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 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 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.

Clique‐width: Harnessing the power of atoms

“…2018/31/D/ST6/00062. An extended abstract of this paper appeared in the proceedings of WG 2020 [34].…”

“…All of the above meta-theorems require a constant-width decomposition of the graph. We can compute such a decomposition in polynomial time for treewidth [4] and clique-width [60], but for other width parameters, such as mim-width, which is even more powerful than cliquewidth [65], it is not known whether this is possible and this problem may turn out to be harder. For instance, unless = NP ZPP, there is no constant-factor approximation algorithm for mimwidth that runs in polynomial time [62].…”

“…Graph width parameters [31,45,47,51,65] help make such results possible. A graph class has bounded width if there is a constant c such that the width of all its members is at most c. As we discuss below, there are several meta-theorems that provide sufficient conditions for a problem to be tractable on a graph class of bounded width.…”