SciELO - Scientific Electronic Library Online

 
vol.50 issue1Exponents of Modular Reductions of Families of Elliptic CurvesFunctional versions of the Caristi-Kirk theorem author indexsubject indexarticles search
Home Pagealphabetic serial listing  

Services on Demand

Article

Indicators

  • Have no cited articlesCited by SciELO

Related links

  • On index processCited by Google
  • Have no similar articlesSimilars in SciELO
  • On index processSimilars in Google

Bookmark


Revista de la Unión Matemática Argentina

Print version ISSN 0041-6932

Rev. Unión Mat. Argent. vol.50 no.1 Bahía Blanca June 2009

 

Exponential families of minimally non-coordinated graphs

Francisco Soulignac* and Gabriel Sueiro

Abstract. A graph G is coordinated if, for every induced subgraph H of G, the minimum number of colors that can be assigned to the cliques of H in such a way that no two cliques with non-empty intersection receive the same color is equal to the maximum number of cliques of H with a common vertex. In a previous work, coordinated graphs were characterized by minimal forbidden induced subgraphs within some classes of graphs. In this note, we present families of minimally non-coordinated graphs whose cardinality grows exponentially on the number of vertices and edges. Furthermore, we describe some ideas to generate similar families. Based on these results, it seems difficult to find a general characterization of coordinated graphs by minimal forbidden induced subgraphs.

* Partially supported by UBACyT Grant X184, Argentina and CNPq under PROSUL project Proc. 490333/2004-4, Brazil.

1. INTRODUCTION

Let G be a graph, with vertex set V (G ) and edge set E(G ) . Denote by -- G the complement of G . Given two graphs G and G′ we say that G contains G ′ if G′ is isomorphic to an induced subgraph of G . If X ⊆ V (G ) , we denote by G \ X the subgraph of G induced by V (G ) \ X .

A complete set or just a complete of G is a subset of vertices pairwise adjacent. A clique is a complete set not properly contained in any other. We may also use the term clique to refer to the corresponding complete subgraph. Let X and Y be two sets of vertices of G . We say that X is complete to Y if every vertex in X is adjacent to every vertex in Y , and that X is anticomplete to Y if no vertex of X is adjacent to a vertex of Y . A complete of three vertices is called a triangle.

The neighborhood of a vertex v is the set N (v) consisting of all the vertices which are adjacent to v . The closed neighborhood of v is N [v] = N (v) ∪ {v } . A vertex of G is simplicial if N (v) is a complete. Equivalently, a vertex is simplicial if it belongs to only one clique.

Given a graph G , we will denote by Q (G ) the set of cliques of G . Also, for every v ∈ V (G ) , we will denote by Q (v) the set of cliques containing v . Finally, define m (v) = |Q (v)| .

The chromatic number of a graph G is the smallest number of colors that can be assigned to the vertices of G in such a way that no two adjacent vertices receive the same color, and is denoted by χ(G ) . An obvious lower bound is the maximum cardinality of the cliques of G , the clique number of G , denoted by ω(G ) .

A graph G is perfect if χ (H ) = ω (H ) for every induced subgraph H of G . Perfect graphs were defined by Berge in 1960 [1] and are interesting from an algorithmic point of view: while determining the chromatic number and the clique number of a graph are NP-hard problems, they are solvable in polynomial time for perfect graphs [8].

A hole is a chordless cycle of length at least 4 . An antihole is the complement of a hole. A hole or antihole is said to be odd if it consists of an odd number of vertices.

Given a graph G , the clique graph K (G ) of G is the intersection graph of the cliques of G . A K-coloring of a graph G is an assignment of colors to the cliques of G such that no two cliques with non-empty intersection receive the same color (equivalently, a K-coloring is a coloring of K (G ) ). A Helly K-complete of a graph G is a collection of cliques of G with common intersection. A Helly K-clique is a maximal Helly K-complete. The K-chromatic number and Helly K-clique number of G , denoted by F (G ) and M (G ) , are the sizes of a minimum K-coloring and a maximum Helly K-clique of G , respectively. It is easy to see by definition that F (G ) = χ(K (G )) and that M (G) = maxv ∈V (G )m (v) . Also, F (G ) ≥ M (G ) for any graph G . A graph G is coordinated if F (H ) = M (H ) for every induced subgraph H of G . Coordinated graphs were defined and studied in [3]. There are three main open problems concerning this class of graphs:

(i) find all minimal forbidden induced subgraphs for the class of coordinated graphs,

(ii) determine the computational complexity of finding the parameters M and F for coordinated graphs and/or some of their subclasses, and

(iii) is there a polynomial time recognition algorithm for the class of coordinated graphs?

Recently, in [2] and [4], questions (i) and (ii) were answered partially. For question (iii), it is shown in [9] that the problem is NP-hard and it is NP-complete even when restricted to a subclass of graphs with M = 3 . In this note, we answer a question related to these problems, which is: how many minimally non-coordinated graphs with n vertices and M = 3 are there? In particular, we show (algorithmic) operations for generating a family Gk of minimally non-coordinated graphs, of size 2k , such that every graph of the family has 16k + 9 vertices and 24k + 14 edges, for every k . It is not difficult to see that the operations we give are not enough for generating every minimally non-coordinated graph with M = 3 , so the question of how to generate every minimally non-coordinated graph is still open.

2. GENERATING NON-COORDINATED GRAPHS

It has been proved recently that perfect graphs can be characterized by two families of minimal forbidden induced subgraphs [7] and recognized in polynomial time [6].

Theorem 1 (Strong Perfect Graph Theorem [7]). Let G be a graph. Then the following are equivalent:

(i) no induced subgraph of G is an odd hole or an odd antihole.

(ii) G is perfect.

Coordinated graphs are a subclass of perfect graphs [3]. Moreover, --- C7 , --- C8 , --- C9 , ---- 2P4 are minimally non-coordinated [35]. Therefore, --- Ck is not coordinated for all k ≥ 7 .

In [24] and manuscript [5], partial characterizations of coordinated graphs by minimal forbidden induced subgraphs were found. In these partial characterizations, the families of minimal forbidden induced subgraphs with k vertices have O(1) size for every k . Another partial characterization which is not difficult to prove (see [9] for a sharper result) is the following.

Theorem 2. Let G be a graph such that M (G ) ≤ 2 . Then the following are equivalent:

(i) G is perfect.

(ii) G does not contain odd holes.

(iii) G is coordinated.

Corollary 3. Let G be a graph with M (G ) ≤ 3 . Then G is coordinated if and only if G does not contain odd holes and F (G ) ≤ 3 .

The aim of this note is to show, for every k , a family of minimally non-coordinated graph of size  k 2 such that every graph has 16k + 9 vertices and 24k + 14 edges. In order to define our families, we are going to use exchanger and preserver graphs which were defined in [9].

A graph G exchanges colors between two different vertices v1,v2 , or simply G is an exchanger between v1,v2 , if G satisfies the following conditions:

(i) No induced subgraph of G is an odd hole.

(ii) F (G ) = M (G ) = 3 .

(iii) m (v1) = m (v2) = 2 .

(iv) Every induced path between v ,v 1 2 has odd length.

(v) In any 3-K-coloring of G the cliques of Q (v1) ∪ Q(v2) are colored with the three colors.

Vertices v 1 and v 2 are called connectors. Call redundant to every simplicial vertex z ∈ V (G) such that K (G \ {z} ) = K (G ) . We say that an exchanger G is a minimal exchanger if and only if G \ {z} does not satisfy condition (v) for every non-redundant vertex z (although this is not the standard way to define minimality, this minimality is useful for "joining" G with other graphs, because redundant vertices of G may be needed so that the cliques of G are also cliques of the joined graph). Please note that conditions (i) and (iv) are hereditary and that if G satisfies (i) and (ii) but G \ {z} does not satisfy (ii) then, by Theorem 2 , G \ {z} does not satisfy (v).

A graph G preserves colors between a set of distinct vertices {v1, ...,vk} , or simply G is a preserver between v1,...,vk , if G satisfies the following conditions:

(i) No induced subgraph of G is an odd hole.

(ii) F (G ) = M (G ) ≤ 3 .

(iii) m (vi) = 1 for every i ∈ {1,...,k } .

(iv) Every induced path between vi,vj has odd length for all 1 ≤ i < j ≤ k .

(v) In any 3-K-coloring of G the cliques of ⋃k i=1 Q(vi) are colored with only one color.

Vertices v1,...,vk are called connectors. We say that a preserver G is a minimal preserver if and only if G \ {z} does not satisfy condition (v) for every non-redundant vertex z . Please note that conditions (i), (ii) and (iv) are hereditary and that condition (v) is satisfied only if condition (iii) is also satisfied.

Let G1, G2 be two graphs (V (G1 ) ∩ V (G2) may be non-empty). The graph G = G1 ∪ G2 has vertex set V(G ) = V (G1) ∪ V(G2 ) and edge set E (G ) = E(G1 ) ∪ E (G2 ) .

We say that graphs G1, G2,...,Gk are compatible when {Q (G1 ), Q (G2 ), ..., Q (Gk )} is a partition of  ⋃ Q ( ki=1Gi ) . We call them minimally compatible if they are compatible and ⋃k G i=1 i contains no redundant vertices.

Theorem 4. Let GX be an exchanger between v,w and GP be a preserver between v,w such that GX, GP are compatible and V (GX ) ∩ V (GP ) = {v, w} . Then G = GX ∪ GP is non-coordinated. Moreover, if both GX and GP are minimal and minimally compatible then G is minimally non-coordinated.

Proof. Since {Q (GP ),Q (GX )} is a partition of Q (G ) and mGX (v) + mGP (v) = mGX (w) + mGP (w) = 3 by definition, then it follows that mG (v) = mG (w ) = 3 , mG (u) = mGX (u) for every u ∈ V (GX ) \ {v,w } and mG (u) = mGP (u) for every u ∈ V (GP ) \ {v,w } . Consequently, M (G ) = 3 .

Suppose, contrary to our claim, that G is coordinated. Then, there exists a 3 -K-coloring cG of G . Since the cliques of GX are also cliques of G , then the K-coloring cGX obtained by restricting the domain of cG to the cliques of GX is a 3 -K-coloring of GX . Analogously, define cGP which is a 3 -K-coloring of GP . Since GX is an exchanger then there exist cliques Q1,Q2, Q3 such that cGX (Qi) = i for i ∈ {1,2,3} where, w.l.o.g., v ∈ Q ∩ Q 1 2 and w ∈ Q 3 . Also, since GP is a preserver, there exist cliques R1,R2 such that cGP (R1) = cGP (R2) where v ∈ R1 and w ∈ R2 . Therefore, c(Qi) = i for i ∈ {1, 2,3} and 1 ≤ c(R1 ) = c(R2) ≤ 3 which is a contradiction because Q1,Q2, Q3, R1 and R2 are pairwise different, v ∈ Q1 ∩ Q2 ∩ R1 and w ∈ Q3 ∩ R2 . Consequently, G is a non-coordinated graph.

From now on, suppose that GP and GX are both minimal and minimally compatible. Let us see that G contains no odd hole. On the contrary, suppose G contains an odd hole C . Since neither GP nor GX contains odd holes and V(GP ) \ {v,w } is anticomplete to V (GX ) \ {v,w } , then it follows that C can be partitioned into two paths P1, P2 from v to w with disjoint interior and such that P1 is an induced path of GP and P2 is an induced path of GX . But this is impossible, because every induced path between v and w has odd length in both GP and GX , by definition.

Let H be any proper induced subgraph of G ; we have to prove that F (H ) = M (H ) . Since G contains no odd hole then, by Corollary 3, if M (H ) ≤ 2 it follows that H is coordinated. Thus, it suffices to prove that if M (H ) = 3 then F (H ) = 3 which is equivalent to prove that F (G \ {u}) ≤ 3 for every vertex u ∈ V(G ) (because F (H ) ≤ F (G) for every induced subgraph H of G ). We divide the proof into three cases:

Case 1: u = v (u = w is analogous). If G \ {v} = GX \ {v} then F (G \ {v}) ≤ 3 . Otherwise, let cGX be a 3 -K-coloring of GX \ {v} where the cliques of Q (w ) are colored using colors from the set {1,2} , and let cGP be a 3 -K-coloring of GP \ {v} where the clique to which w belongs has color 3 . Since G \ {v } ⁄= GX \ {v} and GP \ {v } is connected it follows that the clique to which w belongs in GP has at least one more vertex, thus it is still a clique in G \ {v } . Therefore cGP ∪ cGX is a valid 3 -K-coloring of G \ {v} .

Case 2: u ∈ V (GP ) \ {v, w} . Suppose first that u is a redundant vertex of GP , that is, u is simplicial and K (GP \ {u}) = K (GP ) . Since K (G \ {u }) ⁄= K (G ) (because GP and GX are minimally compatible) and u is simplicial in G , then it follows that N (u) is a complete but not a clique of G \ {u } . Since Q (G ) is the disjoint union of Q (GP ) and Q (GX ) and N (u) is a clique of GP , it follows that N (u) ⊆ V (GX ) and N (u) is complete in GX . Hence N (u) ⊆ {v,w } . Since mGP (v) = mGP (w ) = mGP (u) = 1 and GP is connected, it follows that GP is a triangle, consequently G \ {u} = GX and F(G \ {u }) ≤ 3 . Now, suppose that u is not simplicial or K (GP \ {u}) ⁄= K (GP ) . Then, GP \ {u} has a 3 -K-coloring cGP where the clique containing v has color 1 and the clique containing w has color 2 (because GP is minimal). Let cGX be a 3 -K-coloring of GX where the cliques containing v have colors 1,2 and the cliques containing w have colors 1, 3 . Then c ∪ c GP GX is a valid 3 -K-coloring of G \ {u} .

Case 3: u ∈ V (GX ) \ {v,w } . By minimality, if u is not redundant then GX \ {u } has a 3 -K-coloring cGX where the cliques of QGX (v) ∪ QGX (w ) are colored with at most two colors, say 2, 3 . Let cGP be a 3 -K-coloring of GP where the cliques containing v and w all have color 1 . Then cGP ∪ cGX is a 3 -K-coloring of G \ {u } . If u is redundant then, as before, it follows that N [u] is the clique {u,v,w } in GX and v,w belong to a clique Q in GP . Let cGX be a 3 -K-coloring of GX where {u,v, w} has color 1 and cGP be a 3 -K-coloring of GP where Q has color 1 . Then (cGX ∪ cGP) \ { {u,v,w } ↦→ 1 } is a 3 -K-coloring of G \ {u } .

3. THE EXPONENTIAL FAMILIES

In this section we define a recursive operator for constructing exponentially many non-isomorphic exchangers. Then we use these exchangers and one preserver to generate the exponential families, as in Section 2 . More operators for constructing exchangers and preservers are shown in the next section.

Let u ⁄= v be vertices of a graph G1 and w ⁄= z be vertices of G2 . The graph G = J oin(G1, G2,uw, vz) has V(G ) = V (G1) ∪ V(G2 ) and E (G ) = E(G1 ) ∪ E (G2 ) ∪ {uw, vz} . In particular, when G1 is an exchanger between u,v , G2 is a preserver between w, z , and G1 ∩ G2 = ∅ , we denote J oin(G ,G ) = J oin (G ,G ) = Join (G ,G ,uw, vz) 1 2 2 1 1 2 . Figure 2 shows examples of this operation, using graphs in Figure 1

GP1 GP2 GX

Figure 1. Preservers GP1 and GP2 and exchanger GX . In every graph the connectors are v and w .

Join(Join(GX, GP1 ),GP2 ) Join(Join(GX, GP2 ),GP1 )

Figure 2. Join(J oin (GX, GP1 ),GP2 ) and J oin(Join(GX, GP2 ),GP1 ) are shown. Both graphs are minimal exchangers with connectors v and w .

Lemma 5. Let GX be an exchanger between u, v and GP be a preserver between w, z where w is not adjacent to z . Suppose also that V (GX ) ∩ V (GP ) = ∅ . Then G = Join(GX, GP ) exchanges colors between w, z . Moreover, if both GX and GP are minimal then G is also minimal.

Proof. Arguments similar to those in Theorem 4 show that G contains no odd hole and that M (G ) = 3 . Also, since m (w ) = m (z) = 1 GP GP and {{u, w}, {v,z}} are cliques, it follows that mG (w ) = mG (z) = 2 .

Let P be an induced path between w,z . If P is also a path of GP then it must have odd length. If P is not a path of GX , then there must exists a subpath  ′ P which is a path of GX between u, v such that P = wuP ′vz . Since P ′ has odd length it follows that P also has odd length.

Suppose for a moment that G is 3 -K-colorable and let cG be a 3 -K-coloring of G . Let Q1 (u ) , Q2 (u) and Q1(v ) , Q2 (v) be the cliques of u and v in GX , respectively, and Q (w) and Q (z) be the cliques of w and z in GP , respectively. Since Q (G ) = Q (GX ) ∪ Q(GP ) ∪ {{u, w}, {v,z}} , and this union is disjoint, then it follows that the coloring cGX obtained by restricting the domain of cG to the cliques of GX is a 3 -K-coloring of GX . Analogously, define cGP . Therefore, we may assume without loss of generality that cG (Q1(u )) = 1 , cG (Q2 (u)) = 2 , cG(Q1 (v)) = 1 , c (Q (v)) = 3 G 1 . Then c ({u,w }) = 3 G , c ({v, z}) = 2 G and by definition of GP , cG(Q1 (w )) = cG(Q1 (z)) = 1 . Hence in every 3-K-coloring of G the cliques of Q(w ) ∪ Q (z) are colored with the three colors. By using the conditions derived for cG , it is easy to see that G has at least one 3 -K-coloring as supposed, thus F (G ) = M (G ) = 3 .

From now on, suppose GP and GX are both minimal. Let x ∈ V (GX ) . If x = u then w belongs to only one clique of G . Since GP is a preserver, the clique of w has the same color as one of the cliques of z , therefore, G \ {x } does not satisfy condition (v). The case x = v is analogous. If x ⁄∈ {u,v} is redundant in V (GX ) , then it is also redundant in G . If x ⁄∈ {u, v} is a non-redundant vertex of GX then, by minimality of GX , there exists a K-coloring cGX which contradicts condition (v) for GX \ {x} . As in Theorem 4, it is easy to combine c GX with a K-coloring of GP such that the coloring obtained is a K-coloring of G not satisfying condition (v). Similar arguments can be used to conclude the proof when x ∈ V (GP ) .

We are now ready to show the exponential families of minimally non-coordinated graphs. Let GP1 and GP2 be the preserver graphs and GX be the exchanger graph shown in Figure 1. Let F k (k ≥ 0 ) denote the family of minimal exchangers defined by:

F = {GX }. 0 Fk+1 = {J oin (Join(X, GP1 ),GP2 ),Join (J oin(X, GP2 ),GP1 )}X∈Fk

Both graphs of F 1 are shown in Figure 2. It is easy to see that graphs in F k are pairwise non-isomorphic and that  k |Fk | = 2 . Also, by construction,  ′ |V (G)| = |V (G )| = 16k + 5 and  ′ |E (G )| = |E (G )| = 24k + 7 for every G, G ′ ∈ Fk . Finally, every exchanger in Fk is a minimal exchanger by Lemma 5 , and is minimally compatible with GP1 (where connectors of GP1 are identified with the connectors of the exchanger). Now, define G = {X ∪ GP } k 1 X∈Fk for every k ≥ 0 . By Theorem 4 every graph in G k is minimally non-coordinated, that is, Gk is one of the exponential families.

4. OTHER OPERATIONS

In this section we show other recursive operations for constructing exchangers and preservers. The motivation is to show how exchangers and preservers can be "joined" in a recursive manner. The proofs that these operations generate exchangers and preservers are left to the reader, and can be done in a similar way as the one in the previous section. Instead, we are going to draw sketches showing how vertices of the input graphs are tied together. The components of these sketches are shown in Figure 3 . To represent a preserver P between v1,v2 we are going to draw an oval labeled with P together with two points labeled v1,v2 each one joined to the oval by a line. The oval represents the graph P \ {v1,v2} , the points represent v1 and v2 and the line between v1 (v2 ) and the oval represents the clique of v1 (v2 ). In a similar manner, to represent an exchanger X between v ,v 1 2 we are going to draw an oval labeled with X together with two points labeled v1,v2 each one joined to the oval by two lines. Again, the oval represents X \ {v1,v2} , the points represent vertices {v1,v2} and the lines represent their cliques (in this case, one line of v1 and one of v2 may represent the same clique). Finally, a clique with n vertices is represented by an oval labeled with K n and n points outside the oval, representing each of the n vertices. A line from one point to the oval means that the vertex belongs to the clique. Recall that a clique is a special kind of preserver, thus their vertices are also called connectors. Sometimes we also decorate the lines of the sketches with colors that represent a valid K-coloring.


Figure 3. Examples of sketches. On the left there is a component for an exchanger X , in the middle there is a sketch component for a preserver P between v2,v3 , and on the right there is a sketch for X ∪ P .

Let G ,G 1 2 be graphs which are preservers or exchangers between V 1 and V2 , respectively, where G1 ∩ G2 ⊆ V1 ∩ V2 , and let S1 and S2 be sketches representing G1 and G2 , respectively. We are going to represent the graph G1 ∪ G2 with a sketch formed by S1 and S2 , where for every vertex v of G1 ∩ G2 , the corresponding points of S1 and S2 are drawn as a single point. One of such sketches is shown in Figure 3.

The sketches in Figure 4 represent preservers between two vertices v1,v2 , the sketches in Figure 5 represent exchangers between v ,v 1 2 and the sketch in Figure 6 represents a preserver between v1,v2,v3,v4 . If the set of graphs of a sketch S are minimally compatible and minimal (as preservers or exchangers) then the graph represented by S is also minimal.


Figure 4. Two sketches of preservers between vertices v1 and v2 .


Figure 5. Two sketches of exchangers between v1,v2 .

 

PIC

Figure 6. A sketch of a preserver between v1,v2,v3,v4 .

5. CONCLUSIONS AND FURTHER REMARKS

In this note we have shown one exponential-size family of minimally non-coordinated graphs for every natural number, and several operations for building preservers and exchangers. It is not difficult to define other operations for constructing minimal preservers or exchangers in order to generate different families of minimally non-coordinated graphs. Also, adding edges to some minimally non-coordinated graph may result into another minimally non-coordinated graph. Moreover, it is not clear that preservers and exchangers are enough to define every minimally non-coordinated graph. Perhaps a set of basic graphs together with operations for generating minimally non-coordinated graphs can be defined in a more convenient way.

With all these observations it seems difficult to find a characterization of coordinated graphs by minimal forbidden induced subgraphs, even when we restrict our attention to the class of graphs with M = 3 . This is in turn a complementary result to that one in [9], which states that the problem of determining whether a graph in a very restricted subclass of graphs with M = 3 is coordinated is NP-complete.

References

[1]    Claude Berge. Les problèmes de coloration en théorie des graphes. Publ. Inst. Statist. Univ. Paris, 9:123-160, 1960.        [ Links ]

[2]    Flavia Bonomo, Maria Chudnovsky, and Guillermo Durán. Partial characterizations of clique-perfect graphs. I. Subclasses of claw-free graphs. Discrete Appl. Math., 156(7):1058-1082, 2008.        [ Links ]

[3]    Flavia Bonomo, Guillermo Durán, and Marina Groshaus. Coordinated graphs and clique graphs of clique-Helly perfect graphs. Util. Math., 72:175-191, 2007.        [ Links ]

[4]    Flavia Bonomo, Guillermo Durán, Francisco Soulignac, and Gabriel Sueiro. Partial characterizations of clique-perfect and coordinated graphs: superclasses of triangle-free graphs. Discrete Appl. Math., 2009. In press.        [ Links ]

[5]    Flavia Bonomo, Guillermo Durán, Francisco Soulignac, and Gabriel Sueiro. Partial characterizations of coordinated graphs: line graphs and complements of forests. Math. Methods Oper. Res., 69(2):251-270, 2009.        [ Links ]

[6]    Maria Chudnovsky, Gérard Cornuéjols, Xinming Liu, Paul Seymour, and Kristina Vuskovic. Recognizing Berge graphs. Combinatorica, 25(2):143-186, 2005.        [ Links ]

[7]    Maria Chudnovsky, Neil Robertson, Paul Seymour, and Robin Thomas. The strong perfect graph theorem. Ann. Math. (2), 164(1):51-229, 2006.        [ Links ]

[8]    M. Grötschel, L. Lovász, and A. Schrijver. The ellipsoid method and its consequences in combinatorial optimization. Combinatorica, 1(2):169-197, 1981.        [ Links ]

[9]    Francisco Soulignac and Gabriel Sueiro. Np-hardness of the recognition of coordinated graphs. Ann. Oper. Res., 169(1):17-34, 2009.         [ Links ]

Francisco Soulignac
Departamento de Computación
Facultad de Ciencias Exactas y Naturales
Universidad de Buenos Aires
Ciudad de Buenos Aires, Argentina
fsoulign@dc.uba.ar

Gabriel Sueiro
Departamento de Computación
Facultad de Ciencias Exactas y Naturales
Universidad de Buenos Aires
Ciudad de Buenos Aires, Argentina
gsueiro@dc.uba.ar

Recibido: 6 de agosto de 2007
Aceptado: 10 de julio de 2008