Component Interaction Automata provide a fitting model to capture and analyze the temporal facets of hierarchical-structured component-oriented software systems. However, the rules governing composition, as is typical for all automata-based approaches, suffer from combinatorial state explosion, an effect that can have significant ramifications on the successful application of the Component Inte...

We consider the k-disjoint-clique problem. The input is an undirected graph G in which the nodes represent data items, and edges indicate a similarity between the corresponding items. The problem is to find within the graph k disjoint cliques that cover the maximum number of nodes of G. This problem may be understood as a general way to pose the classical ‘clustering’ problem. In clustering, on...

For a fixed number N of nodes, the number of links L in the line graph H(N,L) can only appear in consecutive intervals, called a band of L. We prove that some consecutive integers can never represent the number of links L in H(N,L), and they are called a bandgap of L. We give the exact expressions of bands and bandgaps of L. We propose a model which can randomly generate simple graphs which are...

In Hell et al. (2004), we have previously observed that, in a chordal graph G, the maximum number of independent r -cliques (i.e., of vertex disjoint subgraphs of G, each isomorphic to Kr , with no edges joining any two of the subgraphs) equals the minimum number of cliques of G that meet all the r -cliques of G. When r = 1, this says that chordal graphs have independence number equal to the cl...

We propose new practical algorithms to find degree-relaxed variants of cliques called s-plexes. An s-plex denotes a vertex subset in a graph inducing a subgraph where every vertex has edges to all but at most s vertices in the s-plex. Cliques are 1-plexes. In analogy to the special case of finding maximum-cardinality cliques, finding maximumcardinality s-plexes is NP-hard. Complementing previou...

We characterize clique trees of a chordal graph in their relation to simplicial vertices and perfect sequences of maximal cliques. We investigate boundary cliques de ned by Shibata[23] and clarify their relation to endpoints of clique trees. Next we de ne a symmetric binary relation between the set of clique trees and the set of perfect sequences of maximal cliques. We describe the relation as ...

Many real networks have cliques as their constitutional units. Here we present a family of scale-free network model consist of cliques, which is established by a simple recursive algorithm. We investigate the networks both analytically and numerically. The obtained analytical solution shows that the networks follow a powerlaw degree distribution, with degree exponent continuously tuned between ...

A fast algorithm to remove proper and homogenous pairs of cliques (while preserving some graph invariants) Abstract We introduce a family of reductions for removing proper and homogeneous pairs of cliques from a graph G. This family generalizes some routines presented in the literature, mostly in the context of claw-free graphs. These reductions can be embedded in a simple algorithm that in at ...

The problem of deciding whether the edge-set of a given graph can be partitioned into at most k cliques is well known to be NP-complete. In this paper we investigate this problem from the point of view of parameterized complexity. We show that this problem is fixed parameter tractable if we choose the number of cliques as parameter. In particular, we show that in polynomial time, a kernel bound...

We present a depth-first search algorithm for generating all maximal cliques of an undirected graph, in which pruning methods are employed as in the Bron–Kerbosch algorithm. All the maximal cliques generated are output in a tree-like form. Subsequently, we prove that its worst-case time complexity is O(3n/3) for an n-vertex graph. This is optimal as a function of n, since there exist up to 3n/3...