We extend the notion of the join of two graphs by replacing the edges connecting the vertices of the two graphs by paths. We also introduce the notion of this type of join to fuzzy graphs. The motivation of this operation comes from applications to human trafficking, modern slavery, and illegal immigration. We then introduce the notion of an (x, y) path fan where x is a vertex and Y is a subset of edges of a graph. We apply this notion to certain trees that arise in applications, where Y is the set of leafs of the tree.

fuzzy graph, join, path fan, fuzzy tree, health consequences, trafficking victims.