Clique-Width: Harnessing the Power of Atoms

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 \(\mathcal{G}\) if they are so on the atoms (graphs with no clique cut-set) of \(\mathcal{G}\). 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 \((H_1,H_2)\)-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 clique-width on \((H_1,H_2)\)-free atoms for all but 18 cases.