth and well-quasiordering. In Graph Structure Theory, Contemporary Mathematics vol. 147, pages 539--564. American Mathematical Society, 1993.

....0 l of P k ;l which is effectively decidable. Finite index corresponds to finite state : there exists a linear time algorithm that decides finite index properties on graphs, given their tree decomposition of bounded treewidth. Moreover, this algorithm is of a special, well described structure [10, 9, 1]. The disadvantage of this algorithm is that a tree decomposition of the input graph is needed. Although for each fixed k, there is a linear time sequential algorithm which, given a graph G, checks if tw(G) k, and if so, computes a minimum width tree decomposition of G [6] this algorithm is ....

K. R. Abrahamson and M. R. Fellows. Finite automata, bounded treewidth and wellquasiordering. In N. Robertson and P. Seymour, editors, Proceedings of the AMS Summer Workshop on Graph Minors, Graph Structure Theory, Contemporary Mathematics vol. 147, pages 539--564. American Mathematical Society, 1993.

....on the maximum degree of vertices in the graph. Finite index corresponds to finite state : there exists a linear time algorithm that decides the property on graphs, given with a tree decomposition of bounded treewidth. Moreover, this algorithm is of a special, well described structure. See e.g. [1]. Note that a reduction rule (H 1 ; H 2 ) 2 R is safe for a property P if and only if H 1 P;l H 2 (if H 1 and H 2 are l terminal graphs) Below, we give a lemma on the existence of subgraphs of a certain size and type in graphs with bounded treewidth. This lemma will be used to show that there ....

K. R. Abrahamson and M. R. Fellows. Finite automata, bounded treewidth and wellquasiordering. In Proceedings of the AMS Summer Workshop on Graph Minors, Graph Structure Theory, Contemporary Mathematics vol. 147, pages 539--564. American Mathematical Society, 1993.

....in O(n) time in total, working bottom up in the decomposition tree. From ff(r) the answer to the problem that is to be solved can be determined in O(1) time. In case each ff(i) contains only a constant bounded number of bits, we call the problem finite state (after Abrahamson and Fellows [1]. Next, we review some useful results on the structure of graphs with treewidth at most 2. A graph has treewidth at most 1, if and only if it is a forest. Consider a graph G = V; E) with treewidth at most 2. Let H = V; E 0 ) be the graph, obtained by adding to G all edges (v; w) 62 E for all ....

K. R. Abrahamson and M. R. Fellows. Finite automata, bounded treewidth and wellquasiordering. In Graph Structure Theory, Contemporary Mathematics vol. 147, pages 539--564. American Mathematical Society, 1993.

....the maximum degree of vertices in the graph. Finite index corresponds to finite state : there exists a linear time algorithm that decides the property on graphs, given with a tree decomposition of bounded treewidth. Moreover, this algorithm is of a special, well described structure. See e.g. [1]. Safe reduction rules are implied by the equivalence relation jP;l : if for l terminal graphs G 1 , G 2 it holds that G 1 jP;l G 2 , then it directly follows from the definitions that the reduction rule G 1 G 2 is safe. Reduction rules for optimization problems We now extend the idea of graph ....

K. R. Abrahamson and M. R. Fellows. Finite automata, bounded treewidth and wellquasiordering. In Graph Structure Theory, Contemporary Mathematics vol. 147, pages 539--564. American Mathematical Society, 1993.

....G P;k H , 8K : P (G Phi K) P (H Phi K) We say that P is of finite index, when for every k, the equivalence relation P;k has a finite number of equivalence classes. One can show that every finite index problem can be solved in linear time on graphs of bounded treewidth (see e.g. [2]) Now, as soon as we have characteristics which need O(1) bits to describe, we know that the problem is finite state: if k terminal graphs G and H have the same full set, then G P;k H, and there are only a constant number of different possible full sets. Graph reduction Another interesting ....

.... which there exists an algorithm that decides whether x; k 2 L in f(k)jxj c time, f a function and c a constant) and a notion of reduction between parameterized languages (that preserves fixed parameterized tractability) Then they introduce a hierarchy of complexity classes FTP W [1] W [2] Delta Delta Delta W [i] Delta Delta Delta W [P ] of parameterized problems. Classes W [i] W [P ] are defined in terms of reductions to certain parameterized problems on Boolean circuits. It is conjectured that the hierarchy is proper. So, hardness for W [1] or any larger class means ....

K. R. Abrahamson and M. R. Fellows. Finite automata, bounded treewidth and well-quasiordering. In N. Robertson and P. Seymour, editors, Proceedings of the AMS Summer Workshop on Graph Minors, Graph Structure Theory, Contemporary Mathematics vol. 147, pages 539--564. American Mathematical Society, 1993.

.... paper as an important intermediate step to obtain explicit and constructive algorithms that solve the treewidth k and pathwidth k problems in linear time (k fixed) Results of a similar nature as ours were independently obtained by Lagergren and Arnborg [31] and by Abrahamson and Fellows [1]. It should be noted, that for k = 1; 2; 3; 4, linear time and space algorithms based on graph rewriting exist for the treewidth k problem [6, 33, 42] We also solve a different, related problem, with basically the same algorithms: for each constant k, we have a polynomial time algorithm, that ....

....Use the partial ordering OE on characterizations in full sets, or stronger forms of such partial orderings, and try to remove many characteristics from full sets that are dominated by other better characteristics in that full set. ffl Use (Myhill Nerode) state reduction techniques. See e.g. [7, 1]. ffl Use memoization ; i.e. do not compute full sets at once, but always try to find characteristics that can be included in a (not yet full) set of a node i that is as close to the root as possible. As soon as we find a characteristic in the set of the root of T we are done. This approach may ....

K. R. Abrahamson and M. R. Fellows. Finite automata, bounded treewidth and well-quasiordering. In Graph Structure Theory, Contemporary Mathematics vol. 147, pages 539--564. American Mathematical Society, 1993.

....and constructive algorithms for the problems. Both Lagergren and Arnborg [91] and Bodlaender and Kloks [31, 82] give such an algorithm, using an involved application of the technique, discussed in section 4. Independently, results of a similar nature were obtained by Abrahamson and Fellows [1]. From these results it follows that a technique of Fellows and Langston [62] can be used to compute the corresponding obstruction set. Bodlaender and Kloks [31] also discuss how in the same time bounds the path or tree decompositions with pathwidth or treewidth at most k can be found, if ....

K. R. Abrahamson and M. R. Fellows. Finite automata, bounded treewidth and well-quasiordering. In Graph Structure Theory, Contemporary Mathematics vol. 147, pages 539--564. American Mathematical Society, 1993.

No context found.

Abrahamson, K. R., and Fellows, M. R. Finite automata, bounded treewidth and well-quasiordering. In Proceedings of the AMS Summer Workshop on Graph Minors, Graph Structure Theory, Contemporary Mathematics vol. 147 (1993), N. Robertson and P. Seymour, Eds., American Mathematical Society, pp. 539--564.

....of Bodlaender [6] we can determine that no such decomposition of width b(k) exists or be given a decomposition of G. In either case, the running time for this procedure is linear in the size of G but exponential only in k. By means of one of several general algorithmic design methodologies (see [1, 4, 5, 15, 20, 54]) we may then answer the original question in time linear in the size of G. Hence, for small values of k, this procedure may lead to algorithms that are practical even for very large graphs G. Examples where these methods have been successful include Treewidth, Pathwidth, Min Cut Linear ....

....h. A parameterized problem L belongs to W [P ] if L reduces to the circuit problem L F , where F is the set of all circuits (no restrictions) We designate the class of fixed parameter tractable problems FPT . These definitions give us the hierarchy of parameterized complexity classes FPT W [1] W [2] Delta Delta Delta W [t] Delta Delta Delta W [P ] for which there are many natural hard or complete problems [37, 23, 24] For example, all of the following problems are now known to be complete for W [1] Square tiling, Independent set, Clique, Bounded post correspondence ....

[Article contains additional citation context not shown here]

K. Abrahamson and M. R. Fellows. Finite Automata, Bounded Treewidth and WellQuasiordering. Contemporary Mathematics, 147:539--563, 1993.

....the technical details of this dynamic programming algorithm are rather detailed and complex. Other problems that have a similar algorithmic solution are the pathwidth problem itself (see [7, 4] and variants on weighted or directed graphs, including directed vertex separation number [2] See also [1]. In this paper, we attempt to present the central ideas in these algorithms in a di erent, more easily accessible manner, by showing that the algorithms can be obtained by a stepwise modi cation of a trivial hypothetical nondeterministic algorithm. Thus, while our resulting algorithms will not ....

....to cutwidth (e.g. directed modi ed cutwidth , or weighted variants) is at most k, and if so, gives the corresponding linear ordering or topological sort of G. Ingredients of the techniques displayed in this paper appeared in the early 1990 s independently in work of Abrahamson and Fellows [1], Lagergren and Arnborg [15] and Bodlaender and Kloks [7] In [6] a relation between decision and construction versions of algorithms running on path decomposition with an eye to nite state automata was established. More background and more references can be found in [10] 2 De nitions In ....

Abrahamson, K. R., and Fellows, M. R. Finite automata, bounded treewidth and well-quasiordering. In Proceedings of the AMS Summer Workshop on Graph Minors, Graph Structure Theory, Contemporary Mathematics vol. 147 (1993), N. Robertson and P. Seymour, Eds., American Mathematical Society, pp. 539-564.

....the technical details of this dynamic programming algorithm are rather detailed and complex. Other problems that have a similar algorithmic solution are the pathwidth problem itself (see [7, 4] and variants on weighted or directed graphs, including directed vertex separation number [2] See also [1]. In this paper, we attempt to present the central ideas in these algorithms in a di erent, more easily accessible manner, by showing that the algorithms can be obtained by a stepwise modi cation of a trivial hypothetical non deterministic algorithm. Thus, while our resulting algorithms will not ....

....to cutwidth (e.g. directed modi ed cutwidth , or weighted variants) is at most k, and if so, gives the corresponding linear ordering or topological sort of G. Ingredients of the techniques displayed in this paper appeared in the early 1990 s independently in work of Abrahamson and Fellows [1], Lagergren and Arnborg [15] and Bodlaender and Kloks [7] In [6] a relation between decision and construction versions of algorithms running on path decomposition with an eye to nite state automata was established. More background and more references can be found in [10] 2 De nitions In ....

Abrahamson, K. R., and Fellows, M. R. Finite automata, bounded treewidth and well-quasiordering. In Proceedings of the AMS Summer Workshop on Graph Minors, Graph Structure Theory, Contemporary Mathematics vol. 147 (1993), N. Robertson and P. Seymour, Eds., American Mathematical Society, pp. 539-564.

....of trees vertex labeled from a finite set of labels) in order to prove the following fixed parameter tractability result. Theorem 2 For every fixed set of parameter values (k; r; m; t) the problem Ball and Trap II can be solved in time linear in the size of the tree. Proof: Using the methods of [AF93], we can represent an input tree T as a labeled binary tree (even though T may not be binary) where the labels (which we will refer to as colors) indicate both the structure of T and the adornments of the vertices of T with balls and traps. The colored binary tree that represents an input tree T ....

....of any given color on T is bounded by t, since 6 otherwise T would be a No instance. We argue that there is a finite state tree automaton that recognizes precisely those labeled binary trees that represent yes instances of the problem. Our argument is based on the method of test sets of [AF93, DFS98]. Since there are at most k2 r types of balls, and since each vertex may have m balls, 2 r (k2 r ) m colors suffice. The input trees are rooted, and we may assume that the parse tree for a given input tree T is rooted compatibly (i.e. at the same vertex) We use the following parsing ....

K. R. Abrahamson and M. R. Fellows. Finite automata, bounded treewidth and wellquasiordering. In Graph Structure Theory, Contemporary Mathematics vol. 147, pp. 539--564. American Mathematical Society, 1993.

....of Bodlaender [7] we can find a decomposition of width b(k) for G or determine that no such decomposition exists. In either case, the running time for this procedure is linear in the size of G but exponential only in k. By means of one of several general algorithmic design methodologies (see [1, 5, 6, 15, 19, 52]) we may then answer the original question in time linear in the size of G. Hence, for small values of k, this procedure may lead to algorithms that are practical even for very large instances. Examples where these methods have been successful include Treewidth, Pathwidth, Min Cut Linear ....

....h. A parameterized problem L belongs to W [P ] if L reduces to the circuit problem L F , where F is the set of all circuits (no restrictions) We designate the class of fixed parameter tractable problems FPT . These definitions give us the hierarchy of parameterized complexity classes FPT W [1] W [2] Delta Delta Delta W [t] Delta Delta Delta W [P ] for which there are many natural hard or complete problems [36, 23, 24, 25, 26] For example, all of the following problems are now known to be complete for W [1] Square tiling, Independent set, Clique, and Bounded post ....

[Article contains additional citation context not shown here]

K. Abrahamson and M. R. Fellows. Finite Automata, Bounded Treewidth and WellQuasiordering. Contemporary Mathematics, 147:539--563, 1993.

....explanation for G1 ; G2 ; Gk requiring the fewest multiple gene duplications. Via a reduction to and from a combinatorial model we call the Ball and Trap Game, we show that the general form of this problem is NP hard and various parameterized versions are hard for the complexity class W [1]. These results immediately imply that Multiple Gene Duplication is similarily hard. We prove that several parameterized variants are in FPT. 1 Introduction to the Model A fundamental problem arising in computational biology is the determination of the (correct) evolutionary topology for a set of ....

.... is NP complete (here the gene trees may contain leaf labels which appear more than once) When each gene tree may contain a leaf label at most once (a gene tree is formed over exactly one gene per taxa) the problem remains NP complete and the parameterized (by k) version is hard for W [1] (see [6] A similar question to Optimal Species Tree arises if we ask for the species tree which implies the minimum number of multiple gene duplications for a given set of gene trees. A duplication event in the genome of an organism involves a stretch of DNA where one or more genes may reside. ....

[Article contains additional citation context not shown here]

K. R. Abrahamson and M. R. Fellows. Finite automata, bounded treewidth and well-quasiordering. In Graph Structure Theory, Contemporary Mathematics vol. 147, pp. 539--564. AMS, 1993.

....of Bodlaender [6] we can determine that no such decomposition of width b(k) exists or be given a decomposition of G. In either case, the running time for this procedure is linear in the size of G but exponential only in k. By means of one of several general algorithmic design methodologies (see [1, 4, 5, 15, 20, 54]) we may then answer the original question in time linear in the size of G. Hence, for small values of k, this procedure may lead to algorithms that are practical even for very large graphs G. Examples where these methods have been successful include Treewidth, Pathwidth, Min Cut Linear ....

....h. A parameterized problem L belongs to W [P ] if L reduces to the circuit problem L F , where F is the set of all circuits (no restrictions) We designate the class of fixed parameter tractable problems FPT . These definitions give us the hierarchy of parameterized complexity classes FPT W [1] W [2] Delta Delta Delta W [t] Delta Delta Delta W [P ] for which there are many natural hard or complete problems [37, 23, 24] For example, all of the following problems are now known to be complete for W [1] Square tiling, Independent set, Clique, Bounded post correspondence ....

[Article contains additional citation context not shown here]

K. Abrahamson and M. R. Fellows. Finite Automata, Bounded Treewidth and WellQuasiordering. Contemporary Mathematics, 147:539--563, 1993.

....all t boundaried graphs. Definition. If F is a family of graphs then the large canonical congruence for F is defined for t boundaried graphs X;Y 2 U t large by X F Y if and only if 8Z 2 U t large : X Phi Z 2 F) Y Phi Z 2 F) The following definition is from Abrahamson and Fellows [AF93]. Definition. A graph family F is fully cutset regular if for every t, the large canonical congruence on U t large has finite index. Courcelle and Lagergren proved in [CL94] that this notion is equivalent to that of recognizable graph families introduced in [Co90] This must be regarded as an ....

....the canonical recognizability congruence for F . It follows from the Graph Minor Theorem and Courcelle s Theorem on MSO graph properties [Co90] that every minor order lower ideal is recognizable. It is interesting that only a few natural graph families are presently known not to be recognizable [AF93, BFW92, FHW93]. The positive results of [FL89b] our Theorem A) apply to many (if not most) natural lower ideals. Normally we have the information (i) about F . Note that since we are concerned here with the issue of whether O can be recursively computed, any algorithm that correctly decides membership in F ....

[Article contains additional citation context not shown here]

K. R. Abrahamson and M. R. Fellows. Finite automata, bounded treewidth and wellquasiordering. In Graph Structure Theory, Contemporary Mathematics vol. 147, pp. 539-- 564. American Mathematical Society, 1993.

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K. R. Abrahamson and M. R. Fellows. Finite automata, bounded treewidth and well-quasiordering. In N. Robertson and P. Seymour, editors, Proceedings of the AMS Summer Workshop on Graph Minors, Graph Structure Theory, Contemporary Mathematics vol. 147, pages 539--564. American Mathematical Society, 1993.

No context found.

K. A. Abrahamson and M. R. Fellows, Finite automata, bounded treewidth and wellquasiordering, in Graph structure theory, no. 147 in Contemporary Mathematics, American Mathematical Society,

No context found.

K. R. Abrahamson and M. R. Fellows. Finite automata, bounded treewidth and well-quasiordering. In N. Robertson and P. Seymour, editors, Proceedings of the AMS Summer Workshop on Graph 34 Minors, Graph Structure Theory, Contemporary Mathematics vol. 147, pages 539--564. American Mathematical Society, 1993.

No context found.

K. R. Abrahamson and M. R. Fellows. Finite automata, bounded treewidth and well-quasiordering. In N. Robertson and P. Seymour, editors, Proceedings of the AMS Summer Workshop on Graph Minors, Graph Structure Theory, Contemporary Mathematics vol. 147, pages 539--564. American Mathematical Society, 1993.

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