## HELLY'S THEOREM AND ITS RELATIVES BY LUDWIG DANZER, BRANKO GRUNBAUM, AND VICTOR KLEE Prologue:

According to this definition, 2,1 -clique-Helly graphs are the clique-Helly graphs. We begin with an example. The general graph G p,q appears in Figure 6 , where a thick line joining two sets means that every vertex of a set is adjacent to all vertices of the other.

Furthermore, for every vertex of Z , there is a dotted line joining it to the only vertex of W which is not adjacent to it. Consequently, for distinct q and t, the classes of graphs p, q -clique-Helly and p, t -clique-Helly are incomparable. The following theorem describes a class of p, q -clique-Helly graphs. Theorem 7. If G is a K r -free graph, then G is p, q -clique-Helly. Our aim is now to characterize p, q -clique-Helly graphs. Then G is a 1,q -clique-Helly graph if and only if G[W] contains q universal vertices.

Let G be a graph and C a p -complete set of G. The p-expansion relative to C is the subgraph of G induced by the vertices w such that w is adjacent to at least p - 1 vertices of C. It is clear that constructing a p -expansion relative to a given p -complete set can be done in polynomial time. Let F be a partial hypergraph of C G. The clique subgraph induced by F in G, denoted by G c [ F ],is the subgraph of G formed exactly by the vertices and edges belonging to the cliques of F.

Lemma 7. Then G c [ C ] is a spanning subgraph of H. As an example see Figure 8. Let G be a graph and C G its clique hypergraph. As a consequence we have:. Corollary 7. The next result is a characterization of 2,2 -clique-Helly graphs. Problem 7. Algorithm 7. Then join two vertices by an edge if the corresponding edges are contained in a same clique of G. If we find one extended triangle which does not have a universal vertex, then we stop as G is not 2, 2 -clique-Helly, otherwise it is. Therefore one can verify if a graph is 2,2 -clique-Helly intime O m 5.

Now we present a generalization of this result. Let C be the partial hypergraph C H that contains at least p vertices of C. Consider a partial p - -hypergraph C ' of C. By Lemma 7. This implies that C is p , 1 -intersecting.

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By Corollary 7. By using Corollary 7. Moreover, by Lemma 7. However, T contains no universal vertex. This is a contradiction. Conversely, assume by contradiction that G is not p, q -clique-Helly. Let C ' be the partial hypergraph of C H formed by the cliques that contain at least p vertices of C. By hypothesis, T contains a universal vertex x. Hence, G is a p, q -clique-Helly graph. From the above theorem one can recognize p, q -clique-Helly graphs in polynomial time if p and q are fixed. Given a graph G, decide whether G is p, q -clique-Helly. If p or q is variable, this procedure does not lead to a polynomial time algorithm.

Indeed, the problem is NP-hard in both cases. Given a graph G and a positive integer q , decide whether G is p, q - clique-Helly. Given a graph G and a positive integer p, decide whether G is p, q - clique-Helly. For any clique-Helly graph, its clique graph is also clique-Helly .

## HELLY'S THEOREM AND ITS RELATIVES BY LUDWIG DANZER, BRANKO GRUNBAUM, AND VICTOR KLEE Prologue:

However, if a graph is not clique-Helly it is still possible for its clique graph to be clique-Helly. This motivated the definition of Helly defect , a parameter that informs how many times the clique operator must be applied for a graph to become clique-Helly. There are graphs with any desired finite Helly defect . However if K i G is not clique-Helly, for any finite i, we say that its Helly defect is infinite.

Trivially, the Helly defect of a clique-Helly graph is 0, while that of a divergent graph is infinity. The Helly defect of a graph is less than or equal to 1 when it or its clique graph is clique-Helly. Given a graph G, decide whether K G is 2,q -clique-Helly.

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Hereditary Helly Property. A hypergraph is strong Helly if for every partial hypergraph H' of H, there exist two hyperedges in H' whose core is equal to the core of H'. A hypergraph H is hereditary Helly if all subhypergraphs of H are Helly. In this section we present algorithms and characterizations on generalizations of these two variants of the Helly property. In fact, we show that these two concepts are equivalent. First we characterize hereditary p -Helly hypergraphs and then consider the hereditary Helly property applied to special families of vertices of a graph, such as cliques, disks, bicliques, open and closed neighbourhoods.

Since the number of partial hypergraphs and of subhypergraphs of a given hypergraph can be exponential in the size of the hypergraph, the definitions do not lead directly to algorithms to verify, in polynomial time, if a hypergraph is strong Helly or hereditary Helly. Problem 8.

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In  it has been shown that a hypergraph H is strong Helly if and only if for every three hyperedges of H there exist two of them whose core equals the core of the three hyperedges. This characterization leads to an algorithm for recognizing strong Helly hypergraphs with time complexity O rm 3 , where r and m are, respectively, the rank and the number of hyperedges of the hypergraph. Also, a hypergraph H is hereditary p-Helly if all subhypergraphs of H are p -Helly. Theorem 8. The following theorem characterizes strong p -Helly and hereditary p -Helly hypergraphs.

It implies that they are equivalent.

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Let H" be a partial hypergraph of H' which is p -intersecting with an empty core. Since any p hyperedges of H" contain one vertex that is not in the core of H" , the same is true for any p hyperedges and the core of H 1.

Therefore H is not strong p -Helly. Denote the core of H' by C'.

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4. Every hyperedge of H ' properly contains C ' because it belongs to a p, q -intersecting partial hypergraph, and C' is a q - 1 - -set. Consider the partial hypergraph of formed by these hyperedges. Note that is p -intersecting and has an empty core. Therefore H ' 1 is not p -Helly. Clearly, H ' is not p , 1 -Helly. We can apply the equivalence i - iv in order to formulate an algorithm for recognizing strong p -Helly graphs, as follows. Given a hypergraph H, decide whether H is strong p-Helly.