Algorithms for maximum induced multicliques and complete multipartite subgraphs in polygon-circle graphs

A graph is a multiclique if its connected components are cliques. A graph is a complete multipartite graph if its vertex set has a partition into independent sets such that every two vertices in different independent sets are adjacent. The complement of a graph G is denoted coG. A graph is a multiclique-multipartite graph if its vertex set has a partition U, W such that G(U) is complete multipartite, G(W) is a multiclique and every two vertices u∈U, v∈W are adjacent. We describe polynomial time algorithms to find in polygon-circle graphs induced maximum and maximum balanced: multicliques and complete multipartite subgraphs. We describe polynomial time algorithms to find in circle graphs induced maximum and maximum balanced multiclique-multipartite subgraphs. These problems have applications for clustering of proteins by PPI criteria, of documents, and of web pages.