Given integers m1, . . . ,ml, the weighted clique graph of G is the clique graph K(G), in which there is a weight assigned to each complete set S of size mi of K(G), for each i = 1, . . . , l. This weight equals the cardinality of the intersection of the cliques of G corresponding to S. We characterize weighted clique graphs in similar terms as Roberts and Spencer’s characterization of clique g...

Many important problems arising in computational biochemistry and genomics have been formulated in terms of underlying combinatorial optimization models. In particular, a number have been formulated as clique-detection models. The proposed article includes an introduction to the underlying biochemistry and genomic aspects of the problems as well as to the graph-theoretic aspects of the solution...

This column addresses the following questions about the complexity of approximating the CLIQUE problem. 1. Is nding the vertices of the largest clique more diicult than merely nding the vertices of a 2-approximate clique (that is, a clique with at least half the size of the largest clique)? 2. What computational resources can be used to nd the vertices of a 2-approximate clique? 3. Could fewer ...

In this paper, we study the relationship between the number of n-vertex graphs in a hereditary class X , also known as the speed of the class X , and boundedness of the clique-width in this class. We show that if the speed of X is faster than n!cn for any c, then the clique-width of graphs in X is unbounded, while if the speed does not exceed the Bell number Bn, then the clique-width is bounded...

In a graph, a clique is a set of vertices such that every pair is connected by an edge. MAX-CLIQUE is the optimization problem of finding the largest clique in a given graph and is NP-hard, even to approximate well. Several real-world and theory problems can be modeled as MAX-CLIQUE. In this paper, we efficiently approximate MAX-CLIQUE in a special case of the Hopfield network whose stable stat...

A bubble model is a 2-dimensional representation of proper interval graphs. We consider proper interval graphs that have bubble models of specific properties. We characterise the maximal such proper interval graphs of bounded clique-width and of bounded linear cliquewidth and the minimal such proper interval graphs whose clique-width and linear cliquewidth exceed the bounds. As a consequence, w...

To push a vertex v of a directed graph −→ G is to change the orientations of all the arcs incident with v. An oriented graph is a directed graph without any cycle of length at most 2. An oriented clique is an oriented graph whose non-adjacent vertices are connected by a directed 2-path. A push clique is an oriented clique that remains an oriented clique even if one pushes any set of vertices of...

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) can be solved efficiently for graphs of certified small clique-width. It is widely believed that determining the clique-width of a graph is NP-h...

Chudnovsky and Seymour proved that every connected claw-free graph that contains a stable set of size 3 has chromatic number at most twice its clique number. We improve this for small clique size, showing that every claw-free graph with clique number at most 3 is 4-choosable and every claw-free graph with clique number at most 4 is 7-choosable. These bounds are tight.

Clique-width is an important graph parameter whose computation is NP-hard. In fact we do not know of any other algorithm than brute force for the exact computation of clique-width on any non-trivial graph class. Results so far indicate that proper interval graphs constitute the first interesting graph class on which we might have hope to compute clique-width, or at least its linear variant line...