Tarjan described a procedure for decomposing a graph using clique separators in "Decomposition by clique separators", RE Tarjan - Discrete mathematics, 1985 - Elsevier.
He also proposed different divide-and-conquer algorithms for various graph problems such as maximum clique, minimum coloring and maximum independent set. An atom is a subgraph of a graph that does not contain a separator. If one can solve the problem efficiently in the atoms (local solutions) then Tarjan describes in the same paper how to extend these local solutions to a global solution. Therefore characterization of the atoms is very important. For example an atom of a chordal graph is a clique and the atoms of an edge intersection graphs of paths in a tree are exactly line graphs of a multigraph.
I am wondering the characterization of atoms for other graph classes like for example disk graphs or AT-free graphs.