A graph is defined to be an atom if no minimal vertex separator induces a complete subgraph; thus, atoms are the graphs that are immune to clique separator decomposition. Atoms are characterized here in two ways: first using generalized vertex elimination schemes, and then as generalizations of 2-connected unichord-free graphs (the graphs in which every minimal vertex separator induces an edgeless subgraph).
@article{bwmeta1.element.doi-10_7151_dmgt_1944,
author = {Terry A. McKee},
title = {Characterizing Atoms that Result from Decomposition by Clique Separators},
journal = {Discussiones Mathematicae Graph Theory},
volume = {37},
year = {2017},
pages = {587-594},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_7151_dmgt_1944}
}
Terry A. McKee. Characterizing Atoms that Result from Decomposition by Clique Separators. Discussiones Mathematicae Graph Theory, Tome 37 (2017) pp. 587-594. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_7151_dmgt_1944/