On the strong p-Helly property

Abstract

The notion of strong p-Helly hypergraphs was introduced by Golumbic and Jamison in 1985 [M.C. Golumbic, R.E. Jamison, The edge intersection graphs of paths in a tree, J. Combin. Theory Ser. B 38 (1985) 8-22]. Independently, other authors [A. Bretto, S. Ubéda, J. Žerovnik, A polynomial algorithm for the strong Helly property. , On maximal clique irreducible graphs. J. Combin. Math. Combin. Comput. 8 (1990) 187-193.] have also considered the strong Helly property in other contexts. In this paper, we characterize strong p-Helly hypergraphs. This characterization leads to an algorithm for recognizing such hypergraphs, which terminates within polynomial time whenever p is fixed. 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. For p = 2, this class is equivalent to that of hereditary clique-Helly graphs [E. Prisner, Hereditary clique-Helly graphs, J. Combin. Math. Combin. Comput. 14 (1993) 216-220]. We describe a characterization for this class and obtain an algorithm for recognizing such graphs. Again, the algorithm has polynomial-time complexity for p fixed, and we show the corresponding recognition problem to be NP-hard, for arbitrary p.

Improved algorithms for recognizing p-Helly and hereditary p-Helly hypergraphs

“…Similarly, the algorithm for recognizing p-Helly hypergraphs is based on a theorem by Berge and Duchet [2], which generalizes Gilmore's theorem. Finally, for recognizing hereditary p-Helly hypergraphs the existing algorithm is based on a characterization by Dourado, Protti and Szwarcfiter [4].…”

A superclass of Edge-Path-Tree graphs with few cliques

“…Both results assert that A is strong Helly if and only if it is O 3 -free. After the result in [2], strong Helly matrices are precisely those matrices that have strong Helly number 2. Hence, strong Helly matrices are necessarily F * 7 -free.…”

“…It is clear that each claw-clique is uniquely determined by any three paths containing, respectively, one of the slices of the corresponding claw in T . Therefore, t = 1 and the number of claw-cliques is at most By a result of [2] stating that a {0, 1}-matrix A has strong Helly number h ≥ 2 if and only if A does not contain the matrix O h+1 = J h+1 − I h+1 as submatrix, (J h and I h , being the all ones and the identity matrix of order h, respectively) it follows straightforwardly that {0, 1}-matrices with no F * 7 submatrix have strong Helly number three (because for h ≥ 4, F * 7 is a submatrix of O h ). Therefore, we can still speak of edge-clique and claw-cliques as in [4].…”

Computational aspects of the Helly property: a survey

“…A p-complete subset C of a (p +1)-complete set C is good, relative to C, if any vertex adjacent to all vertices of C is also adjacent to the vertex in C\C . Theorem 8.7 [34] The following statements are equivalent for any graph G: The number of (p + 1)-complete sets in a graph with n vertices is O(n p+1 ). We need O(np) time to verify, for each one, if it contains a good p-complete set.…”

“…A graph is p-ocular-free if it has not a p-ocular graph as an induced subgraph. Lemma 8.1 [34] Any (p + 1)-ocular graph is not pclique-Helly, p ≥ 2.…”