Topic
K-tree
About: K-tree is a research topic. Over the lifetime, 427 publications have been published within this topic receiving 12096 citations.
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TL;DR: An algorithm for enumerating maximal cliques (complete subgraphs) is proposed, which aims to deal with the difficulties caused by the size of the problem.
Abstract: An algorithm for enumerating maximal cliques (complete subgraphs) is proposed. The aim is to deal with the difficulties caused by the size of the problem.
182 citations
TL;DR: A benchmark network is shown where clique graphs find the overlapping communities accurately while vertex partition methods fail, and how a clique graph may be exploited.
Abstract: It is shown how to construct a clique graph in which properties of cliques of a fixed order in a given graph are represented by vertices in a weighted graph. Various definitions and motivations for these weights are given. The detection of communities or clusters is used to illustrate how a clique graph may be exploited. In particular a benchmark network is shown where clique graphs find the overlapping communities accurately while vertex partition methods fail.
160 citations
TL;DR: A general framework enables maximal clique enumeration to be processed recursively in small subgraphs of the input graph, thus allowing in-memory computation of maximal cliques without the costly random disk access.
Abstract: Maximal clique enumeration is a fundamental problem in graph theory and has important applications in many areas such as social network analysis and bioinformatics. The problem is extensively studied; however, the best existing algorithms require memory space linear in the size of the input graph. This has become a serious concern in view of the massive volume of today's fast-growing networks. We propose a general framework for designing external-memory algorithms for maximal clique enumeration in large graphs. The general framework enables maximal clique enumeration to be processed recursively in small subgraphs of the input graph, thus allowing in-memory computation of maximal cliques without the costly random disk access. We prove that the set of cliques obtained by the recursive local computation is both correct (i.e., globally maximal) and complete. The subgraph to be processed each time is defined based on a set of base vertices that can be flexibly chosen to achieve different purposes. We discuss the selection of the base vertices to fully utilize the available memory in order to minimize I/O cost in static graphs, and for update maintenance in dynamic graphs. We also apply our framework to design an external-memory algorithm for maximum clique computation in a large graph.
138 citations
TL;DR: The following conjecture of T. Gallai is proved: If G is a chordal graph on n vertices, such that all its maximal complete subgraphs have order at least 3, then there is a vertex set of cardinality ⩽n 3 which meets all maximal completeSubgraphs of G.
135 citations
TL;DR: A family of graphs with polynomially solvable maximum weight clique problem is identified using the edge-bicoloring approach developed in a a recent paper by Balas, Chvatal and Nesetril.
Abstract: : This document gives a new bound on the number of maximal cliques in a graph along with a bound on the length of odd antiholes that the graph can contain. Based on these bounds a family of graphs with polynomially solvable maximum weight clique problem is identified using the edge-bicoloring approach developed in a a recent paper by Balas, Chvatal and Nesetril. (Author)
135 citations