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 if they are so on the atoms (graphs with no clique cut-set) of .
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.