TY - GEN A1 - Wagler, Annegret T1 - The Normal Graph Conjecture is true for Circulants N2 - Normal graphs are defined in terms of cross-intersecting set families: a graph is normal if it admits a clique cover $\cal Q$ and a stable set cover $\cal S$ s.t.~every clique in $\cal Q$ intersects every stable set in $\cal S$. Normal graphs can be considered as closure of perfect graphs by means of co-normal products (Körner 1973) and graph entropy (Czisz\'ar et al. 1990). Perfect graphs have been recently characterized as those graphs without odd holes and odd antiholes as induced subgraphs (Strong Perfect Graph Theorem, Chudnovsky et al. 2002). Körner and de Simone observed that $C_5$, $C_7$, and $\overline C_7$ are minimal not normal and conjectured, as generalization of the Strong Perfect Graph Theorem, that every $C_5$, $C_7$, $\overline C_7$- free graph is normal (Normal Graph Conjecture, Körner and de Simone 1999). We prove this conjecture for a first class of graphs that generalize both odd holes and odd antiholes, the circulants, by characterizing all the normal circulants. T3 - ZIB-Report - 04-06 KW - perfect graphs KW - normal graphs KW - circulants Y1 - 2004 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-7813 ER - TY - GEN A1 - Baum, Daniel T1 - Finding All Maximal Cliques of a Family of Induced Subgraphs N2 - 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. T3 - ZIB-Report - 03-53 KW - Bron-Kerbosch algorithm KW - maximal cliques KW - vertex properties KW - family of induced subgraphs KW - backtracking algorithm KW - branch and bound technique KW - re Y1 - 2004 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-7758 ER -