on the $sd_{b}$-critical graphs

A b-coloring of a graph g is a proper coloring of its vertices such that each color classcontains a vertex that has a neighbor in every other color classes. the b-chromatic number of a graphg, denoted by b(g), is the largest integer k such that g admits a b-coloring with k colors. let ge bethe graph obtained from g by subdividing the edge e. a graph g is sdb-critical if b(ge) < b(g) holdsfor any edge e of g. in this paper, we first present several basic properties of sdb-critical graphs andthen we give a characterization of sdb-critical p4-sparse graphs and sdb-critical quasi-line graphs.