# Clique-Width: Harnessing the Power of Atoms

Many NP-complete graph problems are polynomially solvable on graph classes of bounded clique-width. Several of these problems are polynomially 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 G is (H_1, H_2)-free if it is both H_1-free and H_2-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 (H_1, H_2)-free graphs as evidenced by one known example. We prove the existence of another such pair (H_1, H_2) and classify the boundedness of (H_1, H_2)-free atoms for all but 22 cases.

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