Algorithms for GA-H reduced graphs

Let GA be a hereditary family of graphs and H   a hereditary family of acyclically directed family of graphs. A graph G(V,E)G(V,E) is a GA-Hreduced graph   if it can be obtained from a graph GA(V,D)∈GAGA(V,D)∈GA by deleting the edges of an edge subgraph H(V,E′)∈HH(V,E′)∈H. The GA-H reduced graphs are a generalization of the complements of the H-mixed graphs. Examples of such families of GA-H reduced graphs are the interval filament graphs, the subtree filament graphs, the circular-arc filament graphs, the cactus subtree filament graphs, the 3D-interval-filament graphs and the subgraph overlap graphs.We describe polynomial time algorithms for various problems on GA-H reduced graphs, when the families GA and H have specific properties. The algorithms are to find maximum independent sets, maximum K-packings, maximum cliques, maximum induced complete bipartite subgraphs, maximum weight holes of a given parity and antiholes of a given parity.