Breadth first search and Eppstein K shortest paths algorithm
The main reason is that the non-constant degree would imply that the time per result is too high. The number of nodes found by the search, in the end, is going to be k, but all but the last one needs to have their children expanded, so the number of expanded nodes is kd where d is the degree. Unless d is O(1) this will be larger than the eventual time bound of the algorithm.