Maximal graphs with respect to hereditary properties

A property of graphs is a non-empty set of graphs. A property P is called hereditary if every subgraph of any graph with property P also has property P. Let P1, . . . ,Pn be properties of graphs. We say that a graph G has property P1◦ · · · ◦Pn if the vertex set of G can be partitioned into n sets V1, . . . , Vn such that the subgraph of G induced by Vi has property Pi; i = 1, . . . , n. A hereditary property R is said to be reducible if there exist two hereditary properties P1 and P2 such that R = P1◦P2. If P is a hereditary property, then a graph G is called Pmaximal if G has property P but G+e does not have property P for every e ∈ E(G). We present some general results on maximal graphs and also investigate P-maximal graphs for various specific choices of P, including reducible hereditary properties.