Clique‐width: Harnessing the power of atoms

Many NP-complete graph problems are polynomial-time solvable on classes of bounded clique-width. Several these a hereditary class G ${\mathscr{G}}$ if they so the atoms (graphs with no clique cut-set) . Hence, we initiate systematic study into boundedness clique-width classes. A $G$ is H $H$ -free not an induced subgraph , and it ( 1 2 ) $({H}_{1},{H}_{2})$ both ${H}_{1}$ ${H}_{2}$ -free. graphs has only its have this property. This longer true for graphs, as evidenced by one known example. We prove existence another such pair classify all but 18 cases.