We conclude by exhibiting a technique for constructing a central digraph with a prescribed homomorphic image. Suppose H is a central digraph and we wish to construct a central digraph G with H as homomorphic image. The following procedure can be applied. 1. Color the arcs of H such that for each vertex v of H, every 2-path terminating at v has the same sequence of colors. (For maximum flexibility, use as many distinct colors as possible.) 2. To each color associate a positive integer and for each xeH let Γ_1(x) equal the positive integer associated with the common color of all arcs terminating at x, and let Γ_2(x) denote the positive integer associated with the common color of all arcs terminating at some inneighbor of x. 3. To each vertex x∈H assign a distinct set S(x) of points with cardinality Γ_1(x)Γ_2(x). 4. For each x∈H, organize the points of S(x) into an inclass M(x) consisting of Γ_2(x) insets, each with Γ_1 (x) points. Also organize S(x) into an outclass P(x) consisting of Γ_1(x) outsets, each with Γ_2(x) points. Do this, in accordance with Theorem 6iii, so that each inset intersects every outset in exactly one point. 5. Suppose x→y in H. Then by virtue of our coloring, Γ_1(x)=Γ_2(y), and thus the number of outsets in P(x) is equal to the nmber of insets in M(y). So for each arc x→y in H we can form a bijection Φ_(xy) between the collection of outsets in P(x) and the collection of insets in M(y). 6. Form a digraph G with vertex set {∪S(x): x∈H} and arc set determined as follows. If x→y is an arc of H, then for each outset L in P(x) insert an arc from each point of L to every point in the inset Φ_(xy)(L).
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