Gallai's innequality for critical graphs of reducible hereditary properties

In this paper Gallai’s inequality on the number of edges in critical graphs is generalized for reducible additive induced-hereditary properties of graphs in the following way. Let P1,P2, . . . ,Pk (k ≥ 2) be additive induced-hereditary properties, R = P1◦ P2◦ · · · ◦Pk and δ = ∑k i=1 δ(Pi). Suppose that G is an R-critical graph with n vertices and m edges. Then 2m ≥ δn + δ−2 δ2+2δ−2 n + 2δ δ2+2δ−2 unless R = O2 or G = Kδ+1. The generalization of Gallai’s inequality for P-choice critical graphs is also presented.