During the last decade, many problems in social, biological, and financial networks require finding cliques, or quasi-cliques. Cliques or clique partitions have also been used as clustering or classification tools in data sets represented by networks. These networks can be very large and often massive and therefore external (or semi-external) memory algorithms are needed. We discuss four applic...

Our implementation follows the exact algorithm of Fomin et al. [3] which is based on listing the minimal separators, the potential maximal cliques, and then performing dynamic programming based on those structures [1, 2]. The algorithm by Fomin et al. refines the approach of Bouchitté and Todinca on some minor aspects (only considering inclusion-minimal minimal separators, trying only the so-ca...

Given two graphs, a fundamental task faced by matching algorithms consists of computing either the (Connected) Maximal Common Induced Subgraphs ((C)MCIS) or the (Connected) Maximal Common Edge Subgraphs ((C)MCES). In particular, computing the CMCIS or CMCES reduces to reporting so-called connected cliques in product graphs, a problem for which an algorithm has been presented in a recent paper I...

We prove that for all positive integers t, every nvertex graph with no Kt-subdivision has at most 2 n cliques. We also prove that asymptotically, such graphs contain at most 2n cliques, where o(1) tends to zero as t tends to infinity. This strongly answers a question of D. Wood asking if the number of cliques in n-vertex graphs with no Kt-minor is at most 2 n for some constant c.

Problem statement: Protein sequence clustering is a method used to discover relations between proteins. This method groups the proteins based on their common features. It is a core process in protein sequence classification. Graph theory has been used in protein sequence clustering as a means of partitioning the data into groups, where each group constitutes a cluster. Mohseni-Zadeh introduced ...

The detection of communities in social networks is a challenging task. A rigorous way to model communities considers maximal cliques, that is, maximal subgraphs in which each pair of nodes is connected by an edge. State-of-the-art strategies for finding maximal cliques in very large networks decompose the network in blocks and then perform a distributed computation. These approaches exhibit a t...

Clique counts reveal important properties about the structure of massive graphs, especially social networks. The simple setting of just 3-cliques (triangles) has received much attention from the research community. For larger cliques (even, say 6-cliques) the problem quickly becomes intractable because of combinatorial explosion. Most methods used for triangle counting do not scale for large cl...

Many graph-theoretical problems involve the study of cliques, i .e ., complete subgraphs (not necessarily maximal) . In this context the following combinatorial problem arises naturally: For which numbers n and c is there a graph with n vertices and exactly c cliques? For fixed n, let G(n) denote the set of all such `clique numbers' c . Since each singleton and the empty set are always cliques,...

A graph G=(V,E) is monopolar if V can be partitioned into a stable set and inducing the union of vertex-disjoint cliques. Motivated by an application clique partitioning problem on graphs to cosmetic manufacturing, we study complexity computing classical parameters class graphs. We show that partitioning, stability chromatic numbers NP-hard. Conversely, prove every has polynomial number maximal...