On the b-dominating coloring of graphs

The b-chromatic numberϕ(G) of a graph G is defined as the largest number k for which the vertices of G can be colored with k colors satisfying the following property: for each i,1⩽i⩽k, there exists a vertex xi of color i such that for all j≠i,1⩽j⩽k there exists a vertex yj of color j adjacent to xi. A graph G is b-perfect if each induced subgraph H of G has ϕ(H)=χ(H), where χ(H) is the chromatic number of H. We characterize all b-perfect bipartite graphs and all b-perfect P4-sparse graphs by minimal forbidden induced subgraphs. We also prove that every 2K2-free and P5¯-free graph is b-perfect.