Abstract
Many NP-complete graph problems are polynomial-time solvable on graph classes of bounded clique-width. Several of these problems are polynomial-time solvable on a hereditary graph class G if they are so on the atoms (graphs with no clique cut-set) of G.
Hence, we initiate a systematic study into boundedness of clique-width of atoms of hereditary graph classes. A graph G is H-free if H is not an induced subgraph of G, and it is (H1,H2)-free if it is both H1-free and H2-free.
A class of H-free graphs has bounded clique-width if and only if its atoms have this property. This is no longer true for (H1,H2)-free graphs, as evidenced by one known example.
We prove the existence of another such pair (H1,H2) and classify the boundedness of clique-width on (H1,H2)-free atoms for all but 18 cases.