2 Balanced and Stable Directed Graphs Definition 2.1 Suppose D is a reflexive directed graph. 1. D is balanced if for all w, x, y, z ∈ V ( D ) such that wx, xy, yz, wz ∈ A ( D ) there is an arrow from w to y if and only if there is an arrow from x to z . 2. D is stable if D is balanced and ad ∈ A ( D ) for all distinct a, b, c, d ∈ V ( D ) such that ab, ac, bc, bd, cd ∈ A ( D ) . A reflexive directed graph is balanced if and only if does not contain an induced subgraph isomorphic to either (a) or (b) in Figure 2 or either of the directed graphs in Figure 3. Figure 3: Two directed graphs which are not balanced. A balanced directed graph is not stable if and only if it does not contain an induced subgraph isomorphic to (c) in Figure 2. Directed graph (d) in Figure 2 is stable but not preordered. Theorem 2.2 Suppose θ : V ( D$_{2}$ ) → V ( D$_{1}$ ) is a compression map and D$_{1}$ and D$_{2}$ are reflexive directed graphs. 1. If D$_{1}$ is balanced then D$_{2}$ is balanced. 2. If D$_{1}$ is stable then D$_{2}$ is stable. 5