Containment Graphs, Posets, and Related Classes of Graphs

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 = (V, E) a $Z$-containment graph provided one can assign to each vertex $v_i\in V$ a set $S_i\in Z$ such that $v_i{v}_j\in E$ if and only if $S_i\subset S_j$ or $S_j\subset S_i$ . Similarly, we call a (strict) partially ordered set $P=(V,<)$ a $Z$-containment poset if to each $v_i\in V$ we can assign a set $S_i\in Z$ such that $v_i<v_j$ if and only if $S_i\subset S_j$. Obviously, $G$ is the comparability graph of $P$. We give some basic results on containment graphs and investigate the containment graphs of iso-oriented boxes in $d$-space. We present a characterization of those classes of posets and graphs that have containment representations by sets of a specific type, and we extend our results to injective'' containment classes. After that we discuss similar characterizations 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.