On graphs without a C4C4 or a diamond

We consider the class of (C4C4, diamond)-free graphs; graphs in this class do not contain a C4C4 or a diamond as an induced subgraph. We provide an efficient recognition algorithm for this class. We count the number of maximal cliques in a (C4C4, diamond)-free graph and the number of nn-vertex, labeled (C4C4, diamond)-free graphs. We also give an efficient algorithm for finding a largest clique in the more general class of (house, diamond)-free graphs.