Consider a set of three buckets: an $11$ gallon bucket, a $4$ gallon bucket, and a $5$ gallon bucket. We begin with the $4$ and $5$ gallon buckets full, and the $11$ gallon bucket empty. We are allowed to apply a single operation to these buckets: pour the contents of a bucket into another bucket until either the source bucket is empty, or the target bucket is full.

$(5$ pts$)$ Generally speaking, what are the states related to the above system, and the transitions between them?

$(10$ pts$)$ Define a graph which models the above states and transitions. Provide specific definitions of the vertex and edge sets. Remember that a graph vertex can contain any information you like$--$you aren't restricted to storing a single number.

$(10$ pts$)$ We'd like to devise an algorithm that would allow us to determine if it is possible to construct a sequence of pourings, from the given initial state, that would leave $x$ gallons of water in the bucket of $C$ capacity, where $xC.$ Using the graph defined above, what property of the graph would be present in the case that a solution exists to this problem? Devise an algorithm that would allow you to determine if that property holds, and