Clique-Width: Harnessing the Power of Atoms

TL;DR

This paper investigates the conditions under which the clique-width of atoms in hereditary graph classes is bounded, providing a near-complete classification for $(H_1,H_2)$-free graphs.

Contribution

It offers a systematic classification of bounded clique-width in atoms of hereditary graph classes, especially for $(H_1,H_2)$-free graphs, extending previous understanding.

Findings

Classified boundedness of clique-width for most $(H_1,H_2)$-free atoms

Identified a new pair $(H_1,H_2)$ where boundedness does not transfer from atoms to the class

Provided a near-complete taxonomy for clique-width in hereditary graph classes

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 ${\cal G}$ if they are so on the atoms (graphs with no clique cut-set) of ${\cal G}$. 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.