Elimination schemes and lattices

Perfect vertex elimination schemes are part of the characterizations for several classes of graphs, including chordal and cop-win. Partial elimination schemes reduce a graph to an important subgraph, for example, k-cores and robber-win graphs. We are interested in those partial elimination schemes, in which once a vertex can be eliminated it is always ready to be eliminated. In such a scheme, the sets of subsets of eliminated vertices, when ordered by inclusion, form an upper locally distributed lattice. We also consider the cop-win orderings having this property, the lattices obtained from the process of cleaning graphs, and raise the following question: which graphs are associated with distributive lattices?