Extremal perfect graphs for a bound on the domination number

We examine classes of extremal graphs for the inequality γ(G)⩽|V|-max{d(v)+βv(G)}γ(G)⩽|V|-max{d(v)+βv(G)}, where γ(G)γ(G) is the domination number of graph G  , d(v)d(v) is the degree of vertex vv, and βv(G)βv(G) is the size of a largest matching in the subgraph of G   induced by the non-neighbours of vv. This inequality improves on the classical upper bound |V|-maxd(v)|V|-maxd(v) due to Claude Berge. We give a characterization of the bipartite graphs and of the chordal graphs that achieve equality in the inequality. The characterization implies that the extremal bipartite graphs can be recognized in polynomial time, while the corresponding problem remains NP-complete for the extremal chordal graphs.