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 ${\displaystyle ₁,₂,...,ₖ}$ (k ≥ 2) be additive induced-hereditary properties, ${\displaystyle =₁\circ ₂\circ ...\circ ₖ}$ and ${\displaystyle \delta ={\sum }_{i=1}^k\delta (\_i)}$. Suppose that G is an -critical graph with n vertices and m edges. Then 2m ≥ δn + (δ-2)/(δ²+2δ-2)*n + (2δ)/(δ²+2δ-2) unless = ² or ${\displaystyle G=K_{\delta +1}}$. The generalization of Gallai's inequality for -choice critical graphs is also presented.