## Pesquisa Operacional

##
*Print version* ISSN 0101-7438*On-line version* ISSN 1678-5142

### Pesqui. Oper. vol.24 no.3 Rio de Janeiro Sept./Dec. 2004

#### https://doi.org/10.1590/S0101-74382004000300006

**Computational complexity of classical problems for hereditary clique-helly graphs**

**Flavia Bonomo ^{I}; Guillermo Durán^{*}^{, II}**

^{I}Departamento de Computación, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Buenos Aires – Argentina, fbonomo@dc.uba.ar

^{II}Departamento de Ingeniería Industrial, Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile, Santiago – Chile, gduran@dii.uchile.cl

**ABSTRACT**

A graph is clique-Helly when its cliques satisfy the Helly property. A graph is hereditary clique-Helly when every induced subgraph of it is clique-Helly. The decision problems associated to the stability, chromatic, clique and clique-covering numbers are NP-complete for clique-Helly graphs. In this note, we analyze the complexity of these problems for hereditary clique-Helly graphs. Some of them can be deduced easily by known results. We prove that the clique-covering problem remains NP-complete for hereditary clique-Helly graphs. Furthermore, the decision problems associated to the clique-transversal and the clique-independence numbers are analyzed too. We prove that they remain NP-complete for a smaller class: hereditary clique-Helly split graphs.

**Keywords:** computational complexity; hereditary clique-Helly graphs; split graphs.

**1. Introduction**

All graphs in this paper are finite, without loops or multiple edges. For a graph *G* we denote by *V*(*G*) and *E*(*G*) the vertex set and the edge set of *G*, respectively.

A graph is complete if every pair of vertices is connected by an edge. A complete in a graph *G* is a subset of pairwise adjacent vertices of *G*. A clique in a graph is a complete maximal under inclusion. The clique number of a graph *G* is the cardinality of a maximum clique of *G* and is denoted by w(*G*).

The chromatic number c(*G*) of a graph *G* is the smallest number of colours that can be assigned to the vertices of *G* in such a way that no two adjacent vertices receive the same colour.

A clique cover of a graph *G* is a subset of cliques covering all the vertices of *G*. A clique-transversal is a set of vertices intersecting all the cliques of *G*. The clique-covering number *k*(*G*) and the clique-transversal number t_{C}(*G*) are the cardinalities of a minimum clique cover and a minimum clique-transversal of *G*, respectively.

A stable set in a graph *G* is a subset of pairwise non-adjacent vertices of *G*. A clique-independent set is a subset of pairwise disjoint cliques of *G*. The stability number a(*G*) and the clique-independence number a_{C}(*G*) are the cardinalities of a maximum stable set and a maximum clique-independent set of *G*, respectively.

Consider a finite family of non-empty sets. The intersection graph of this family is obtained by representing each set by a vertex, two vertices being connected by an edge if and only if the corresponding sets intersect. The clique graph *K*(*G*) of *G* is the intersection graph of the cliques of *G*.

A family *S* of subsets satisfies the Helly property when every subfamily of *S* consisting of pairwise intersecting subsets has a common element. A graph is clique-Helly (*CH*) when its cliques satisfy the Helly property. A graph *G* is hereditary clique-Helly (*HCH*) when *H* is clique-Helly for every induced subgraph *H* of *G*. These graphs have been characterized in [Pr93] as the graphs which contains none of the four graphs in Figure 1 as an induced subgraph. This characterization leads to a polynomial time recognition algorithm for hereditary clique-Helly graphs.

An interesting survey on clique-Helly and hereditary clique-Helly graphs appears in [Fa02].

A graph is split if its vertices can be partitioned into a clique and a stable set.

The neighborhood of a vertex *v* in a graph *G* is the set *N*(*v*) consisting of all the vertices that are adjacent to *v*. The closed neighborhood of *v* is *N*[*v*]=*N(v*) È {*v*}. A vertex *v* of *G* is called simplicial when *N*[*v*] is a complete of *G*, and universal when *N*[*v*]=*V*(*G*).

It is easy to see that the decision problems associated to the stability, chromatic, clique and clique-covering numbers are NP-complete for clique-Helly graphs. The reduction is trivial: we have to add a universal vertex to the general graph *G* in order to generate a clique-Helly graph *G*^{+}.

However, w(*G*) can be obtained in polynomial time for *HCH* graphs. The number of cliques is bounded by the number of edges [Pr93] and all the cliques can be generated in *O*(*nmk*), where *m* is the number of edges, *n* the number of vertices and *k* the number of cliques of the graph [TIAS77].

The stable set and the colorability problems remain NP-complete for *HCH* graphs. These results are direct corollaries of the NP-completeness of these problems for triangle-free graphs [Pol74], [MP96]. For triangle-free graphs, a subclass of *HCH* graphs, the clique-covering number can be obtained in polynomial time [GJ79].

So, the following question arises naturally: what happens with the complexity of the clique-cover problem for hereditary clique-Helly graphs?

The decision problems associated to the problems of finding the clique-independence number and the clique-transversal number are NP-complete [CFT93] and NP-hard [EGT92], respectively. This last problem is not known to be in NP, in fact the problem of determining if a subset of vertices is a clique-transversal is NP-hard [DLS02].

The clique-transversal problem is NP-complete for *HCH* graphs. Again, this result is a consequence of the NP-completeness of this problem for triangle-free graphs. In this class of graphs, the clique-transversal problem is equivalent to vertex cover, and vertex cover is NP-complete for triangle-free graphs [Pol74]. Remember that in this case the problem is in NP for the property of *HCH* graphs above mentioned. This problem remains NP-complete for split graphs [GP00].

However, the clique-independence number can be obtained in polynomial time for triangle-free graphs, because it is equivalent in this case to maximum matching. This problem is NP-complete for split graphs [GP00] but, to our knowledge, it was not known its complexity for clique-Helly graphs.

Again, the following question appears naturally: what happens with the complexity of the clique-independence problem for hereditary clique-Helly graphs?

In this note, we prove that clique-cover and clique-independence problems remain NP-complete for *HCH* graphs. Additionally, it is proved that clique-transversal and clique-independence problems remain NP-complete for a smaller class: the intersection between *HCH* and split graphs.

**2. Preliminaries**

There are some relations between the parameters defined in the introduction in a graph *G* and its clique graph *K*(*G*).

**Theorem 2.1** *Let G be a graph. Then:*

(i)a

_{C}(G) = a(K(G)).(ii)

If G is a clique-Helly graph thent_{C}(G) =k(K(G)).

*Proof: *(i) It follows from the fact that independent cliques of *G* correspond to non adjacent vertices in *K*(*G*), and conversely, non adjacent vertices in *K*(*G*) correspond to independent cliques in *G*.

(ii) Let *v*_{1} be a clique-transversal set of *G*. For each *i*, analyze the vertices in *K*(*G*) corresponding to the cliques in *G* that contain the vertex *v*_{i}. They form a complete of *K*(*G*). This complete must be included in some clique *L*_{i} of *K*(*G*). Observe that these cliques *L*_{i} (*i* = 1,...,t_{C}(G)) are not all necessarily different. Let us see that these at most t_{C}(G) cliques are a clique cover of *K*(*G*). Let *w* be a vertex of *K*(*G*). Then *w* corresponds to some clique *M _{w}* of

*G*. As the set

*v*

_{1},..., intersects all the cliques of

*G*, there is some vertex

*v*

_{j}that belongs to

*M*. This means that the corresponding vertex of

_{w}*M*in

_{w}*K*(

*G*) belongs to the clique

*L*

_{j}, i.e,

*w*Î

*L*

_{j}. Then, the size of the minimum clique cover of

*K*(

*G*) is at most the size of this clique cover which is at most t

_{C}(G).

All we need to prove is that if *G* is clique-Helly, then t_{C}(G)* < k*(

*K*(

*G*)). By the Helly property, each clique

*L*of

*K*(

*G*) has an associated vertex

*v*

_{L}in

*G*such that

*v*

_{L}belongs to all the cliques of

*G*corresponding to the vertices of

*L*in

*K*(

*G*).

Let *L*_{1} ,...,*L*_{k(K(G))} be a clique cover of *K*(*G*). Let be the vertices in *G* associated to those *k*(*K*(*G*)) cliques. Let us see that they form a clique-transversal set of *G*. Let *M* be a clique of *G* and *w _{M}* its corresponding vertex in

*K*(

*G*). Then there is an index

*j*such that

*w*belongs to the clique

_{M}*L*

_{j}in

*K*(

*G*). It follows that belongs to

*M*in

*G*.

Let *M*_{1},...,*M*_{K} and *v*_{1},...,*v*_{n }be the cliques and vertices of a graph *G*, respectively. A clique matrix *A _{G}* Î of

*G*is a 0-1 matrix whose entry

*a*

_{ij}is 1 if

*v*

_{j}Î

*M*

_{i}, and 0, otherwise. Another characterization of

*HCH*graphs is the following [Pr93]: a graph

*G*is

*HCH*if and only if

*A*does not contain a vertex-edge incidence matrix of a triangle as a submatrix.

_{G}Let *M*_{1},...,>*M*_{K} and *v*_{1},...,*v*_{n } be the cliques and vertices of a graph *G*, respectively. Define the graph *H*(*G*) where *V*(*H*(*G*)) ={*q*_{1},...*q*_{w}, *w*_{1},...*w*_{n}} each *q*_{i} corresponds to the clique *M*_{i} of *G*, and each *w*_{j} corresponds to the vertex *v*_{j} of *G*. The edges of *H*(*G*) are the following: the vertices *q*_{1},...*q*_{w} induce the graph *K*(*G*), the vertices *w*_{1},...*w*_{n} are a stable set and *w*_{j} is adjacent to *q*_{i} if and only if *v*_{j} belongs to the clique *M*_{i} in *G*.

Let *A* Î and *B* Î be two matrices. We define the matrix *A | B* Î as (*A | B*)(*i, j*) for *i*=1,¼,*n*, *j*=1,¼,*m* and *A | B*)(*i, m + j*) for *i*=1,¼,*n*, *j*=1,¼,*k*. Let *I*_{n} be the *n × n* identity matrix.

**Theorem 2.2** [Ham68] *Let G be a clique-Helly graph and H(G) as it is defined above. Then the cliques of H(G) are N*[*w*_{i}]* for each i, **w*_{i}* is a simplicial vertex of H(G) for every i, and K(H(G)) = G.*

**Corollary 2.1** *Let G be a clique-Helly graph*, |*V*(*G*)| = *n*. *Then A*_{H(G)} = | *I*_{n}.

*Proof:* It follows directly from the fact that *N*[*w*_{i}] (*i*=1,¼,*n*) are the cliques of *H*(*G*) and each clique contains the vertex *w*_{i} and the vertices *q*_{j} whose corresponding cliques *M*_{j} contain the vertex *v*_{i} in *G*.

This corollary leads us to prove the following result:

**Theorem 2.3 ** *Let G be an HCH graph. Then H(G) is HCH.*

*Proof: *Let *G* be an *HCH* graph, |*V*(*G*)| = *n*. By Corollary 2.1, *A*_{H(G)} = | *I*_{n}. Let us suppose that *A*_{H(G)} contains a vertex-edge incidence matrix of a triangle as a submatrix. Since it has two 1's in each column, it must be a submatrix of *A*_{H(G)}, but then *A _{G}* contains a vertex-edge incidence matrix of a triangle as a submatrix, which is a contradiction.

**3. Clique Cover**

The decision problem associated to the problem of finding the clique-covering number of a graph is the following:

**CLIQUE COVER**

INSTANCE: A graph *G* = (*V*,*E*), a positive integer *K* __<__|*V*|.

QUESTION: Are there *k* __<__ *K* cliques of *G* covering all the vertices of *G* ?

To prove that CLIQUE COVER is NP-complete for *HCH* graphs, we will use that the following problem is NP-complete [GJ79]:

**EXACT COVER BY 3-SETS (X3C)**

INSTANCE: A set *X* such that |*X*|=3*q* and a collection *C* of 3-element subsets of *X*.

QUESTION: Does *C* contain an exact cover (by *q* sets) of *X* ?

**Theorem 3.1 ** *The problem CLIQUE COVER is NP-complete for HCH graphs.*

*Proof: *The transformation from X3C to CLIQUE COVER on *HCH* graphs is based on the transformation given in [GJ79] from X3C to PARTITION INTO TRIANGLES and is the following: let the set *X* with |*X*|=3*q* and the collection *C* of 3-element subsets of *X* be an arbitrary instance of X3C. We will construct an *HCH* graph *G*=(*V*,*E*), with |*V*|=3*q*', such that *G* has a clique cover of size at most *q*' if and only if *C* contains an exact cover of *X*.

We will replace each subset *c*_{i} = {*x*_{i}, *y*_{i}, *z*_{i}} in *C* by the graph of Figure 2. Let *E*_{i} be the set of 18 edges of the graph corresponding to *x*_{i}, *y*_{i}, *z*_{i}}.

Thus *G*=(*V*,*E*) is defined by

It is easy to see that *G* does not contain any graph of Figure 1 as an induced subgraph, thus *G* is an *HCH* graph, |*V*| = |*X*| + 9|*C*| (*q*' = *q* + 3|*C*|) and the transformation can be made in polynomial time. Figure 3 shows an example of this transformation from an instance of X3C to an instance of CLIQUE COVER.

Let us suppose that *C* contains an exact cover of *X*, then we construct a clique cover of *G* of size *q*', by taking for each 1__<__ *i*__<__|*C*|

{*a*_{i}[1],*a*_{i}[2],*x*_{i}}, {*a*_{i}[4],*a*_{i}[5],*y*_{i}}, {*a*_{i}[7],*a*_{i}[8],*z*_{i}}, {*a*_{i}[3],*a*_{i}[6],*a*_{i}[9]},

whenever *c*_{i} = {*x*_{i}, *y*_{i}, *z*_{i}} is in the exact cover and

{*a*_{i}[1],*a*_{i}[2],*a*_{i}[3]}, {*a*_{i}[4],*a*_{i}[5],*a*_{i}[6]}, {*a*_{i}[7],*a*_{i}[8],*a*_{i}[9]},

otherwise.

Let us now suppose that *G* has a clique cover of size at most *q*'. Since the cliques of *G* are triangles, the number of cliques in the clique cover must be *q* and each vertex of *G* must be covered exactly once.

In the graph of Figure 2, the only two ways of covering by triangles each vertex *a*_{i}[*j*] (*j*=1,¼,9) exactly once are the above mentioned, covering or not *x*_{i}, *y*_{i} and *z*_{i}, respectively. Then the exact cover of *X* is given by choosing those *c*_{i} Î *C* such that {*a*_{i}[3],*a*_{i}[6], *a*_{i}[9]}, belongs to the clique cover of *G*.

Finally, the membership in NP for the restricted problem follows from that for the general problem.

**4. Clique Transversal and Clique-Independent Set**

The decision problems associated to the problems of finding the clique-independence number and the clique-transversal number of a graph, respectively, are the following:

**CLIQUE-INDEPENDENT SET**

INSTANCE: A graph *G* = (*V*,*E*), a positive integer *K* __<__|*V*|>.

QUESTION: Is there a set of *K* or more pairwise disjoint cliques of *G* ?

**CLIQUE-TRANSVERSAL**

INSTANCE: *G* = (*V*,*E*), a positive integer *K* __<__|*V*|.

QUESTION: Is there a set of *K* or fewer vertices of *G* intersecting all the cliques of *G* ?

**Theorem 4.1** *The problems CLIQUE-TRANSVERSAL and CLIQUE-INDEPENDENT SET are NP-complete for HCH split graphs.*

*Proof: *We will show a polynomial time transformation from CLIQUE COVER on *HCH* graphs (by Theorem 3.1 it is NP-complete) to CLIQUE-TRANSVERSAL on *HCH* split graphs.

Define the graph *G*^{+} where *V*(*G*^{+}) = *V*(*G*) È {*u*}, *V*(*G*), induces the graph *G* and *u* is a universal vertex. Since for any graph *G* all the cliques of *G*^{+}> share the vertex *u*, the graph *K*(*G*^{+}) is complete and thus the graph *H*(*G*^{+}) is a split graph.

Let *G* be an *HCH* graph. As the set of cliques of an *HCH* graph has polynomial size and can be computed in polynomial time, *H*(*G*^{+}) can be built in polynomial time. By Theorem 2.3, since *G*^{+} is an *HCH* graph, *HCH* is an *HCH*(*G*^{+}) graph. By Theorem 2.2 *K*(*H*(*G*^{+})) = *G*^{+}, and by Theorem 2.1 *K*(*G*) = *K*(*G*^{+}) = t_{C}> (*H*(*G*^{+})). Finally, the problem of determining if a subset of vertices is a clique-transversal is solvable in polynomial time for *HCH* graphs, and therefore CLIQUE-TRANSVERSAL is NP-complete for *HCH* split graphs.

In a similar way, using the equality (*G*) = a (*G*^{+}) = a_{C}(*H*(*G*^{+})) instead of *K*(*G*) = *K*(*G*^{+}) = t_{C} (*H*(*G*^{+})), and the NP-completeness of the STABLE SET problem for *HCH *graphs, CLIQUE-INDEPENDENT SET is NP-complete for *HCH* split graphs.

**Corollary 4.1** *The problem CLIQUE-INDEPENDENT SET is NP-complete for HCH graphs.*

**Acknowledgements**

To the anonymous referees for their careful reading and valuable suggestions which improved this work.

The first author was partially supported by UBACyT Grant X184, PICT ANPCyT Grant 11-09112 and PID CONICET Grant 644/98, Argentina and "International Scientific Cooperation Program CONICyT/SETCIP", Chile-Argentina.

The second author was partially supported by FONDECyT Grant 1030498 and Millennium Science Nucleus "Complex Engineering Systems", Chile and "International Scientific Cooperation Program CONICyT/SETCIP", Chile-Argentina.

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Recebido em 04/2004; aceito em 09/2004

Received April 2004; accepted September 2004

* *Corresponding author* / autor para quem as correspondências devem ser encaminhadas