This paper presents a parallel algorithm to solve the Clique Partitioning Problem, an NP-complete problem. Given a graph G = (V, E) , a clique is a complete subgraph in G. The clique partitioning problem is to partition the vertices in G into a number of cliques such that each vertex appears in one and only one clique. The clique partitioning problem has important applications in many areas inc...

We consider a branch-and-bound algorithm for maximum clique problems. We introduce cost based filtering techniques for the so-called candidate set (i.e. a set of nodes that can possibly extend the clique in the current choice point). Additionally, we present a taxonomy of upper bounds for maximum clique. Analytical results show that our cost based filtering is in a sense as tight as most of the...

It is well-known that any chordal graph can be represented as a clique tree (acyclic hypergraph, join tree). Since some chordal graphs have many distinct clique tree representations, it is interesting to consider which one is most desirable under various circumstances. A clique tree of minimum diameter (or height) is sometimes a natural candidate when choosing clique trees to be processed in a ...

The clique graph K(G) of a simple graph G is the intersection graph of its maximal complete subgraphs, and we define iterated clique graphs by K(G) = G, K(G) = K(K(G)). We say that two graphs are homotopy equivalent if their simplicial complexes of complete subgraphs are so. From known results it can be easily inferred that Kn(G) is homotopy equivalent to G for every n if G belongs to the class...

It is well-known that any chordal graph can be represented as a clique tree (acyclic hypergraph, join tree). Since some chordal graphs have many distinct clique tree representations, it is interesting to consider which one is most desirable under various circumstances. A clique tree of minimum diameter (or height) is sometimes a natural candidate when choosing clique trees to be processed in a ...

A clique in an undirected graph G= (V, E) is a subset V ' ⊆ V of vertices, each pair of which is connected by an edge in E. The clique problem is an optimization problem of finding a clique of maximum size in graph. The clique problem is NP-Complete. We have succeeded in developing a fast algorithm for maximum clique problem by employing the method of verification and elimination. For a graph o...

Clique-width is an important graph parameter due to its algorithmic and structural properties. A graph class is hereditary if it can be characterized by a (not necessarily finite) set H of forbidden induced subgraphs. We initiate a systematic study into the boundedness of clique-width of hereditary graph classes closed under complementation. First, we extend the known classification for the |H|...

A clique colouring of a graph is a colouring of the vertices such that no maximal clique is monochromatic (ignoring isolated vertices). The least number of colours in such a colouring is the clique chromatic number. Given n points x1, . . . ,xn in the plane, and a threshold r > 0, the corresponding geometric graph has vertex set {v1, . . . , vn}, and distinct vi and vj are adjacent when the Euc...

In this paper, we introduce DLS-MC, a new stochastic local search algorithm for the maximum clique problem. DLS-MC alternates between phases of iterative improvement, during which suitable vertices are added to the current clique, and plateau search, during which vertices of the current clique are swapped with vertices not contained in the current clique. The selection of vertices is solely bas...

Clique-width is a graph parameter that measures in a certain sense the complexity of a graph. Hard graph problems (e.g., problems expressible in Monadic Second Order Logic with second-order quantification on vertex sets, that includes NP-hard problems such as 3-colorability) can be solved in polynomial time for graphs of bounded clique-width. We show that the clique-width of a given graph canno...