Lemma 1.3 shows that if a reflexive directed graph has a preordered compression then it must be a preordered directed graph. Example 1.2 shows a preordered directed graph may have a compression which is not preordered. Figure 2 shows there are also directed graphs which are not compressions of preordered directed graphs. Our main result, Theorem 3.2, describes a class of directed graphs which are compressions of directed graphs. Figure 2: Directed graphs which are not compressions of transitive directed graphs. 4 θ ( x ) = a , θ ( y ) = b , θ ( z ) = c . Assume ( a, b, c ) = ( θ ( x ) , θ ( y ) , θ ( z )) ∈ Trans ( D$_{1}$ ). Then a, b, c are distinct since ab, bc, ac ∈ A ( D ∗ $_{1}$) by part 3 of Definition 1.1. By part 2 of Definition 1.1 there exist x $^{′}$, y $^{′}$, z ′ ∈ V ( D$_{2}$ ) such that ( x $^{′}$, y $^{′}$, z $^{′}$) ∈ Trans ( D ), θ ( x $^{′}$) = a , θ ( y $^{′}$) = b , and θ ( z $^{′}$) = c . Moreover x $^{′}$, y $^{′}$, z ′ are distinct since a, b, c are distinct. Then x $^{′}$y $^{′}$, y $^{′}$z ′ ∈ A ( D ∗ $_{2}$) so θ $^{∗}$( xy ) = ab = θ $^{∗}$( x $^{′}$y $^{′}$) and θ $^{∗}$( yz ) = bc = θ $^{∗}$( y $^{′}$z $^{′}$). Therefore x ′ = x , y ′ = y , and z ′ = z since θ ∗ is bijective by part 3 of Definition 1.1. We assumed ( x $^{′}$, y $^{′}$, z $^{′}$) ∈ Trans ( D ) and proved x ′ = x and z ′ = z so xz ∈ A ( D$_{2}$ ) . Therefore ( x, y, z ) ∈ Trans ( D$_{2}$ ) as desired.