- Email: [email protected]
The Helly property on subhypergraphs
Electronic Notes in Discrete Mathematics 19 (2005) 71–77 www.elsevier.com/locate/endm
The Helly property on subhypergraphs Mitre C. Dourado a,1,3 , F´abio Protti b,2,4 , Jayme L. Szwarcfiter a,b,2,5 a b
COPPE - Sistemas
IM, NCE, Caixa Postal 2324, 20001-970
Universidade Federal do Rio de Janeiro, Brasil
Abstract The celebrated theorem by Helly (1923) motivated the study of p-Helly hypergraphs. In this context, Golumbic and Jamison introduced the notion of strong p-Helly hypergraphs (1985). In this paper we characterize strong p-Helly hypergraphs. This characterization leads to a polynomial-time algorithm, whenever p is fixed, for recognizing such hypergraphs. In contrast, we show that the recognition problem is co-NP-complete, for arbitrary p. Further, we apply the concept of strong p-Helly hypergraphs to the cliques of a graph, leading to the class of strong p-clique-Helly graphs. We describe a characterization for this class and obtain an algorithm for recognizing such graphs with polynomial-time complexity for p fixed, and we show the corresponding recognition problem to be NP-hard, for arbitrary p. Keywords: strong Helly property, hereditary clique-Helly graphs, computational complexity
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Supported by the Conselho Nacional de Desenvolvimento Cient´ıfico e Tecnol´ogico - CNPq. Partially supported by CNPq, and Funda¸ca˜o de Amparo a` Pesquisa do Estado do Rio de Janeiro - FAPERJ, Brasil. 3 Email:[email protected] 4 Email:[email protected] 5 Email:[email protected] 2
1571-0653/2005 Published by Elsevier B.V. doi:10.1016/j.endm.2005.05.011
72
1
M.C. Dourado et al. / Electronic Notes in Discrete Mathematics 19 (2005) 71–77
Introduction
Helly hypergraphs form a classical topic of combinatorics, since the celebrated Helly’s theorem (1923). On the other hand, the notion of strong p-Helly hypergraphs was introduced by Golumbic and Jamison [6]. In this paper, we characterize strong p-Helly hypergraphs. The characterization leads to an algorithm for recognizing hypergraphs of this class. The algorithm has polynomial-time complexity, whenever p is fixed. In contrast, we show that the recognition problem is co-NP-complete, for arbitrary p. We remark that [2] and [10] describe polynomial-time algorithms for the case p = 2. Further, we apply the concept of strong p-Helly graphs to the cliques of a graph, leading to the class of strong p-clique-Helly graphs. For p = 2, this class is equivalent to that of hereditary clique-Helly graphs, introduced by Prisner [8]. We describe a characterization for strong p-clique-Helly graphs, leading to a recognition algorithm. Again, the algorithm has polynomial-time complexity for fixed p, and we prove the recognition problem to be NP-hard whenever p is arbitrary. Let V be a finite set of vertices. A hypergraph on V is a family H = {E1 , E2 , . . . , Em } of nonempty subsets of V , called hyperedges or simply edges, such that the union of the edges of H is equal to V . The core of H is E1 ∩ E2 ∩ . . . ∩ Em . A hypergraph H is a partial hypergraph of H if every edge of H is also an edge of H; and H is a subhypergraph of H (induced by V ) if H contains exactly the nonempty sets Ei ∩ V , for every Ei ∈ H. We say that S is a p-set if |S| = p, a p− -set when |S| ≤ p, and a p+ -set when |S| ≥ p. Throughout the work, this notation will be applied to any term standing for a set. Given integers p and q, H is a p− -hypergraph if |H| ≤ p and H is (p, q)-intersecting if every partial p− -hypergraph of it has a q + -core. A hypergraph is (p, q, s)-Helly if every (p, q)-intersecting partial hypergraph of it has an s+ -core. The p-Helly property corresponds to! (p, 1, 1)-Helly and the Helly property to (2, 1, 1)-Helly. This generalization was introduced in [9] and characterized in [3]. The case (p, 1, 1)-Helly was characterized by Berge and Duchet [1]. A hypergraph H is strong p-Helly [6] if for every partial (p+1)+ -hypergraph H of H, there exist p edges in H whose core equals the core of H . A property formulated by Wallis and Zhang [10] leads to an algorithm recognizing strong 2-Helly hypergraphs with time complexity O(rm3 ), where r and m are, respectively, the rank and the number of edges of the hypergraph. A hypergraph H is hereditary p-Helly if all subhypergraphs of H are pˇ Helly. Bretto, Ub´eda and Zerovnik [2] have shown an algorithm for recognizing
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hereditary 2-Helly hypergraphs that needs O(m∆r 4 ) time and O(mr 2 ) space, where ∆ is the maximum degree of the hypergraph. A hypergraph is a graph if every edge has exactly two vertices and there are no repeated edges. We denote an edge of G containing the vertices u and v by uv and say that these vertices are adjacent. A graph G is a subgraph of a graph G if every edge of G is also an edge of G. Given a set C ⊂ V (G) we say that C is an independent set if any two vertices of C are not adjacent; that C is a complete set if any two vertices of C are adjacent; and a clique if C is a maximal complete set. The clique hypergraph of a graph G is the hypergraph whose edges are the cliques of G. A graph G is p-clique-Helly if its clique hypergraph is p-Helly, see [4]. We say that a graph is strong p-clique-Helly if its clique hypergraph is strong p-Helly and that it is hereditary p-clique-Helly if all induced subgraphs of it are p-clique-Helly. The hereditary cliq! ue-Helly graphs (p = 2) were characterized by Prisner [8]. It is important to note that a subhypergraph or a partial hypergraph of the clique hypergraph of a graph G is not necessarily the clique hypergraph of an induced subgraph of G.
2
Hypergraphs
Theorem 2.1 The following statements are equivalent for a hypergraph H = {E1 , E2 , . . . , Em }: (i) H is strong p-Helly; (ii) H is hereditary p-Helly; (iii) H is (p, q, q)-Helly, for every q; (iv) every partial (p + 1)-hypergraph of H is (p, q, q)-Helly for every q; (v) for any (p + 1)-subset V ∗ = {v1 , v2 , . . . , vp+1 } ⊂ V (H) and any partial (p+1)-hypergraph H∗ = {E1 , E2 , . . . , Ep+1 } of H such that V ∗ \{vi } ⊂ Ei , at least one vertex of V ∗ is in the core of H∗ . Proof. (i) ⇒ (ii) Suppose that H contains a subhypergraph H that is not p-Helly. Let H be a partial hypergraph of H which is p-intersecting with empty core. Define a partial hypergraph H∗ of H choosing for every edge E ∈ H the edge of H that originated it. Since any p edges of H contain one vertex that is not in the core of H , the same can be said to any p edges and the core of H∗ . Therefore H is not strong p-Helly. (ii) ⇒ (iii) Suppose that H is not (p, q, q)-Helly, for some q. Let H be a (p, q)-intersecting partial hypergraph of H without q-core. Denote the core of H by C . Every edge of H properly contains C because it belongs to a
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(p, q)-intersecting partial hypergraph and C is a (q − 1)− -set. Hence, in the subhypergraph H1 of H induced by V (H) \ C , there is one edge for every edge of H . Consider the partial hypergraph H1 of H1 formed by these edges. Note that H1 is (p, 1)-intersecting with empty core. Therefore H1 is not p-Helly. (iii) ⇒ (iv) Trivial. (iv) ⇒ (v) Consider V ∗ = {v1 , v2 , . . . vp+1 } and the partial hypergraph H∗ = {E1 , E2 , . . . Ep+1 } of H such that V ∗ \{vi } ⊂ Ei , but none of the vertices of V ∗ is in the core of H∗ . Denote by c the cardinality of the core of H∗ . Hence H∗ is (p, c + 1)-intersecting with c vertices in the core. Hence H ∗ is a (p + 1)-partial hypergraph that is not (p, c + 1, c + 1)-Helly. (v) ⇒ (i) Suppose that H is not strong p-Helly. So, it contains a partial hypergraph H1 such that the core of every p edges of H1 properly contain C1 , the core of H1 . Do the following process, starting with i = 1: if Hi contains one edge E such that the core of Hi \ {E} is C1 , define Hi+1 = Hi \ {E}, add 1 to i, and repeat; otherwise stop. Denote by Hj the last hypergraph obtained by the above iteration. The core of Hj is C1 , because in the process a hypergraph is built only if its core is C1 . This implies that Hj contains at least p + 1 edges, because it is a partial hypergraph of H1 and the core of every p or less edges of this hypergraph properly contains C1 . In addition, for every E ∈ Hj , the core of Hj \ {E} also properly contains C1 , because of the stop condition of the process. This implies that in the core of Hj \ {Ek }, for any Ek ∈ Hj , there is one vertex vk ∈ Ek . This allows us to write V ∗ = {v1 , v2 , . . . , vp+1 } and H∗ = {E1 , E2 , . . . , Ep+1 } such that none of the vertices of V ∗ is in the core of H∗ . 2 We can apply the equivalence (i)-(iv) in order to formulate an algorithm for recognizing strong p-Helly graphs, as follows. A (p + 1)-hypergraph H is (p, q, q)-Helly, for every q, if and only if the greatest q for which H is (p, q )-intersecting is equal to the cardinality of the core of H. This can be checked in O(p2 r) steps. Since the number of partial (p + 1)-hypergraphs of a hypergraph is O(mp+1 ), the corresponding algorithm has time complexity O(p2 rmp+1 ). Equivalence (i)-(iv) of Theorem 2.1 also leads to a polynomial time algorithm, for fixed p. We show that this problem is co-NP-complete for variable p. Theorem 2.2 It is co-NP-complete to decide if a hypergraph is strong p-Helly, for variable p. If a hypergraph is strong p-Helly, it is also strong (p + 1)-Helly. So we say that p is the strong Helly number of a hypergraph if this hypergraph is strong
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p-Helly but is not strong (p − 1)-Helly. Other variations of the Helly number are presented in [5]. Corollary 2.3 Given a hypergraph H and an integer p, it is NP-complete to decide if the strong Helly number of H is greater than p.
3
Cliques of graphs
The results of Theorem 2.1 are valid for general hypergraphs, and in particular for the clique hypergraph of a graph. However, since the number of cliques of a graph may be exponential in the size of the graph [7] these results do not lead to a polynomial-time algorithm for recognizing strong p-clique-Helly graphs. Similarly, the algorithm for recognizing p-clique-Helly graphs is also not suitable for recognizing hereditary p-clique-Helly graphs because the number of induced subgraphs may also be exponential in the size of the graph. If a graph is p-clique-Helly, then it is (p + 1)-clique-Helly. So if a graph is hereditary p-clique-Helly, then it is hereditary (p + 1)-clique-Helly. For every integer p ≥ 3 a graph G is p-ocular if V (G) is the union of the disjoint sets W = {w1 , w2 , ..., wp } and U = {u1 , u2 , ..., up }, where W is a complete set, U induces an arbitrary subgraph, and wi , uj are adjacent precisely when i = j. The 3-ocular graph corresponds to the ocular graph defined by Wallis and Zhang [10]. A graph is p-ocular free if it has not a p-ocular graph as an induced subgraph. Lemma 3.1 Any (p + 1)-ocular graph is not p-clique-Helly, p ≥ 2. Let G be a graph and C be a p-complete set of G. The p-expansion relative to C is the subgraph of G induced by the vertices that are adjacent to at least p − 1 vertices of C. A p-complete subset C of a (p + 1)-complete set C is good if any vertex adjacent to all vertices of C is also adjacent to the vertex of C\C . A vertex is universal if it is adjacent to any other vertex of the graph. Theorem 3.2 [4] A graph G is p-clique-Helly if and only if every (p + 1)expansion of G contains a universal vertex. Theorem 3.3 The following statements are equivalent for any graph G: (i) G is strong p-clique-Helly; (ii) G is hereditary p-clique-Helly; (iii) G is (p + 1)-ocular free; (iv) every (p + 1)-complete set of G contains a good p-complete subset.
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Proof. (i) ⇒ (ii) Suppose that G is not hereditary p-clique-Helly. So there is an induced subgraph G that is not p-clique-Helly. By Theorem 3.2, G contains a (p + 1)-expansion T , relative to a (p + 1)-complete set C = {u1 , u2 , . . . , up+1 }, without a universal vertex. This means for every vertex ui ∈ C that there is one vertex vi ∈ V (T ) adjacent to all vertices of C except ui . This implies the existence of p+1 cliques Q1 , Q2 , . . . , Qp+1 of G such that Qi ⊃ C \ {ui } ∪ {vi } and ui ∈ Qi . Therefore the cliques Q1 , Q2 , . . . , Qp+1 and the set V ∗ = {v1 , v2 , . . . , vp+1 } imply, via Theorema 2.1, that the clique hypergraph of G is not strong p-Helly. (ii) ⇒ (iii) Consequence of Lemma 3.1. (iii) ⇒ (iv) Suppose that G contains a (p + 1)-complete set which does not contain a good p-complete subset, denote it by W = {w1 , . . . , wp+1 }. Hence, for every vertex wi ∈ W there is a vertex ui ∈ W that is adjacent to all vertices of W except Wi . The vertex set {w1 , . . . , wp+1 , u1 , . . . , up+1 } induces a (p + 1)-ocular subgraph in G. (iv) ⇒ (i) Suppose that G is not strong p-clique-Helly. By Theorem 2.1 there is a (p + 1)-subset V ∗ = {v1 , v2 , . . . vp+1 } ⊂ V and the set of cliques H∗ = {Q1 , Q2 , . . . Qp+1 } such that V ∗ \{vi } ⊂ Qi and none of the vertices of V ∗ is in the core of H∗ . So vi ∈ Qi and, by its maximality, Qi contains a vertex adjcent to all vertices of V ∗ but not adjacent to vi . Since p ≥ 2 every two vertices of V ∗ are in a same clique, hence V ∗ is a (p + 1)-complete set which does not contain a good p-complete set. 2 To determine all (p + 1)-complete sets of a graph with n vertices we require O(n(p+1) ) steps. In order to verify for each one if it contains a good p-complete set, we need O(np) time. Therefore the complexity of the algorithm for recognizing hereditary p-clique-Helly graphs is O(pnp+2 ), which is polynomial for fixed p. We prove that, for variable p, this problem is NP-hard. Theorem 3.4 It is NP-hard to decide if a graph is hereditary p-clique-Helly, for variable p.
References [1] C. Berge and P. Duchet. A generalization of Gilmore’s theorem. In M. Fiedler, editor, Recent Advances in Graph Theory, pages 49–55. Acad. Praha, Prague, 1975. ˇ [2] A. Bretto, S. Ub´eda, and J. Zerovnik. A polynomial algorithm for the strong Helly property. Information Processing Letters, (81):55–57, 2002.
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[3] M. C. Dourado, F. Protti, and J. L. Szwarcfiter. On Helly hypergraphs with predescribed intersection sizes. Submitted. [4] M. C. Dourado, F. Protti, and J. L. Szwarcfiter. Characterization and recognition of generalized clique-Helly graphs. Proc. WG 2004 - 30th International Workshop on Graph-Theoretic Concepts in Computer Science, jun 2004. Lecture Notes in Computer Science, to appear. [5] M. C. Dourado, F. Protti, and J. L. Szwarcfiter. generalized Helly hypergraphs. Manuscript, 2005.
Complexity aspects of
[6] M. C. Golumbic and R. E. Jamison. The edge intersection graphs of paths in a tree. Journal of Combinatorial Theory, Series B, (38):8–22, 1985. [7] J. W. Moon and L. Moser. On cliques in graphs. Israel Journal of Mathematics, (3):23–28, 1965. [8] E. Prisner. Hereditary clique-Helly graphs. Journal of Combinatorial Mathematics and Combinatorial Computing, (14):216–220, 1993. [9] V. I. Voloshin. On the upper chromatic number of a hypergraph. Australasian Journal of Combinatorics, (11):25–45, 1995. [10] W. D. Wallis and Guo-Hui Zhang. On maximal clique irreducible graphs. The Journal of Combinatorial Mathematics and Combinatorial Computing, (8):187– 193, 1990.
The Helly property on subhypergraphs Mitre C. Dourado a,1,3 , F´abio Protti b,2,4 , Jayme L. Szwarcfiter a,b,2,5 a b
COPPE - Sistemas
IM, NCE, Caixa Postal 2324, 20001-970
Universidade Federal do Rio de Janeiro, Brasil
Abstract The celebrated theorem by Helly (1923) motivated the study of p-Helly hypergraphs. In this context, Golumbic and Jamison introduced the notion of strong p-Helly hypergraphs (1985). In this paper we characterize strong p-Helly hypergraphs. This characterization leads to a polynomial-time algorithm, whenever p is fixed, for recognizing such hypergraphs. In contrast, we show that the recognition problem is co-NP-complete, for arbitrary p. Further, we apply the concept of strong p-Helly hypergraphs to the cliques of a graph, leading to the class of strong p-clique-Helly graphs. We describe a characterization for this class and obtain an algorithm for recognizing such graphs with polynomial-time complexity for p fixed, and we show the corresponding recognition problem to be NP-hard, for arbitrary p. Keywords: strong Helly property, hereditary clique-Helly graphs, computational complexity
1
Supported by the Conselho Nacional de Desenvolvimento Cient´ıfico e Tecnol´ogico - CNPq. Partially supported by CNPq, and Funda¸ca˜o de Amparo a` Pesquisa do Estado do Rio de Janeiro - FAPERJ, Brasil. 3 Email:[email protected] 4 Email:[email protected] 5 Email:[email protected] 2
1571-0653/2005 Published by Elsevier B.V. doi:10.1016/j.endm.2005.05.011
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M.C. Dourado et al. / Electronic Notes in Discrete Mathematics 19 (2005) 71–77
Introduction
Helly hypergraphs form a classical topic of combinatorics, since the celebrated Helly’s theorem (1923). On the other hand, the notion of strong p-Helly hypergraphs was introduced by Golumbic and Jamison [6]. In this paper, we characterize strong p-Helly hypergraphs. The characterization leads to an algorithm for recognizing hypergraphs of this class. The algorithm has polynomial-time complexity, whenever p is fixed. In contrast, we show that the recognition problem is co-NP-complete, for arbitrary p. We remark that [2] and [10] describe polynomial-time algorithms for the case p = 2. Further, we apply the concept of strong p-Helly graphs to the cliques of a graph, leading to the class of strong p-clique-Helly graphs. For p = 2, this class is equivalent to that of hereditary clique-Helly graphs, introduced by Prisner [8]. We describe a characterization for strong p-clique-Helly graphs, leading to a recognition algorithm. Again, the algorithm has polynomial-time complexity for fixed p, and we prove the recognition problem to be NP-hard whenever p is arbitrary. Let V be a finite set of vertices. A hypergraph on V is a family H = {E1 , E2 , . . . , Em } of nonempty subsets of V , called hyperedges or simply edges, such that the union of the edges of H is equal to V . The core of H is E1 ∩ E2 ∩ . . . ∩ Em . A hypergraph H is a partial hypergraph of H if every edge of H is also an edge of H; and H is a subhypergraph of H (induced by V ) if H contains exactly the nonempty sets Ei ∩ V , for every Ei ∈ H. We say that S is a p-set if |S| = p, a p− -set when |S| ≤ p, and a p+ -set when |S| ≥ p. Throughout the work, this notation will be applied to any term standing for a set. Given integers p and q, H is a p− -hypergraph if |H| ≤ p and H is (p, q)-intersecting if every partial p− -hypergraph of it has a q + -core. A hypergraph is (p, q, s)-Helly if every (p, q)-intersecting partial hypergraph of it has an s+ -core. The p-Helly property corresponds to! (p, 1, 1)-Helly and the Helly property to (2, 1, 1)-Helly. This generalization was introduced in [9] and characterized in [3]. The case (p, 1, 1)-Helly was characterized by Berge and Duchet [1]. A hypergraph H is strong p-Helly [6] if for every partial (p+1)+ -hypergraph H of H, there exist p edges in H whose core equals the core of H . A property formulated by Wallis and Zhang [10] leads to an algorithm recognizing strong 2-Helly hypergraphs with time complexity O(rm3 ), where r and m are, respectively, the rank and the number of edges of the hypergraph. A hypergraph H is hereditary p-Helly if all subhypergraphs of H are pˇ Helly. Bretto, Ub´eda and Zerovnik [2] have shown an algorithm for recognizing
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hereditary 2-Helly hypergraphs that needs O(m∆r 4 ) time and O(mr 2 ) space, where ∆ is the maximum degree of the hypergraph. A hypergraph is a graph if every edge has exactly two vertices and there are no repeated edges. We denote an edge of G containing the vertices u and v by uv and say that these vertices are adjacent. A graph G is a subgraph of a graph G if every edge of G is also an edge of G. Given a set C ⊂ V (G) we say that C is an independent set if any two vertices of C are not adjacent; that C is a complete set if any two vertices of C are adjacent; and a clique if C is a maximal complete set. The clique hypergraph of a graph G is the hypergraph whose edges are the cliques of G. A graph G is p-clique-Helly if its clique hypergraph is p-Helly, see [4]. We say that a graph is strong p-clique-Helly if its clique hypergraph is strong p-Helly and that it is hereditary p-clique-Helly if all induced subgraphs of it are p-clique-Helly. The hereditary cliq! ue-Helly graphs (p = 2) were characterized by Prisner [8]. It is important to note that a subhypergraph or a partial hypergraph of the clique hypergraph of a graph G is not necessarily the clique hypergraph of an induced subgraph of G.
2
Hypergraphs
Theorem 2.1 The following statements are equivalent for a hypergraph H = {E1 , E2 , . . . , Em }: (i) H is strong p-Helly; (ii) H is hereditary p-Helly; (iii) H is (p, q, q)-Helly, for every q; (iv) every partial (p + 1)-hypergraph of H is (p, q, q)-Helly for every q; (v) for any (p + 1)-subset V ∗ = {v1 , v2 , . . . , vp+1 } ⊂ V (H) and any partial (p+1)-hypergraph H∗ = {E1 , E2 , . . . , Ep+1 } of H such that V ∗ \{vi } ⊂ Ei , at least one vertex of V ∗ is in the core of H∗ . Proof. (i) ⇒ (ii) Suppose that H contains a subhypergraph H that is not p-Helly. Let H be a partial hypergraph of H which is p-intersecting with empty core. Define a partial hypergraph H∗ of H choosing for every edge E ∈ H the edge of H that originated it. Since any p edges of H contain one vertex that is not in the core of H , the same can be said to any p edges and the core of H∗ . Therefore H is not strong p-Helly. (ii) ⇒ (iii) Suppose that H is not (p, q, q)-Helly, for some q. Let H be a (p, q)-intersecting partial hypergraph of H without q-core. Denote the core of H by C . Every edge of H properly contains C because it belongs to a
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(p, q)-intersecting partial hypergraph and C is a (q − 1)− -set. Hence, in the subhypergraph H1 of H induced by V (H) \ C , there is one edge for every edge of H . Consider the partial hypergraph H1 of H1 formed by these edges. Note that H1 is (p, 1)-intersecting with empty core. Therefore H1 is not p-Helly. (iii) ⇒ (iv) Trivial. (iv) ⇒ (v) Consider V ∗ = {v1 , v2 , . . . vp+1 } and the partial hypergraph H∗ = {E1 , E2 , . . . Ep+1 } of H such that V ∗ \{vi } ⊂ Ei , but none of the vertices of V ∗ is in the core of H∗ . Denote by c the cardinality of the core of H∗ . Hence H∗ is (p, c + 1)-intersecting with c vertices in the core. Hence H ∗ is a (p + 1)-partial hypergraph that is not (p, c + 1, c + 1)-Helly. (v) ⇒ (i) Suppose that H is not strong p-Helly. So, it contains a partial hypergraph H1 such that the core of every p edges of H1 properly contain C1 , the core of H1 . Do the following process, starting with i = 1: if Hi contains one edge E such that the core of Hi \ {E} is C1 , define Hi+1 = Hi \ {E}, add 1 to i, and repeat; otherwise stop. Denote by Hj the last hypergraph obtained by the above iteration. The core of Hj is C1 , because in the process a hypergraph is built only if its core is C1 . This implies that Hj contains at least p + 1 edges, because it is a partial hypergraph of H1 and the core of every p or less edges of this hypergraph properly contains C1 . In addition, for every E ∈ Hj , the core of Hj \ {E} also properly contains C1 , because of the stop condition of the process. This implies that in the core of Hj \ {Ek }, for any Ek ∈ Hj , there is one vertex vk ∈ Ek . This allows us to write V ∗ = {v1 , v2 , . . . , vp+1 } and H∗ = {E1 , E2 , . . . , Ep+1 } such that none of the vertices of V ∗ is in the core of H∗ . 2 We can apply the equivalence (i)-(iv) in order to formulate an algorithm for recognizing strong p-Helly graphs, as follows. A (p + 1)-hypergraph H is (p, q, q)-Helly, for every q, if and only if the greatest q for which H is (p, q )-intersecting is equal to the cardinality of the core of H. This can be checked in O(p2 r) steps. Since the number of partial (p + 1)-hypergraphs of a hypergraph is O(mp+1 ), the corresponding algorithm has time complexity O(p2 rmp+1 ). Equivalence (i)-(iv) of Theorem 2.1 also leads to a polynomial time algorithm, for fixed p. We show that this problem is co-NP-complete for variable p. Theorem 2.2 It is co-NP-complete to decide if a hypergraph is strong p-Helly, for variable p. If a hypergraph is strong p-Helly, it is also strong (p + 1)-Helly. So we say that p is the strong Helly number of a hypergraph if this hypergraph is strong
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p-Helly but is not strong (p − 1)-Helly. Other variations of the Helly number are presented in [5]. Corollary 2.3 Given a hypergraph H and an integer p, it is NP-complete to decide if the strong Helly number of H is greater than p.
3
Cliques of graphs
The results of Theorem 2.1 are valid for general hypergraphs, and in particular for the clique hypergraph of a graph. However, since the number of cliques of a graph may be exponential in the size of the graph [7] these results do not lead to a polynomial-time algorithm for recognizing strong p-clique-Helly graphs. Similarly, the algorithm for recognizing p-clique-Helly graphs is also not suitable for recognizing hereditary p-clique-Helly graphs because the number of induced subgraphs may also be exponential in the size of the graph. If a graph is p-clique-Helly, then it is (p + 1)-clique-Helly. So if a graph is hereditary p-clique-Helly, then it is hereditary (p + 1)-clique-Helly. For every integer p ≥ 3 a graph G is p-ocular if V (G) is the union of the disjoint sets W = {w1 , w2 , ..., wp } and U = {u1 , u2 , ..., up }, where W is a complete set, U induces an arbitrary subgraph, and wi , uj are adjacent precisely when i = j. The 3-ocular graph corresponds to the ocular graph defined by Wallis and Zhang [10]. A graph is p-ocular free if it has not a p-ocular graph as an induced subgraph. Lemma 3.1 Any (p + 1)-ocular graph is not p-clique-Helly, p ≥ 2. Let G be a graph and C be a p-complete set of G. The p-expansion relative to C is the subgraph of G induced by the vertices that are adjacent to at least p − 1 vertices of C. A p-complete subset C of a (p + 1)-complete set C is good if any vertex adjacent to all vertices of C is also adjacent to the vertex of C\C . A vertex is universal if it is adjacent to any other vertex of the graph. Theorem 3.2 [4] A graph G is p-clique-Helly if and only if every (p + 1)expansion of G contains a universal vertex. Theorem 3.3 The following statements are equivalent for any graph G: (i) G is strong p-clique-Helly; (ii) G is hereditary p-clique-Helly; (iii) G is (p + 1)-ocular free; (iv) every (p + 1)-complete set of G contains a good p-complete subset.
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Proof. (i) ⇒ (ii) Suppose that G is not hereditary p-clique-Helly. So there is an induced subgraph G that is not p-clique-Helly. By Theorem 3.2, G contains a (p + 1)-expansion T , relative to a (p + 1)-complete set C = {u1 , u2 , . . . , up+1 }, without a universal vertex. This means for every vertex ui ∈ C that there is one vertex vi ∈ V (T ) adjacent to all vertices of C except ui . This implies the existence of p+1 cliques Q1 , Q2 , . . . , Qp+1 of G such that Qi ⊃ C \ {ui } ∪ {vi } and ui ∈ Qi . Therefore the cliques Q1 , Q2 , . . . , Qp+1 and the set V ∗ = {v1 , v2 , . . . , vp+1 } imply, via Theorema 2.1, that the clique hypergraph of G is not strong p-Helly. (ii) ⇒ (iii) Consequence of Lemma 3.1. (iii) ⇒ (iv) Suppose that G contains a (p + 1)-complete set which does not contain a good p-complete subset, denote it by W = {w1 , . . . , wp+1 }. Hence, for every vertex wi ∈ W there is a vertex ui ∈ W that is adjacent to all vertices of W except Wi . The vertex set {w1 , . . . , wp+1 , u1 , . . . , up+1 } induces a (p + 1)-ocular subgraph in G. (iv) ⇒ (i) Suppose that G is not strong p-clique-Helly. By Theorem 2.1 there is a (p + 1)-subset V ∗ = {v1 , v2 , . . . vp+1 } ⊂ V and the set of cliques H∗ = {Q1 , Q2 , . . . Qp+1 } such that V ∗ \{vi } ⊂ Qi and none of the vertices of V ∗ is in the core of H∗ . So vi ∈ Qi and, by its maximality, Qi contains a vertex adjcent to all vertices of V ∗ but not adjacent to vi . Since p ≥ 2 every two vertices of V ∗ are in a same clique, hence V ∗ is a (p + 1)-complete set which does not contain a good p-complete set. 2 To determine all (p + 1)-complete sets of a graph with n vertices we require O(n(p+1) ) steps. In order to verify for each one if it contains a good p-complete set, we need O(np) time. Therefore the complexity of the algorithm for recognizing hereditary p-clique-Helly graphs is O(pnp+2 ), which is polynomial for fixed p. We prove that, for variable p, this problem is NP-hard. Theorem 3.4 It is NP-hard to decide if a graph is hereditary p-clique-Helly, for variable p.
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