This question is an extension of this one: Min path cover for a three-layer graph with all paths traversing all layers.
I’m designing fictional fruits. Each fruit has three attributes; color, taste and smell. Also, each of the values of the attributes have some compatibility with the values of the other attributes. So, we get a tri-partite graph. An example is shown below. Here, 1 and 2 are color attributes, 3,4 and 5 are taste attributes and 6 and 7 are smell attributes. Also, color-1 is compatible with taste-3 and smell-6 and so on.
I want to design a minimal number of fruits while still covering all attributes (all 2 colors, all 3 tastes and all 2 smells in this case). For example, the example graph above has the following solution (shown with pink lines, fruit-1 has color-1, taste-3 and smell-6; fruit-2 has color-1, taste-4 and smell-7 while fruit-3 has color-2, taste-5 and smell-6; hence covering all levels of all attributes with 3 fruits):
We know this is optimal since there are three tastes and we couldn’t have used less than 3 fruits in this case.
The question is: how to design an algorithm to get the minimum number of fruits required and their configurations given a general graph like the one specified above.
Note that it might not have been possible to cover all attributes even if the graph has no isolated vertices and I asked a question on feasibility here: https://math.stackexchange.com/questions/3998648/possible-to-cover-all-vertices-of-a-tri-partite-graph-with-triangles