Infinitely many minimal classes of graphs of unbounded clique-width.
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Abstract
The celebrated theorem of Robertson and Seymour states that in the family of minor-closed graph classes, there is a unique minimal class of graphs of unbounded tree-width, namely, the class of planar graphs. In the case of tree-width, the restriction to minor-closed classes is justified by the fact that the tree-width of a graph is never smaller than the tree-width of any of its minors. This, however, is not the case with respect to clique-width, as the clique-width of a graph can be (much) smaller than the clique-width of its minor. On the other hand, the clique-width of a graph is never smaller than the clique-width of any of its induced subgraphs, which allows us to be restricted to hereditary classes (that is, classes closed under taking induced subgraphs), when we study clique-width. Up to date, only finitely many minimal hereditary classes of graphs of unbounded clique-width have been discovered in the literature. In the present paper, we prove that the family of such classes is infinite. Moreover, we show that the same is true with respect to linear clique-width.Citation
Collins, A. et al (2017) 'Infinitely many minimal classes of graphs of unbounded clique-width', Discrete Applied Mathematics, DOI: 10.1016/j.dam.2017.02.012Publisher
ElsevierJournal
Discrete Applied MathematicsDOI
10.1016/j.dam.2017.02.012Additional Links
http://linkinghub.elsevier.com/retrieve/pii/S0166218X17300975https://arxiv.org/abs/1701.08857
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enISSN
0166218Xae974a485f413a2113503eed53cd6c53
10.1016/j.dam.2017.02.012
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