On the strong p-Helly property |
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Authors: | Mitre C Dourado Fábio Protti Jayme L Szwarcfiter |
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Institution: | a COPPE - Sistemas, Brazil b IM, NCE, Universidade Federal do Rio de Janeiro, Caixa Postal 2324, 20001-970, Brazil |
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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. ?erovnik, A polynomial algorithm for the strong Helly property. Inform. Process. Lett. 81 (2002) 55-57, E. Prisner, Hereditary clique-Helly graphs, J. Combin. Math. Combin. Comput. 14 (1993) 216-220, W.D. Wallis, Guo-Hui Zhang, 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. |
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Keywords: | Strong Helly property Hereditary clique-Helly graphs Computational complexity |
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