Strongly invertible knots, rational-fold branched coverings and   hyperbolic spatial graphs
  A construction of a spatial graph from a strongly invertible knot was developed by the second author, and a necessary and sufficient condition for the given spatial graph to be hyperbolic was provided as well. The condition is improved in this paper. This enable us to show that certain classes of knots can yield hyperbolic spatial graphs via the construction. 
Kazuhiro Ichihara and Akira Ushijima