Clique-width is one of the most important graph parameters, as many NP-hard graph problems are solvable in linear time on graphs of bounded clique-width. Unfortunately the computation of clique-width is among the hardest problems. In fact we do not know of any other algorithm than brute force for the exact computation of clique-width on any nontrivial large graph class. Another difficulty about...

In this paper we compare and illustrate the algorithmic use of graphs of bounded treewidth and graphs of bounded clique-width. For this purpose we give polynomial time algorithms for computing the four basic graph parameters independence number, clique number, chromatic number, and clique covering number on a given tree structure of graphs of bounded tree-width and graphs of bounded clique-widt...

Let G = (V, E) be a simple undirected graph. A forest F ⊆ E of G is said to be clique-connecting if each tree of F spans a clique of G. This paper adresses the clique-connecting forest polytope. First we give a formulation and a polynomial time separation algorithm. Then we show that the nontrivial nondegenerate facets of the stable set polytope are facets of the clique-connecting polytope. Fin...

Clique-width of graphs is defined algebraically through operations on graphs with vertex labels. We characterise the clique-width in a combinatorial way by means of partitions of the vertex set, using trees of nested partitions where partitions are ordered bottom-up by refinement. We show that the correspondences in both directions, between combinatorial partition trees and algebraic terms, pre...

Cutset conditioning and clique-tree propagation are two popular methods for exact probabilistic inference in Bayesian belief networks. Cutset conditioning is based on decomposition of a subset of network nodes, whereas clique-tree propagation depends on aggregation of nodes. We characterize network structures in which the performances of these methods differ. We describe a means to combine cuts...

We consider the problem of determining the size of a maximum clique in a graph, also known as the clique number. Given any method that computes an upper bound on the clique number of a graph, we present a sequential elimination algorithm which is guaranteed to improve upon that upper bound. Computational experiments on DIMACS instances show that, on average, this algorithm can reduce the gap be...

Royal Road functions manipulate the t-ness landscape to provide controlled experiments into genetic algorithm (GA) theory Mitchell et al., 1992]. Variations of clique detection in a graph, e.g., nding both the max clique Soule et al., 1996] and the clique cover Haynes, 1996], have been proposed as naturally occurring Royal Road functions. Soule and Foster have tried to positively correlate diie...

A clique in a graph is a complete subgraph maximal under inclusion. The clique graph of a graph is the intersection graph of its cliques. A graph is self-clique when it is isomorphic to its clique graph. A circular-arc graph is the intersection graph of a family of arcs of a circle. A Helly circular-arc graph is a circular-arc graph admitting a model whose arcs satisfy the Helly property. In th...

The edge clique graph of a graph H is the one having the edge set of H as vertex set, two vertices being adjacent if and only if the corresponding edges belong to a common complete subgraph of H. We characterize the graph classes {edge clique graphs} ∩ {interval graphs} as well as {edge clique graphs} ∩ {probe interval graphs}, which leads to polynomial time recognition algorithms for them. Thi...

A graph G is k-clique replete if G has clique number k and every elementary homomorphism of G has clique number greater than k . Results on the order of k-clique replete graphs are presented, and bounds for the minimum degree and the maximum degree of such graphs are discussed.