Characterization and Recognition of Generalized Clique-Helly Graphs

• Published in: Graph-Theoretic Concepts in Computer Science pp. 344 - 354
• Main Authors: Dourado, Mitre Costa, Protti, Fábio, Szwarcfiter, Jayme Luiz
• Format: Book Chapter Conference Proceeding
• Language: English
• Published: Berlin, Heidelberg Springer Berlin Heidelberg 2004; Springer
• Series: Lecture Notes in Computer Science
• Subjects: Applied sciences; Computer science; control theory; systems; Exact sciences and technology; Information retrieval. Graph; Theoretical computing; Lead time; Terminology; Cardinal number; Graph clique; NP hard problem; Graph theory; Polynomial method; Polynomial time
• ISBN: 9783540241324; 3540241329
• ISSN: 0302-9743; 1611-3349
• DOI: 10.1007/978-3-540-30559-0_29

Let p ≥ 1 and q ≥ 0 be integers. A family of sets \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\mathcal F}$\end{document} is (p,q)-intersecting when every subfamily \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\mathcal F}' \subseteq {\mathcal F}$\end{document} formed by p or less members has total intersection of cardinality at least q. A family of sets \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\mathcal F}$\end{document} is (p,q)-Helly when every (p,q)-intersecting subfamily \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\mathcal F}' \subseteq {\mathcal F}$\end{document} has total intersection of cardinality at least q. A graph G is a (p,q)-clique-Helly graph when its family of (maximal) cliques is (p,q)-Helly. According to this terminology, the usual Helly property and the clique-Helly graphs correspond to the case p=2, q=1. In this work we present a characterization for (p,q)-clique-Helly graphs. For fixed p,q, this characterization leads to a polynomial-time recognition algorithm. When p or q is not fixed, it is shown that the recognition of (p,q)-clique-Helly graphs is NP-hard.