Clique neighborhoods and nearly chordal graphs

We study two new special families of complete subgraphs of a graph. For chordal graphs, one of these reduces to the family of minimal vertex separators while the other is empty. When the intersection characterization of chordal graphs is extended from acyclic (i.e., k(3)-free chordal) hosts to K-4-free chordal hosts, these new families are as fundamental as minimal vertex separators are for chordal graphs. Every graph satisfies certain inequalities involving the cardinalities of these families, with interesting questions arising when equality holds.