Abstract

ARTICLES

Algebraic theory for the clique operator^* * The authors acknowledge the financial support of Argentinian agency Fund for Quality Improvement of High Education (FOMEC) and Brazilian agencies: the State of São Paulo Research Foundation (FAPESP), the National Council for Scientific and Technological Development (CNPq), and Vitae.

Marisa Gutierrez^I; João Meidanis^II

^IDepartamento de Matemática, Universidad Nacional de La Plata, C. C. 172, (1900) La Plata, Argentina, marisa@mate.unlp.edu.ar

^IIInstituto de Computação, Universidade Estadual de Campinas, Cx. Postal 6176, 13084-971 Campinas-SP, Brazil, meidanis@ic.unicamp.br

ABSTRACT

In this text we attempt to unify many results about the K operator based on a new theory involving graphs, families and operators. We are able to build an ''operator algebra'' that helps to unify and automate arguments. In addition, we relate well-known properties, such as the Helly property, to the families and the operators.

As a result, we deduce many classic results in clique graph theory from the basic fact that CS = I for conformal, reduced families. This includes Hamelink's construction, Roberts and Spencer theorem, and Ban-delt and Prisner's partial characterization of clique-fixed classes [2]. Furthermore, we show the power of our approach proving general results that lead to polynomial recognition of certain graph classes.

Keywords: Helly graphs, intersection graphs

1 Introduction

In this text graphs are finite, undirected, and simple. In addition, we will not be concerned with the particular representation of graphs, so isomorphic graphs will be the same graph for us.

The clique operator (denoted by K) takes a graph G and returns the intersection graph of the maximal cliques of G. Besides being among the best studied graph operators (such as the line graph, the block graph and the power graph operators, for instance — see Prisner's book Graph Dynamics for a review [14]), K is far beyond the most interesting one.

Many questions about K are still open. Probably the most important of them is the one related to the computational complexity of recognizing clique graphs (i.e., graphs that are K(G) for some 67). Even though a characterization of clique graphs due to Roberts and Spencer [16] implies that the problem is in NP, still nobody knows whether it is polynomially solvable or NP-complete.

A good portion of the research on the clique operator has focused on graph classes. Images under K of chordal graphs, clique-Helly graphs, trees, and many others have been characterized. Many properties of these images have also been discovered. For instance, Escalante [3] has shown that the class of clique-Helly graphs is clique-fixed.

In the last few years [8, 9, 10] we have been trying to unify the results about the K operator based on the following:

As a result of our research we have reached the conclusion that many important results in clique graph theory rely on the fact that CS = I for conformal, reduced families. This includes Hamelink's construction, Roberts and Spencer theorem, and Bandelt and Pris-ner's partial characterization of clique-fixed classes [2]. We demonstrate the power of our approach proving general results that lead to known polynomial recognition of certain classes.

The remaining of this paper is organized as follows. Section 2 contains the basic definitions, including those about the operators. Sections 3 and 4 contain definitions and results about several graph classes previously studied, but with proofs in the context of our new theory. Section 5 presents our concluding remarks.

2 Definitions and basic results

2.1 Graphs, families, and their classes

Next we introduce graphs, families, and their classes. To minimize misunderstanding throughtout the text, we try and use a somehow consistent notation, summarized as follows.

Vertices: lowercase román letters (u, v, ...).

Graphs: uppercase román letters (G, H,...).

Graph Classes: boldface names (Chordal, ...).

Elements: lowercase román letters (u, v, ...).

Sets: uppercase román letters (A, B,...).

Families: calligraphic letters (, , , ...).

Family Classes: slanted names (Conformal, ...).

In addition, the following diagram shows ''hierarchical'' roles of these objects.

2.1.1 Graphs

For a set V, let [V]² denote the set of all two-element subsets of V. A graph is a pair (V, E) where F is a finite, nonempty set and E Í [V]². The elements of V are vertices and the elements of E are edges. Notation for edges: {u,v} = uv or vu.

Two graphs (V,E) and (V',E') are isomorphic when there is a bijection a : V

A class of graphs is a subset of graphs closed under isomorphism.

A graph (V,E) is a subgraph of a graph (V',E') when V Í V and E Í E'. Notation: (V, E)<(V', E'). In addition, (V, E) is an induced subgraph of (V',E') when E = E' Ç [V]². Notation: (V, E) Í (V', E').

Notice that G Í H implies G < H .

A set C of vertices of a graph (V, E) is complete when any two vertices of C are adjacent. A maximal complete subset of V is called a clique.

2.1.2 Families

A family is a pair (I, F) where I is a finite, nonempty set and F is a mapping from I to the class of all sets such that F(i) is a finite, nonempty set for all i Î I. We usually denote the set F(i) by F(i) and a family (I, F) by F_iÎI. We will call elements the elements of _iÎI F_i and members the sets F_i.

Two families = (F_i)_iÎI and = (A_j)_jÎJ are isomorphic when there are two bijections a : I J and b : _iÎI F_i

A class of families is a subset of families closed under isomorphism.

A family (F_i)_iÎI is a subfamily of (or: is contained in) a family (A_j )_jÎJwhen I Í J and F_i = A_ifor all i Î I. Notation: (F_i)_iÎI Í (A_j )_jÎJ.

There is still another important relation among families:

A family (F_i)_iÎI is below another family (A_j )_jÎJwhen there is a mapping a : I J such that F_i Í A_a(i) for all i Î I. Notation: (F_i)_iÎI< (A_j )_jÎJ.

In this case we also say that (A_j)_jÎJ is above (F_i)_iÎI.

This relation is a preorder, that is, it is reflexive and transitive, but not antisymmetric. However, it is antisymmetric (and hence a partial order) for reduced families, defined later in Section 2.3.

Notice that Í implies < .

2.2 Operators

Let Graph be the class of all graphs and Family be the class of all families.

We define the intersection operator L : Family

We define the f amily-of- diques operator C : Graph

We define the dual operator D : Family

It is important that families do not have empty members. Otherwise all we would get is that D²() would be a subfamily of instead of the stronger result in Theorem 1.

We define the two-section operator S : Family

Theorem 1 If, are families, G and H are graphs, we have:

2.3 Reduced and separating families

A family (F_i)_iÎI is reduced when i ¹ j Þ F_i F_j for all pairs i, j Î I .

It is straightforward to verify that this property is invariant under isomorphism, so we can speak of the class of all reduced families, namely, Reduced.

Theorem 2 If and are reduced families with < and < , then = .

Proof: Let = (F_i)_iÎI and = (A_j)_jÎJ. If < and < there are mappings a : I J and a' : J such that F_iÍ A_a(i) for all i Î I, and A_jÍ F_a'(j) for all j Î I.

Then we have that

and

But is a reduced family then i = a' (a(i)) for all i Î I. Exchanging the roles of and we can also obtain that aa' is the identity. Hence a and a' are bijections. By (1), F_i = A_a(i). Therefore (F_i)_iÎI and (A_j )_jÎJare isomorphic. Hence = .

Let (F_i)_iÎIbe a family. We say that u Î È_iÎI F_i is separated by the family when _iÎI F_i = {u}. A family is separating when it separates every element in _iÎI F_i.

It is straightforward to verify that this property is invariant under isomorphism , so we can speak of the class of all separating families, namely, Separating.

The following theorem tells us that both properties, reduced and separating are dual.

Theorem 3 A family is separating if and only if its dual is reduced.

2.4 Operators for reduction and separation

In this section we introduce operators that make a family either reduced or separating, without changing its image under either S or L. So, we are looking at four operators: one that reduces maintaining the image under S, another that separates maintaining the image under S, and two others that do these things maintaining the image of the family under L.

The first operator, called M (for ''maximal sets''), acts in the following way. Given = (F_i)_iÎI, throw away all f¿ properly contained in another F_i', then remove duplicates (if any) , creating a subfamily. This operator has the following properties:

Theorem 4 For any family we have: is reduced.

• M () is reduced.

• M () Í < M ()

• SM () = S ()

Thus M reduces a family, maintaining its image under S.

The second operator, called U, acts as follows. Given a family F = (F_i)_iÎI, it adds members of the form {u} for each u Î _iÎIF_i. It has the following properties:

Theorem 5 For any family we have:

• U() is separating.

• U() <Í U()

• SU() = S()

Thus U separates a family, maintaining its image under S.

The analogous operators for L can be readily obtained from M and U since LD = S and SD = L.

Theorem 6 For any family we have:

• DMD() is separating.

• LDMD() = L()

• DUD() is reduced.

• LDUD() = L().

2.5 Helly and conformal families

A family (F_i)_iÎI is called intersecting when F_i Ç F_j¹ for all pairs i,j Î I. A family (F_i)_iÎI is Helly or has the Helly property when all its intersecting subfamilies of the form (F_i)_iÎI', for ¹ I' Í I, have a nonempty intersection.

It is straightforward to verify that this property is invariant under isomorphism, so we can speak of the class of Helly families. We denote by Helly this class.

Theorem 7 (Three-Point Condition) A family (F_i)_iÎI is Helly if and only if for each triple x,y,z Î _iÎI F_i there is an element w Î _iÎI F_i with

for all i Î I.

A family is conformal when its dual is a Helly family. We denote by conformal the class of all conformal families.

Theorem 8 (Three-Set Condition) A family (F_i)_iÎI is conformal if and only if for each triple i,j, k Î I there is an index I Î I with

2.6 Operators for conformalization and hellization

The reader may have noticed that a family can be made Helly by addition of elements, and can be made conformal by addition of sets. This section formalizes these results. It turns out that one can always make a family Helly maintaining its image under L, but not always under S. For conformal, the situation is dual: one can easily conformize maintaining the image under S, but not, in general, maintaing the image under L.

Let us start with conformization. The composite operator C S performs the desired task. First, a useful lemma.

Lemma 1 If is conformal then CS() Í .

Proof: Notice that a clique R in S() is a clique in LD(). Since D() is Helly, and R is a maximal intersecting subfamily of D(), there is an element in its common intersection, which is a set in . This set contains R, and, by maximality of R, this set is actually equal to R. Then R is a member of .

Theorem 9 For any family we have:

• CS() is conformal and reduced.

• SCS() = S()

• Family is conformal if and only if CS() = M()

• If is conformal and reduced, then CS() =

• Ij is any conformal family with S(') = S(), then M(') =CS().

Proof: The first statement is true because the family of cliques of any graph is conformal and reduced. In fact, it satisfies the Three-Set Condition, because (F_i Ç F_j) È (F_j Ç F_k ) È (F_kÇ F_i ) is a complete set whenever F_i, F_j, and F_k are cliques. And every complete set is contained in some clique. Also, cliques form a reduced family because they are maximal complete sets.

The second statement is an immediate consequence of SC = I. For the third one, first note that < CS() for every family . Together with Lemma 1 and Theorem 2, this shows that families and CS() are each one below the other one, that is, differ only by contained sets. Hence they have the same reduction, but since CS() is reduced, this implies CS() = M(). The reverse follows by noticing that is conformal if and only if M() is conformal, by the Three-Set Condition. Of course, if T is also reduced, then CS() = , which is the fourth statement.

For the last one, notice that S(') = S() implies CS(') = CS(). Since' is conformal, CS becomes M, and the result follows.

We make a little digression here to offer the converse of the result

Theorem 10 If is conformal, then S() < S() implies < .

Proof: We have

The first relation is a general property of CS, the second is true by hypothesis and because C preserves <, and the third is a consequence of the conformality of .

Coming back to our main theme, we have seen that CS produces actually a conformal, reduced family while maintaining S. This shows that S (conformal) = Graph. Moreover, unlike operators M and U, this is essentially the only way to make a family conformal maintaining S, since any other such family differs from the one given by CS by contained sets.

On the other hand, there is no general technique for conformalization maintaining L. Not all families can be fixed in this way.

Let us now look at hellization procedures. The situation is analogous. The operator DCSD = DCL produces a Helly family with the same L (showing that L(Helly) = Graph), and there is no way, in general, to produce a Helly family with the same S as a given family. Deciding which families are hellizable maintaining 5 is equivalent to deciding which graphs are clique graphs.

2.7 Composition of operators

Compositions of the operators L, C, D, S have several important properties. We already had the opportunity to see some trivial: DD is the identity for families, SC is the identity for graphs, LD = S, and SD = L. Table 1 shows the other possible compositions of these operators and what we know about them.

In this table, I denotes the identity operador. We use the same letter for the identity in Graph as in Family.

The composition LC is K by definition, of course.

The compositions C S and CL are very interesting. Under certain conditions they can be simplified to I and D, respectively, and this will play a key role in determining the image under K of several graph classes. In Theorem 9 we saw that for CS previously. The result can be easily passed to CL by composing with D.

Theorem 11 We have

• Family is Helly if and only if CL() = MD().

• If is Helly and separating then CL() = D().

The behavior of these compositions leads to a trivial result about the injectivity of S and L.

Theorem 12 S is an injective operator for conformal, reduced families. L is an injective operator for Helly, separating families.

In addition, the following is true: C(Graph) = conformal Ç Reduced.

2.8 Complete edge covers

During our study of the clique operator, we often need to search for ''a family of complete sets that covers all the edges of a graph''. For instance, Roberts and Spencer [16] look for such a family that has the Helly property. It is interesting to note that there are other ways of expressing this concept, one of them using the < relation for families, and the other using the S operator.

A technical detail: to handle isolated vertices correctly, we require that the family covers the edges and the vertices. For graphs without isolated vertices, covering the edges is enough.

Let G be a graph and a family. We say that = (F_i)_iÎI is a complete edge cover of G = (V, E) when V = _iÎI F_i, F_i is a complete set of G for each i Î I, and uv G Î Þ $i Î I with u, v Î F_i.

Define also A(G) for G a graph as follows. If G = (V,E), A(G) is the family (F_i)_iÎI where I = V È E, F_i = {v} for i Î V, and F_i = {u, v} for i = uv ÎE. The family A(G) is the family of all vertices and edges of G.

Theorem 13 For a graph G and a family the following are equivalent:

1. is a complete edge cover of G.

2. A(G) << C(G).

3. G = S()

Proof: 1) Þ 2): A(G) < because all vertices and edges of G must be contained in a member of . On the other hand each member of is a complete set of G and it is contained in a clique of G. Hence < C(G).

2) Þ 3): By Theorem 1 we have that SA(G)< S()< SC(G). But both extremes of the inequality are G, and then S() = G.

3) Þ 1): Since G < S(G) all vertices and edges of G are contained in some member of . But also S()< G then all members of are complete sets of G. Hence is a complete edge cover of G.

3 Classes

In this work we will be interested in several particular classes of graphs and families. This section summarizes the definitions and some properties of these classes.

The operators of Section 2.2 were defined for graphs and families, but they can be extended to classes in the usual way. For instance,

L(Class) = {L() | Î Class}

and so on.

3.1 Family classes

Table 2 presents most of the family classes we study. We have the following class containments:

Interval Í RDTP-V Í DTP-V Í TP-V Í Subtree

Interval Í RDTP-E Í DTP-E Í TP-EÇHelly Í TP-E

Interval Í CircularArc

Apart from these we have the already defined classes Reduced, Separating, Helly, and conformal. It is interesting to see how these classes relate to one another. Table 3 shows some of this information.

3.2 Graph classes

Each family class defined above can potentially lead to two graph classes, by taking its image under the operators S and L. Sometimes S (Class) = L(Class), but this is not true for any of the above families.

Table 4 shows the corresponding graph classes.

In general by A (respectively DuallyA) we denote the class of graphs L (A) (respectively S (A)).

There is also a class Helly of graphs whose family of cliques satisfies the Helly property, that is, Helly = C^–1 (Helly).

4 Using the operators

Now we will prove several known results in the setting of the present paper, and relate the constructions with the operators.

4.1 Intersection and two-section classes

We try to unify properties of all classes of graphs defined as intersection or two-section graphs of a family class. Let A be a family class and L(A) (resp. S (A)) the class of all graphs obtained by intersection (resp. two-section) of families in A. First, we will obtain trivial characterizations for classes L(A) and S(A).

As we saw in Theorem 13, G = S(F) is equivalent to ''F is a complete edge cover of G''. It follows that G Î S(A) is equivalent to ''there is a complete edge cover of G in A.''

Theorem 14 If A Í conformal and M(A) Í A then

G Î S(A) Û C(G) Î A.

Proof: If G = S () with Î A, then C(G) = CS() = M(), because is conformal (see Theorem 9). However, this last family is in A, since A is closed under M. The converse is true because SC = I.

The dual of this Theorem is:

Theorem 15If A Í Helly and DMD(A) Í A then

G Î L(A) Û DC(G) Î A.

Example 1 It is easy to see that Helly, Helly Ç Conformal, Subtree, TP-E Ç Helly, DTP-V, DTP-E, RDTP-V, and Interval are Helly. To prove that all of them are closed under DMD, a hint is to think that DMD() is the family obtained from by deleting all dominated elements in . Then, applying Theorem 15 we obtain several results that have been obtained in different works by different authors:

Fulkerson-Gross's Theorem [4], which characterizes interval graphs, Gavril's results [5, 6, 7] for chordal graphs, UV-graphs, and RDV-graphs, and the Clique-tree Theorem [13] for DV-graphs and other families.

4.2 The behaviour of K

In this section we will show how we can obtain results related to the clique operator. First, we will obtain a trivial result about clique-families of graphs in L(A), from Theorem 11.

Theorem 16 If A Í Helly then CL(A) = MD(A).

The following corollary gives a sufficient condition for all the graphs obtained as intersection graphs of families in A to have the Helly property in their clique-family.

Corollary 1If A Í Helly Ç conformal then L(A) Í Helly.

Proof: By Theorem 16 we have CL(A) = MD(A). But D(A) is Helly because A is conformal, and subfamilies of Helly families are also Helly families.

Example 2 Since DTP-V, DTP-E, RDTP-V and Interval are Helly and conformal families, we obtain that the clique family of graphs in each of DV, DE, RDV and Interval have the Helly property. In other words, the classes DV, DE, RDV and Interval are contained in Helly.

In the following theorem we will see how the behaviour of the clique operator is similar in some classes of intersection graphs.

Theorem 17 [8, 9] If (i) A Í Helly, (ii) DMD(A) Í A, and (iii) U (A) Í A then KL(A) = S(A).

Proof:

Conversely

Example 3 As we said previously Helly, Helly Ç Conformal, Subtree, TP-En Helly, DTP-V, DTP-E, RDTP-V , and Interval are Helly families closed under DMD. It is trivial that all of them are also closed under U. Then, applying Theorem 17 we obtain several results that have been obtained in different works by different authors.

• K(Graph) = S (Helly)

• K(Helly) = Helly [3]

• K(Chordal) = DuallyChordal [1, 17]

• K(UEH) = DuallyUEH [9].

• K(DV) = DuallyDV [15]

• K(DE) = DuallyDE [10]

• K(RDV) = DuallyRDV [15]

• K(Interval) = Indifference [12]

Clearly we can obtain a dual result from Theorem 17.

Theorem 18 [8] If A Í Conformal, M (A) Í A, and DUD(A) Í A then KS(A) = L(A).

Then we can obtain the Bandelt-Prisner result about clique fixed classes [2], as well as prove the behaviour of the K operator in several classes.

• K(DuallyUEH) = UEH.

• K(DuallyDV) = DV [15]

• K(DuallyDE) = DE

• K(DuallyRDV) = RDV [15]

• K(Indifference) = Indifference [12]

In Figures 1,2, and ³ we show the effect of K and its iterations on some classes.

Some of these new classes, such as Dually A, have been characterized by properties of a complete edge cover of their graphs using the equivalence between G Î S (A) and ''there is a complete edge cover of G in A.'' In particular, since K(Graph) = S(Helly), we can also obtain Roberts and Spencer's Theorem for clique graphs [16].

Unfortunately this characterization does not lead to a good algorithm for recognition of these new classes of graphs. Nevertheless, for some particular classes a general polynomial time algorithm works, as we will see in the following section.

4.3 An algorithm for recognizing two-sections

In this section we rephrase the techniques of Prisner and Szwarcfiter [15] in terms of operators and apply them to a generic class of graphs A. Prisner and Szwarcfiter define the graph G' obtained from G by adding a new vertex v' and an edge vv' for each v Î V(G). The result we are interested in focuses on class DV, the class of intersection graphs of paths of directed trees, viewed as sets of vertices [13], and on DuallyDV, its image under K. Prisner and Szwarcfiter show that

This result in some sense reduces the recognition of DuallyDV to the recognition of DV. We will try to generalize the idea for the other classes that appear in the last column of Table 2. All these classes, with the exception of Indifference and Helly, lack a polynomial time recognition algorithm. Since they were defined as the image under the clique operator K of a recognized class, a natural idea is study an ''inverse'' of K. More clearly, if we want to know whether a graph G is in DuallyA it is sufficient to find a graph H in A such that K(H) = G. However, the inverse image of each graph is an infinite set. What element of this inverse image is convenient to select?

Prisner and Szwarcfiter used K(G') for this purpose. In fact, it is not difficult to prove directly that KK(G') is G for every clique-Helly graph G. We will use the same construction, but we would like to point out two interesting facts here. First, K(G') can be written in terms of operators as LUC(G). Second, the construction of K(G'), or LUC(G), as we will call it from here on, is the very one used by Hamelink in his celebrated proof that all clique-Helly graphs are clique graphs [11].

In the sequel we rewrite result (3) above, but replacing DV by a generic class A which shares with DV some fundamental properties. The proofs use operator techniques, and the fundamental properties mentioned were deduced from the fact that they are the ones needed for the proofs to work.

Theorem 19 If (i) A Í Helly, (ii) CS(A) Í A, (iii) DMD(A) Í A, and (iv) U (A) Í A then

Proof: Since G Î DuallyA = S(A) there is a family Î A such that G = S(). But C(G) = CS() Î A by (ii). Hence C(G) is Helly by (i), and conformal and reduced like all families of cliques. Then LUC(G) e L(A), by (iv).

To prove the converse we will show that KLUC(G) = G and thus G will be a graph in S(A) by Theorem 17. Indeed

The classes Interval, RDTP-V, DTP-V, RDTP-E, DTP-E, Subtree and TP-E Ç Helly satisfy the hypotheses of this theorem. But in order to obtain a polynomial time algorithm to recognize S(A) we need a polynomial upper bound on the number of cliques of these graphs.

Theorem 20 If class A is Interval, RDTP-V, DTP-V, RDTP-E, DTP-E, or TP-E Ç Helly, then we have that:

G Î S(A) Û |C(G)|< n(n + 1)/2,

where n = |V(G)|.

Proof: Recall that all these classes are conformal. Hence if G Î S(A) , there is a family in A such that G = S(). Then C(G) = CS() = M() since A is conformal. In other words, the maximal members of are exactly the cliques of S(). But, in these particular cases, it is clear that there are at most n(n + 1)/2 different maximal members of because each one is a path and therefore is determined by two elements: the end points.

Unfortunately S (Subtree) is a graph class for which there is no polynomial bound on the number of cliques, because any graph with a universal vertex is in this class.

Thus, these results show that if A is a class of graphs recognizable in polynomial time that fulfills the hypotheses of Theorems 19 and 20, then K(A) = DuallyA will be recognizable in polynomial time as well. As seen, this is the case of the classes UEH,DV,DE,RDV and Interval.

5 Conclusion

In this text we attempted to unify many results about the K operator based on a new theory involving graphs, families and operators. We were able to build an ''operator algebra'' that helps to unify and automate arguments. In addition, we related well-known properties, such as the Helly property, to the families and the operators.

As a result, we deduced many classic results in clique graph theory from the basic fact that CS = I for conformal, reduced families. This includes Hamelink's construction, Roberts and Spencer theorem, and Ban-delt and Prisner's partial characterization of clique-fixed classes [2]. Furthermore, we showed the power of our approach proving general results that lead to polynomial recognition of certain graph classes.

6 Acknowledgments

We acknowledge the financial support of Argentinian agency Fund for Quality Improvement of High Education (FOMEC) and of Brazilian agencies: the State of Sao Paulo Research Foundation (FAPESP), the National Council for Scientific and Technological Development (CNPq), and Vitae Foundation. Vitae does not necessarily share the concepts and opinions expressed in this work, for which the authors are solely responsible.

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