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Super-colored paths in digraphs

We work in an Anthropology application where it is desired to enumerate colored rings (structures that look like cycles) present in kinship net-works. For this goal, we came across the following question: for all vertex v of a vertex-colored digraph, how many colors (in maximum) a path starting in v can have? The answer for this question would help us to enumerate the colored rings since we would know how many colors a ring evolving some vertices could have, in maximum. Here, we call a path as v-super-colored if it starts in vertex v and it has the maximum amount of colors among all paths starting in v. We show that the problem to find the number of colors of v-super-colored paths for all v is NP-hard when the input digraph is general. We describe a simple algorithm which demonstrates that the problem is tractable if the input digraphis acyclic and the number of colors is small.