In this paper we study the edge-thickness and the clique-thickness of a graph. The edge-thickness of a graph is defined as the thickness of the family of edges. The clique-thickness of a graph is defined as the thickness of the family of maximal cliques. Edges and maximal cliques of a graph are both considered as a collection of subsets of the vertex set. On one hand, we introduce a new paramet...

We present efficient branch and bound algorithms for solving the Maximum Clique Problem. First strategy is to store the size of clique consisting of a seed vertex in a subset induced by candidate set of the seed vertex. Results stored are further used to reduce the computation involved in computing clique consisting of other seed vertices in another subset of the graph. This approach along with...

Basic chordal graphs arose when comparing clique trees of chordal graphs and compatible trees of dually chordal graphs. They were defined as those chordal graphs whose clique trees are exactly the compatible trees of its clique graph. In this work, we consider some subclasses of basic chordal graphs. One of them is the class of hereditary basic chordal graphs, which will turn out to have many p...

This paper introduces Phased Local Search (PLS), a new stochastic reactive dynamic local search algorithm for the maximum clique problem. PLS interleaves sub-algorithms which alternate between sequences of iterative improvement, during which suitable vertices are added to the current clique, and plateau search, where vertices of the current clique are swapped with vertices not contained in the ...

The problem of finding a maximum clique in a graph is prototypical for many clustering and similarity problems; however, in many real-world scenarios, the classical problem of finding a complete subgraph needs to be relaxed to finding an almost complete subgraph, a so-called quasi-clique. In this work, we demonstrate how two previously existing definitions of quasi-cliques can be unified and ho...

The clique problem consists in determining whether an undirected graph G of order n contains a clique of order `. In this paper we are concerned with the decremental version of clique problem, where the property of containing an `-clique is dynamically checked during deletions of nodes. We provide an improved dynamic algorithm for this problem for every fixed value of ` ≥ 3. Our algorithm natur...

Avoiding the partitioning of a cluster into cliques that are not able to communicate with each other is an important issue in the time-triggered communication protocol TTP/C. This is achieved by a mechanism called Clique Avoidance. The Clique Avoidance algorithm always selects one partition (clique) to win and causes all nodes of other partitions to shut down. In this paper, we are going to inv...

Abstract. A graph is an {r, s}-graph if the set of degrees of their vertices is {r, s}. A clique of a graph is a maximal complete subgraph. The clique graph K(G) of a graph G is the intersection graph of all its cliques. A graph G is self-clique if G is isomorphic to K(G). We show the existence of self-clique {5, 6}-graphs whose cliques are all triangles, thus solving a problem posed by Chia an...

Given an undirected graph G = (V,E) with vertex set V = {1, ..., n} and edge set E ⊆ V × V . The maximum clique problem is to determine in G a clique (i.e., a complete subgraph) of maximum cardinality. This paper presents an effective algorithm for the maximum clique problem. The proposed multistart tabu search algorithm integrates a constrained neighborhood, a dynamic tabu tenure mechanism and...

In social network analysis, a k-clique is a relaxed clique, i.e., a kclique is a quasi-complete sub-graph. A k-clique in a graph is a sub-graph where the distance between any two vertices is no greater than k. The visualization of a small number of vertices can be easily performed in a graph. However, when the number of vertices and edges increases the visualization becomes incomprehensible. In...