Extending the Tractability of the Clique Problem via Graph Classes Generalizing Treewidth

The study of the Clique problem in algorithmic graph theory is important both because it is a central problem in complexity theory, almost at the same level as SAT [8], but also because its practical resolution has many applications, notably in Artificial Intelligence (e.g. checking consistency of a binary CSP is equivalent to check the size of maximum clique in its microstructure [7]). A great deal of work has therefore been carried out, and this paper attempts to extend the results obtained in this field. It starts from the observation that this problem is tractable in polynomial time on graphs whose treewidth [10] are bounded by a constant (see [5]). Although this type of result is very interesting from a theoretical point of view, it often remains limited in terms of application. So, we propose here an extension of such these approaches based on the definition of graph classes wider than those of bounded treewidth for which we would be able to propose polynomial time algorithms. These graph classes denoted C-W(k) include graphs of treewidth W + k, but also contain, even if k and W are constants, graphs known to be of unbounded treewidth. These C-W(k) classes are introduced, their fundamental properties are given and the associated algorithms are presented and analysed.