Containment Graphs, Posets, and Related Classes of Graphs zy MARTIN CHARLES GOLUMBIC“ AND EDWARD R. SCHEINERMAN~ zyxw “I.B.M.Israel Scient$c Center 7echnion City, Haifa 32000, Israel bDepartment zyxwv of Mathematical Sciences The Johns Hopkins University Baltimore, Maryland 21218 INTRODUCTION In this paper we introduce the notion of the containment graph of a family of sets and containment classes of graphs and posets. Let Z be a family of nonempty sets. We call a (simple, finite) graph G zyxwv = (V, E) a Z-containment graph provided one can assign to each vertex vi zyxwvu E V a set Si E C such that uivj E E if and only if Si z c Sj or zyxwvut Si 3 Sj. Similarly, we call a (strict) partially ordered set P = (V, <) a X- containment poset if to each vi E V we can assign a set Si E C such that vi < vj if and only if Si c Sj . Obviously, G is the comparability graph of P. The function$ V + C, which assigns sets of C to elements of V by f(vi) = Si, is called a C-containment representation for G and P, or simply a E-representation. There are two approaches that one might take in investigating containment graphs. In the first, one restricts the family of sets Z to a certain type (such as intervals on a line, circles in the plane, paths in a tree) and then asks what class of graphs is obtained. In the second approach, one begins with a class C of compara- bility graphs and asks whether C can be characterized as the family of containment graphs of some family of sets. In the next section we give some basic results on containment graphs and inves- tigate the containment graphs of iso-oriented boxes in d-space. In the section follow- ing we present a characterization of those classes of posets and graphs that have containment representations by sets of a specific type. In the next section we present an alternative characterization, and then in the following section we extend our results to “injective” containment classes. After that we discuss similar character- izations for intersection, overlap, and disjointedness classes of graphs. Finally, in the last section we discuss the nonexistence of a characterization theorem for “strong” containment classes of graphs. Unless specifically stated otherwise, all graphs and posets are assumed to be finite. CONTAINMENT GRAPHS AND POSETS A binary relation < on a set V is called a strict partial order if it is irreflexive and transitive. The comparability graph of a partially ordered zyx set (poset) P = (V, <) 192