Representing Hyper-Graphs by Regular Languages ?
Salvatore La Torre and Margherita Napoli Dipartimento di Informatica ed Applicazioni Universita degli Studi di Salerno 84081 Baronissi, Italy. e-mail: fsallat,
[email protected]
Abstract. A new compact representation of in nite graphs is investigated. Regular languages are used to represent labelled hyper-graphs which can be also multi-graphs. Our approach is similar to that used by A. Ehrenfeucht et al. for nite graphs since we use a regular pre x-free language as set of vertices, but it di ers from that in the representation of the edges. In fact, we use a regular language for the edges instead of a nite loop-free graph. Our approach preserves the nite representation of the edges and of the corresponding labelling mapping and yields to a higher expressive power. As a matter of fact, our graph representation results to be more powerful than the equational graphs introduced by B. Courcelle. Moreover, the use of a regular pre x-free language to represent the vertices allows ( xed the language of the edges) to express a graph by a labelled tree. The advantage to represent graphs by trees is that properties of graphs can be veri ed by induction on the tree, often leading to ecient algorithms.
1 Introduction A lot of e orts have been made in the last decade to obtain small speci cations of graphs. A well supported idea has been that of representing graphs by expressions or trees [1, 6, 12]. Recently, A. Ehrenfeucht et al. [9] have introduced a representation of nite graphs by nite pre x-free languages of strings whose alphabets have themselves a graph structure. The strings of the language represent the vertices and there is an edge between two vertices if and only if the pair of the rst two symbols, at which the two corresponding strings di er, is an edge in the alphabet. Another way to specify nite graphs was introduced by M. Bauderon et al. [3]. They de ne nite hyper-graphs in a compositional way, that is they use graph expressions built from basic ones by applications of simple graph operations corresponding to hyper-edge replacements. Systems of graph equations with this ?
Partially supported by the M.U.R.S.T. in the framework of \Tecniche formali per la speci ca, l'analisi, la veri ca, la sintesi e la trasformazione di sistemi software" project.
kind of expressions provide a way to de ne in nite hyper-graphs, called equational graphs [7]. In the recent paper [2], the class of simple equational hypergraphs is extended by allowing vertex replacement. In this way a characterization is obtained of the class of simple graphs de ned in [5] whose monadic secondorder theory is decidable and strictly containing the simple graphs among the equational graphs. In this paper we introduce a new way of specifying in nite hyper-graphs through regular languages. Our approach is similar to that used in [9] for nite graphs since we use a regular pre x-free language as set of vertices, but it di ers from that in the representation of the edges. Actually, the nite loop-free graphs turn out to be not sucient when the goal is the representation of nite graphs. We use a regular language P for the edges instead of a nite loop-free graph with the meaning that the graph has an edge linking the ordered sequence of vertices x1; : : :; xk if their \sux" belongs to P. Intuitively, by sux we mean the tuple obtained from x1 ; : : :; xk by cutting their longest common pre x, if there exists i and j such that xi 6= xj , and the tuple itself, otherwise. Our approach preserves the nite representation of the (possibly in nite) graphs and allows to specify a meaningful class of in nite hyper-graphs. As a matter of fact, our graph representation results to be more powerful than the equational graphs introduced by B. Courcelle. In a similar way as in [9], the use of regular languages allows us to inherit concepts and ideas from the formal language theory and to use them for graphs. In particular we are interested in the relationships between language operations and graph operations: the relevance of this investigation is due to the possibility of performing graph trasformations by manipulating the regular languages used for the graph representation. Moreover, the use of a regular pre x-free language to represent the vertices allows ( xed the language of the edges) to express a graph by a labelled tree. The advantage to represent graphs by trees is that properties of graphs can be veri ed by induction on the tree, often leading to ecient algorithms [4, 8, 10, 13]. In section 2 we give some preliminary de nitions. In section 3 the graph representation is introduced, some properties of this representation are shown and the relationships between graph substitution and language concatenation is stated. The main result of section 4 is the proof that the graph representation introduced in section 3 is more expressive than the equational graphs de ned in [7]. The paper ends with some conclusions in section 5, where we remark the differences between our approach and that in [9] and mention some directions for future works. Due to the lack of space the proofs are omitted, for a full version of the paper see the URL [14].
2 Preliminaries In this section we give some basic de nitions. We suppose that the reader is familiar with the basic concepts of the formal languages (see for example [11]). We only recall that L is said pre x-free if for every x; y 2 L it holds that x is
not a pre x of y and remark that in this paper with Ln we denote the Cartesian product L1 : : : Ln when L1 = : : : = Ln = L. In the following we will use N to denote the set of the positive integers. A multiset ms over a nite set is a mapping from into N [ f0g [ f1g and ms(a) is said the multiplicity of a for each a 2 . The set of the multisets over a given nite set is denoted by MS(). We denote with 0 the multiset mapping a into 0 for each a 2 . Given two multisets ms and ms0 , we say that ms ms0 if ms(a) ms0 (a) for every a 2 and we denote with ms + ms0 the multiset which maps each a 2 into ms(a)+ms0 (a). Moreover, if ms0 (a) 6= 1 for each a 2 , then with ms ? ms0 we denote the multiset which maps a into ms(a) ? ms0 (a), if ms(a) ? ms0 (a) 0, and 0, otherwise. Obviously, 1 + k = k + 1 = 1 + 1 = 1 ? k = 1 holds. Sometimes in the following we denote the multisets which are also sets with the usual set notation. In this paper we cope with directed labelled hyper-graphs which can be also multi-graphs, that is they can have many hyper-edges linking any ordered tuple of vertices. We consider a directed labelled hyper-edge as given by a sequence of vertices (v1 ; : : :; vk ) and a multiset ms, with the meaning that there are exactly ms(a) directed hyper-edges linking v1 ; : : :; vk and labelled by a 2 . We denote each of them with ((v1 ; : : :; vk ); a) and we say that an hyper-edge is incident to each of the vertices v1 ; : : :; vk that it links. Note that we are not interested in distinguishing among hyper-edges linking the same tuple of vertices and having the same label. Labels are taken from a ranked alphabet, that is a pair (; ) where is an alphabet and is a mapping from into N.
De nition1. Let (; ) be a ranked alphabet. A labelled n-hyper-graph is a tuple g = (V; E; lab; src) where: ? V is the set of the vertices; k ? lab : S1 k=1 V ! MS() is a total mapping such that, for every a 2 and v1 ; : : :; vk 2 V , lab(v1 ; : : :; vk )(a) > 0 implies (a) = k; k ? E = f(v1 ; : : :; vk ) 2 S1 6 0g. k=1 V =lab(v1; : : :; vk ) = ? src is a total mapping from f1; : : :; ng into V . The mapping src de nes a sequence of n vertices which is called the sequence of sources of G and the integer n is the type of g. From now on, we will consider
only labelled n-hyper-graphs, so we use the word n-graph (or simply graph, when the speci cation of n is useless ) for a labelled n-hyper-graph and its hyper-edges are simply called edges. A subgraph of a graph g = (V; E; lab; src) is a graph g0 = (V 0; E 0; lab0; src0 ) such that V 0 V , lab0(v1 ; : : :; vk ) lab(v1; : : :; vk ) for each (v1 ; : : :; vk ) and src0 (i) = src(i). In this case we say that g0 g. Let g = (V; E; lab; src) be a graph and be an equivalence relation on V . We de ne the quotient graph, denoted by g=, the graphP(V=; E=; lab=; src=) where: V== f[v]=v 2 V g, lab= ([v1]; : : :; [vk ]) = v 2[v ] lab(v1 ; : : :; vk ) and src= (i) = [src(i)]. Let gi = (Vi ; Ei; labi; srci ) for i = 1; 2 be graphs, an isomorphism between graphs : V1 ! V2 is a bijective mapping such that: lab1(v1 ; : : :; vk ) = lab2( (v1 ); : : :; (vk )) and src2 (i) = (src1 (i)). Finally we de ne the limit of a succession of nite graphs by referring to the intuitive 0
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concept of limit. For the sake of simplicity we omit a formal de nition and we use the intuitive concept of \arbitrarily close". Then, let fgn gn>0 be a succession of nite graphs, we say that a (possibly in nite) graph, denoted by limn gn, is the limit for n ! 1 of the succession fgngn>0 if it is always possible to nd a graph in the succession which is arbitrarily close to limn gn. Given a succession of nite graphs fgn gn> S0 such that S gn = (Vn; En; labn; src) and gn gn+1, we have that limn gn = ( n>0 Vn ; n>0 En ; limn labn; src). Note that flabn gn>0 is a monotonic succession of functions, then its limit always exists. Moreover, it is easy to show the following result. Lemma 2. Given two successions of graphs fgngn>0 and fgn0 gn>0 such that for each n > 0 gn is isomorphic to gn0 , gn gn+1 and gn0 gn0 +1, then limn gn is isomorphic to limn gn0 .
3 Graph representation over a regular language of tuples In this section we introduce a new way of representing graphs which is obtained from the representation introduced in [9]. The new representation is as powerful as the previous one when nite graphs are dealt with. The main di erence between them concerns the representation of the edges. In fact, we use a regular language instead of a nite loop-free graph. Moreover, our approach preserves some agreeable features of the previous one and the nite representation of the edges and of the corresponding labelling mapping also for in nite hyper-graphs. To introduce the new graph representation we de ne rst a notion of regularity for languages of tuples which we call parallel regularity. Let be an alphabet, \ 62 and x1; : : :; xk 2 , we denote with Matrix\ (x1 ; : : :; xk ) the word [a11 : : :a1 k ]; : : :; [ah 1; : : :; ah k ] over the alphabet ( [ f\g)k where: ? h = maxfjxj j=j = 1; : : :; kg; ? for j = 1; : : :; k: xj = a1 j : : :ajx j j and ar j = \ for jxj j < r h. That is the i-th symbol [ai 1 : : :ai k ] of the word Matrix\ (x1 ; : : :; xk) is the ordered tuple of the i-th symbols of the words x1; : : :; xk . De nition3. Let be an alphabet, \ 62 and s be a positive integer. Then, S P si=1 ( )i is said to be regular in parallel if the language Matrix\ (P) = fMatrix\ (x1 ; : : :; xk)=(x1 ; : : :; xk ) 2 P g is regular. Then, we formalize the notion of sux of a tuple of strings. De nition4. Let be an alphabet and L be a pre x-free language. For each x1; : : :; xk 2 we de ne the sux of the k-tuple (x1 ; : : :; xk), denoted by suf(x1 ; : : :; xk), as the k-tuple (a1 y1 ; : : :; akyk ), if xi = xai yi , for ai 2 , and 9j; m such that aj 6= am , and as the k-tuple (x1; : : :; xk), otherwise. Intuitively, by sux we mean the tuple obtained from x1; : : :; xk by cutting their longest common pre x, if there exists i and j such that xi 6= xj , and the tuple itself, otherwise. The fact that suf(x1 ; : : :; xk ) = (x1; : : :; xk ) whenever x1 = : : : = xk allows us to represent the graphs with loops. In fact, we represent j
a graph with a pre x-free language (the vertices), a set of tuples P and a labelling function with the meaning that there is an edge with vertices x1; : : :; xk if the sux of the tuple (x1 ; : : :; xk) belongs to P and the labelling function maps this sux in a multiset which di ers from 0. Formally:
De nition5. Let be an alphabet, (; ) be a ranked alphabet, LSm S 1 be a regular pre x-free language, P j =1 ( )j be such that P = i=1 Pi where regular in parallel for all i = 1; : : :; m, lab be a total mapping from S1 (Pi )isj into MS() such that jlab(Pi)j = 1 for all i = 1; : : :; m and src be j =1 a total mapping from f1; : : :; ng into L. We denote with gra(L; P; lab; src) the n-graph g = (V; E; lab0; src) where: ? V = L;S1 ? lab0 : j =1 ( )j ! MS() is the mapping de ned as lab0 (x1; : : :; xk) = lab(suf(x1 ; : : :; xk)) if suf(x1 ; : : :; xk ) 2 P and lab0(x1 ; : : :; xk) = 0, otherwise. In this case we say that the graph g is representable by regular languages .
Example 1. Let g = (N; E; lab; src) be such that:
1. E = f(2i ? 1; 2i + 1)=i 2 N g [ f(2i ? 1; 2i)=i 2 N g [ f(2i + 2; 2i)=i 2 N g, 2. lab(2i ? 1; 2i + 1) = fag; lab(2i ? 1; 2i) = fbg and lab(2i + 2; 2i) = fcg and 3. src is the sequence 1; 2. We show that g is representable by regular languages. In fact, consider the graph g0 = gra(1 (v + y + w + z); f((v; y); (v; w); (w; z); (z; y); (w; 1w); (1z; z)g; lab0; src0 ) where lab0(v; y) = lab0 (w; z) = fbg, lab0 (v; w) = lab0 (w; 1w) = fag, lab0(z; y) = lab0 (1z; z) = fcg and src0 de nes the sequence v; y (see Figure 1). Then, it easy to see that g0 is isomorphic to g.
000 111 111 000 111 000 111 000 01 01 0 1 0 1 111 000 111 000 111 000 111 000 01 01 0 1 0 1
v b
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a
b
y
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1w
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1z
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Fig. 1. A graphical representation of g . 0
By using this new graph representation, as for the graph representation introduced in [9], the following substitution of graphs corresponds to a language concatenation.
De nition6. Let g = (V; E; lab; src) be an n-graph and gv = (Vv ; Ev; labv ; srcv ) be an nv -graph, for v 2 V . Then, the graph obtained by substituting gv for v in
g, denoted by g[v
gv ]v2V , is the graph (V 0; E 0; lab0; src0 ) where:
? V 0 = Sv2V fvgVv ; ? lab0(x1 y1 ; : : :; xkyk ) is equal to labx1 (y1 ; : : :; yk ), if x1 = : : : = xk , and is equal to lab(x1 ; : : :; xk ), otherwise;
? src0 de nes the sequence s1 ; : : :; sn where si = xi srcx (1); : : :; xisrcx (nx ) i
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and x1; : : :; xn is the sequence de ned by src. This graph substitution is said uniform if we substitute a unique graph for every vertex v. The following theorem states that uniform graph substitution corresponds to language concatenation in the representation of the graph. Theorem 7. Let g = gra(L; P; lab; src) and gx = gra(Lx ; P; lab; srcx), for x 2 LS. It holds that g[x gx ]x2L =gra(L0 ; P; lab; src0), where L0 is the language 00 0 00 x2L fxgLx. Moreover, if L = Lx for every x 2 L then L = LL . Another interesting property of our notation is related to the continuity of the function gra. In fact, the following result holds. Theorem 8. Let fgra(Ln ; Pn; labn; src)gn>0 be a succession of graphs such that Ln SLn+1 , PS n Pn+1 and labn labn+1. Then, limn gra(Ln ; Pn; labn; src) = gra( n>0 Ln ; n>0 Pn; limn labn ; src).
4 Equational graphs vs. graphs representable by regular languages In this section we rst brie y recall the notion of the equational graphs as de ned in [7] and then we compare the expressive power of the equational graphs with our representation. In the following,some operations on graphs are de ned. Let g = (V; E; lab; src) and g0 = (V 0 ; E 0; lab0; src0 ) be respectively an n-graph and an n0-graph such that V \ V 0 = ; and E \ E 0 = ;. The disjoint union of g and g0 , denoted by g g0 , is de ned as the (n + n0 )-graph (V [ V 0 ; E [ E 0 ; lab [ lab0; src00) where src00 is the mapping de ning the sequence which is the concatenation of the sequences de ned by src and src0 . Moreover, let f : f1; : : :; mg ! f1; : : :; ng be a total mapping, the source rede nition map f is de ned as f (g) = (V; E; lab; src f) where is the usual composition of functions. Let be an equivalence relation over f1; : : :; ng, the source fusion map is de ned as (g) = g=, where g= is the quotient graph with respect to the equivalence relation on V de ned as: v v0 , v = v0 or (v = src(i); v0 = src(j) and (i; j) 2 ): Finally, let (; ) be a ranked alphabet, (v1 ; : : :; vn ) be a tuple belonging to E and a 2 be such that lab(v1 ; : : :; vn )(a) 6= 0. The graph, which is obtained by substituting g0 for an edge ((v1 ; : : :; vn ); a) in g and which we denote as g[((v1 ; : : :; vn ); a) g0 ], is the graph g00 = where: ? g00 = f (g g0 ) with f : f1; : : :; ng ! f1; : : :; n + n0 g de ned as f(i) = i for i = 1; : : :; n and g = (V; E; lab; src) where lab(w1; : : :; wk ) = lab(w1 ; : : :; wk) for (w1; : : :; wk ) 6= (v1 ; : : :; vn ) and lab(v1; : : :; vn ) = lab(v1 ; : : :; vn ) ? fag; ? is an equivalence relation on V [ V 0 de ned as: v v0 , v = v0 or (v = vi and v0 = src0 (i)): 0
0
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Roughly speaking, the substitution of g0 for the edge ((v1 ; : : :; vn ); a) in g consists of the deletion in g of an edge linking the tuple (v1 ; : : :; vn ) and labelled by a and the gluing of g0 and g by the fusion of the sources of g0 and the vertices in (v1 ; : : :; vn ), in the order. Note that the graph g is the result of the deletion of one edge among those linking (v1 ; : : :; vn ) and labelled by a in g. The above graph substitution can be generalized by de ning a substitution of all the edges which have a given label. Thus, for i = 1; : : :; m let gi be an ni -graph and ai 2 be such that (ai ) = ni . We denote with g[a1 g1 ; : : :; am gm ] the graph which is obtained by simultaneously substituting gi for every edge which is labelled by ai in g. The next de nition gives the notion of graph expression. De nition9. Let (; ) be a ranked alphabet. The graph expressions are de ned as: ? a is a graph expression of type (a) for all a 2 ; ? n is a graph expression of type n for all n 2 N [ f0g; ? e1 ^ e2 is a graph expression of type n1 + n2 for all the graph expressions e1 of type n1 and e2 of type n2 ; ? ^f (e) is a graph expression of type m for all the graph expressions e of type n and for all the total mappings f : f1; : : :; mg ! f1; : : :; ng; ? ^ (e) is a graph expression of type n for all the graph expressions e of type n and for all the equivalence relations on f1; : : :; ng. The meaning of the operators ^ ; ^ and ^f is obtained by the meaning of the corresponding operators on graphs. In fact, to each graph expression e of type n it is possible to associate a n-graph, denoted with val(e), which represent the evaluation of the expression e. By considering some unknowns in the set , the graph equations and the systems of graph equations are de ned in an obvious way. A solution to such a system is a tuple of graph expressions whose depth (i.e. the nesting of the operators) may be in nite. An evaluation mapping exists associating an unique (up to an isomorphism) graph to an in nite graph expression and we denote it again with val. More details can be found in [7]. Let S be the system of graph equations hu1 = e1 ; : : :; um = em i in the unknowns u1; : : :; um . If S satis es the Greibach condition (that none of the ei is equal to an unknown), then S has a unique solution (U1 ; : : :; Um ) and the elements of the m-tuple (val(U1 ); : : :; val(Um )) are said equational S graphs [7]. We can de ne, for each h, a regular per x-free language L, a set P = pi=1 Pi where each Pi is regular in parallel , a labelling mapping lab such that jlab(Pi)j = 1 and src, in such a way that val(Uh ) is equal to gra(L; P; lab; src). So we have the following theorem. 0
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Theorem10. Every equational graph is representable by regular languages.
The proof of the above theorem gives us a way to obtain a description of every equational graph. For example, if we consider the system u = ^f (^ (c^ a^ b^ u)) where is f(1; 3); (2; 6); (4; 7);(5; 8)g and f : f1; 2g ! f1; 2; 3; 4;5; 6; 7; 8g is de ned as f(i) = i for i = 1; 2, then the graph obtained in that way is the one in Figure 1.
The vice-versa of the above theorem is not true. In fact there is a class of graphs which are representable by regular languages but are not equational. Theorem 11. Let g = (V; E; lab; src) be a graph with a subgraph g0 = (V 0 ; E 0; lab0 ; src0) such that: a) V 0 is in nite and b) for every v; v0 2 V 0 there exists (v1 ; : : :; vk ) 2 E 0 such that (v1 ; : : :; vk ) is incident on v and v0 . Then, g is not equational.
Corollary 12. The class of the equational graphs is strictly included in the class of the graphs representable by regular languages.
5 Conclusions In this paper we have introduced a new way of specifying in nite hyper-graphs through regular languages and have compared the class of representable graphs with the equational graphs considered in [7]. Our approach is similar to that used in [9] where the authors introduce a new representation of nite graphs. They use nite pre x-free languages of strings on alphabets which have themselves a graph structure. The strings of the language represent the vertices of the graph and there is an edge between two vertices if and only if the pair of the rst two symbols, at which the two corresponding strings di er, is an edge in the alphabet. One can prove that in the above approach the class of in nite graphs which can be represented through an in nite pre x-free regular language and a nite loopfree graph contains only graphs such that either they have an in nite degree (that is, there is a vertex with in nite edges incident on it) or they are the disjoint union of in nitely many maximal connected subgraphs. So, when the goal is the representation of in nite hyper-graphs, that approach turns out to be strictly less powerful than the one presented in this paper. Two interesting aspects in [9] are the use of pre x-free languages, which can be viewed as trees so that this approch presents the advantages of representing graphs by trees, and the relationships between graph operations and language operations (and then operations on the representation itself). Our aim was that of preserving these advantages also when in nite graphs are dealt with. We have also proved that the class of the equational graphs is strictly contained in the class of graphs that we have considered. For the case of simple graphs, this result could also be derived by comparing the simple graphs in the class considered in our paper with the graphs de nedin [5]. However, our direct proof provides a way to obtain the representation by regular languages of a given equational hyper-graph. By introducing some constraints on P and lab (such as, the number of di erent multisets ), it is possible to determine some hierarchies whose investigation could give interesting hints on the use of this representation. Another worthwhile aspect to study concerns the comparisons among the di erent classes of hyper-graphs de ned by di erent choices of the set P. It would be interesting to check how the relationships among di erent P's re ect on the corresponding classes. Acknowledgment We thank the referees for helpful comments and for pointing us to the paper of Caucal.
References 1. S. Arnborg, J. Lagergren and D. Seese, \Easy problems for tree-decomposable graphs", Journal of Algorithms , 12 (1991) 308{340. 2. K. Barthelmann, \When Can an Equational Simple Graph Be Generated by Hyperedge Replacement?", this Volume. 3. M. Bauderon and B. Courcelle, \Graph Expressions and Graph Rewritings", Mathematical System Theory , 20 (1987) 83{127. 4. H. L. Bodlaender and R. H. Mohring, \The pathwidth and treewidth of cographs", SIAM Journal on Discrete Mathematics , 6 (2) (1993) 181{188. 5. D. Caucal, \On in nite transition graphs having a decidable monadic theory", Proc. of ICALP'96, (F. M. auf der Heide and B. Monien, Eds.), Lecture Notes in Computer Science, vol. 1099 (1996) 194{205. 6. D. G. Corneil, H. Lerchs and L. Stuart Burlingham, \Complement reducible graphs", Discrete Applied Mathematics , 3 (1981) 163{174. 7. B. Courcelle, \The monadic second-order logic of graphs. II. In nite graphs of bounded width", Mathematical System Theory , 21 (1989) 187{121. 8. B. Courcelle, \The monadic second-order logic of graphs. III. Tree-width, forbidden minors and complexity issues", RAIRO Inform. Theor. Appl., 26 (1992) 257{286. 9. A. Ehrenfeucht, J. Engelfriet and G. Rozenberg, \Finite Languages for the Representation of Finite Graphs", Journal of Computer and System Sciences , 52 (1996) 170{184. 10. J. Engelfriet, T. Harju, A. Proskurowski and G. Rozenberg, \Characterization and Complexity of Uniformly Non-primitive Labeled 2-Structures", Theoretical Computer Science , 154 (1996) 247{282. 11. J. Hopcroft and J. Ullman, \Introduction to Automata Theory, Formal Languages and Computation" Addison-Wesley Series in Computer Science (Addison-Wesley Publishing Company) (1979). 12. N. Robertson and P. Seymour, \Graph Minors. II Algorithmic aspects of treewidth", Journal of Algorithms , 7 (1986) 309{322. 13. J. Valdez, R. E. Tarjan and E. Lawler, \The recognition of series parallel digraphs", SIAM Journal of Computing , 11 (1982) 298{313. 14. www.unisa.it/papers/g.ps.gz
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