CliqueWidth: Harnessing the Power of Atoms
Abstract
Many NPcomplete graph problems are polynomialtime solvable on graph classes of bounded cliquewidth. Several of these problems are polynomialtime solvable on a hereditary graph class ${\cal G}$ if they are so on the atoms (graphs with no clique cutset) of ${\cal G}$. Hence, we initiate a systematic study into boundedness of cliquewidth 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 cliquewidth 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 cliquewidth on $(H_1,H_2)$free atoms for all but 18 cases.
 Publication:

arXiv eprints
 Pub Date:
 June 2020
 arXiv:
 arXiv:2006.03578
 Bibcode:
 2020arXiv200603578D
 Keywords:

 Computer Science  Discrete Mathematics;
 Computer Science  Computational Complexity;
 Mathematics  Combinatorics;
 05C75
 EPrint:
 37 pages, 32 figures, an extended abstract of this paper appeared in the proceedings of WG 2020