Geodesic
Characterizing Atoms that Result from Decomposition by Clique Separators
Discussiones Mathematicae. Graph Theory, Tome 37 (2017) no. 3, pp. 587-594

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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).
Keywords: clique separator, minimal separator, unichord-free graph 
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McKee, Terry A. Characterizing Atoms that Result from Decomposition by Clique Separators. Discussiones Mathematicae. Graph Theory, Tome 37 (2017) no. 3, pp. 587-594. http://geodesic.mathdoc.fr/item/DMGT_2017_37_3_a6/