Graphs maximal with respect to hom-properties

For a simple graph H, → H denotes the class of all graphs that admit homomorphisms to H (such classes of graphs are called homproperties). We investigate hom-properties from the point of view of the lattice of hereditary properties. In particular, we are interested in characterization of maximal graphs belonging to → H. We also provide a description of graphs maximal with respect to reducible homproperties and determine the maximum number of edges of graphs belonging to → H.