On the $sd_{b}$-critical graphs

A b-coloring of a graph G is a proper coloring of its vertices such that each color class contains a vertex that has a neighbor in every other color classes. The b-chromatic number of a graph G, denoted by b(G), is the largest integer k such that G admits a b-coloring with k colors. Let Ge be the graph obtained from G by subdividing the edge e. A graph G is sdb-critical if b(Ge) b(G) holds for any edge e of G. In this paper, we first present several basic properties of sdb-critical graphs and then we give a characterization of sdb-critical P4-sparse graphs and sdb-critical quasi-line graphs.