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I have to prove that if $D$ is a m-coloured digraph, then $K(D) = K\big(C(D)\big)$

Where $C(D)$ is the chromatic transitive closure of $D$, which is a multidigraph with the same nodes and arcs of $D$ plus those arcs of the form $(a,b)$ iff there is a monochromatic directed path from $a$ to $b$ in $D$.

I'm only asking for the definition of $K(D)$, which might be called something like the Chromatic class digraph of a m-coloured digraph.

I know this is a vague definition, but any help would be highly appreciated.

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