Numerous problems can be modeled as clique partitioning problems in digital design synthesis. In this paper, we present two new polynomial time heuristic algorithms for efficient clique partitioning with or without limiting the maximum clique size. The goal of clique partitioning is to partition a graph into a minimum number of cliques. The basic approach of the new algorithm is to find small c...

Since Razborov, based on the approximation method, succeeded to obtain a superpolynomial lower bound on the size of monotone circuits computing the clique function, much effort has been devoted to explore the method and derive good lower bounds[K, NM, R1, R2, RR]. Employing the approximation method, Alon and Boppana[AB] obtained an exponential lower bound for monotone circuits computing the cli...

Clique-width is a graph parameter with many algorithmic applications. For a positive integer k, the k-th power of a graph G is the graph with the same vertex set as G, in which two distinct vertices are adjacent if and only if they are at distance at most k in G. Many graph algorithmic problems can be expressed in terms of graph powers. We initiate the study of graph classes of power-bounded cl...

A k-clique-coloring of a graph G is an assignment of k colors to the vertices of G such that every maximal (i.e., not extendable) clique of G contains two vertices with different colors. We show that deciding whether a graph has a k-clique-coloring is Σ 2 -complete for every k ≥ 2. The complexity of two related problems are also considered. A graph is k-clique-choosable, if for every k-list-ass...

The celebrated theorem of Robertson and Seymour states that in the family of minor-closed graph classes, there is a unique minimal class of graphs of unbounded tree-width, namely, the class of planar graphs. In the case of tree-width, the restriction to minor-closed classes is justified by the fact that the tree-width of a graph is never smaller than the tree-width of any of its minors. This, h...