3 Clasps and Soloists The next definition helps us identify vertices where the transitive relation fails. Definition 3.1 Suppose D is a reflexive directed graph and x ∈ V ( D ) . 1. We say x is a clasp if there exist w, y ∈ V ( D ) \ { x } such that wx, xy ∈ A ( D ) and there is no arrow from w to y . 2. We say x is a locked clasp if there exist u, v,w, y ∈ V ( D ) \ { x } such that ( u, x,y ) , ( u, x,v ) , ( w, x,v ) ∈ Trans ( D ) and there is no arrow from w to y . 3. An unlocked clasp is a clasp which is not locked. Directed graph (d) in Figure 2 contains a locked clasp determined by the vertex in the lower right corner. We are now able to state our main Theorem. Theorem 3.2 Let D be a stable directed graph. Then D is the compression of a preordered directed graph if and only if every clasp in D is unlocked. Remark 3.3 Suppose D is a reflexive directed graph. Theorem 3.2 shows D is the compression of a preordered directed graph if D does not contain an induced subgraph isomorphic to one of the directed graphs in Figure 2 or in Figure 3. This is reminiscent of Kuratowki’s characterization of planar graphs (see [2]). Theorem 3.2 is not a complete classification since both directed graphs in Figure 3 are compressions of preordered directed graphs. But both directed graphs in Figure 3 contain directed cycles so we can give a complete classification if D ∗ is acyclic. 7