Corollary 3.4 Suppose D is a reflexive directed graph such that D ∗ is acyclic. Then D is the compression of a preordered directed graph if and only if D does not contain an induced subgraph isomorphic to one of the directed graphs in Figure 2. Definition 3.5 Suppose D is a reflexive directed graph. If r, s ∈ V ( D ) satisfy rs, sr ∈ A ( D ) then r and s are said to be paired in V ( D ) . We say an element s ∈ V ( D ) is a soloist in D if r and s are not paired for all r ∈ V ( D ) \ { s } . Lemma 3.6 Suppose D is a reflexive directed graph. 1. If x ∈ V ( D ) is a clasp then x is a soloist. 2. Suppose D$_{2}$ is a stable directed graph and θ : V ( D$_{2}$ ) → V ( D ) is a compression map. There is a locked clasp in D$_{2}$ if and only if there is a locked clasp in D . 3. If D contains a locked clasp then D is not the compression of a preordered directed graph. Proof. (1) Since x is a clasp there exist w, y ∈ V ( D ) such that w, y ∈ V ( D ) \ { x } , wx, xy ∈ A ( D ), and there is no arrow from w to y . Suppose there exists z ∈ V ( D ) \ { x } such that x is paired with z . Applying the balance property to w, x, z, x and to x, z, x, y yields wz, zy ∈ A ( D ). This gives w = y , y = z , and w = z since there is no arrow from w to y . Applying the stable property to w, x, z, y gives wy ∈ A ( D ), which is a contradiction. (2) If x ∈ V ( D$_{2}$ ) is a locked clasp then there exist u, v, w, y ∈ V ( D$_{2}$ ) \ { x } such that ( u, x, y ) , ( u, x, v ) , ( w, x, v ) ∈ Trans ( D$_{2}$ ) and there is no arrow from w to y in D$_{2}$ . Properties (1) and (3) of Definition 1.1 give θ ( u ) , θ ( v ) , θ ( w ) , θ ( y ) ∈ V ( D ) \ { θ ( x ) } 8