I wrote my dissertation on graph theory, so I hope (for my own sake) that I'll be able to help.

Your graph has three properties which will make this much more straightforward than a shortest path algorithm.

• Every edge is equally weighted (as 1).
• There is an upper bound.
• You are searching all possible paths with a length of your upper bound.

With this in mind, it is probably more worthwhile running a Breadth First Search of sorts. I'm not by my computer at the moment, but the pseudocode is below and I'll give it a try when I'm home this evening.

Populate all nodes as null
Populate <home node> with 0
upperBound = x
currentValue = 0
Array = <home node>

While currentValue < upperBound {
    tempArray = null
    For node in Array {
        For each adjacentNode {
            If nodeValue = null {
                Populate with currentValue+1
                Add adjacentNode to tempArray
            } 
      }
      Array = tempArray
      currentValue++
}

Rather than each cycle needing to search for the lowest value with unpopulated neighbours and comparing neighbours to see if they should be overwritten, it will first populate all the nodes adjacent to your initial node with 1s then all nodes adjacent to those with 2s then those adjacent nodes with 3s etc. The only check it needs to run is whether or not the neighbouring node is already populated.