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Finding All Maximal Cliques of a Family of Induced Subgraphs

Please always quote using this URN: urn:nbn:de:0297-zib-7758
  • Many real world problems can be mapped onto graphs and solved with well-established efficient algorithms studied in graph theory. One such problem is to find large sets of points satisfying some mutual relationship. This problem can be transformed to the problem of finding all cliques of an undirected graph by mapping each point onto a vertex of the graph and connecting any two vertices by an edge whose corresponding points satisfy our desired relationship. Clique detection has been widely studied and there exist efficient algorithms. In this paper we study a related problem, where all points have a set of binary attributes, each of which is either 0 or 1. This is only a small limitation, since all discrete properties can be mapped onto binary attributes. In our case, we want to find large sets of points not only satisfying some mutual relationship; but, in addition, all points of a set also need to have at least one common attribute with value 1. The problem we described can be mapped onto a set of induced subgraphs, where each subgraph represents a single attribute. For attribute $i$, its associated subgraph contains those vertices corresponding to the points with attribute $i$ set to 1. We introduce the notion of a maximal clique of a family, $\mathcal{G}$, of induced subgraphs of an undirected graph, and show that determining all maximal cliques of $\mathcal{G}$ solves our problem. Furthermore, we present an efficient algorithm to compute all maximal cliques of $\mathcal{G}$. The algorithm we propose is an extension of the widely used Bron-Kerbosch algorithm.

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Metadaten
Author:Daniel BaumORCiD
Document Type:ZIB-Report
Tag:Bron-Kerbosch algorithm; backtracking algorithm; branch and bound technique; family of induced subgraphs; maximal cliques; re; vertex properties
MSC-Classification:05-XX COMBINATORICS (For finite fields, see 11Txx) / 05Cxx Graph theory (For applications of graphs, see 68R10, 81Q30, 81T15, 82B20, 82C20, 90C35, 92E10, 94C15) / 05C69 Dominating sets, independent sets, cliques
68-XX COMPUTER SCIENCE (For papers involving machine computations and programs in a specific mathematical area, see Section -04 in that area) / 68Wxx Algorithms (For numerical algorithms, see 65-XX; for combinatorics and graph theory, see 05C85, 68Rxx) / 68W05 Nonnumerical algorithms
92-XX BIOLOGY AND OTHER NATURAL SCIENCES / 92Cxx Physiological, cellular and medical topics / 92C40 Biochemistry, molecular biology
CCS-Classification:G. Mathematics of Computing / G.2 DISCRETE MATHEMATICS / G.2.2 Graph Theory (F.2.2)
Date of first Publication:2004/02/12
Series (Serial Number):ZIB-Report (03-53)
ZIB-Reportnumber:03-53
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