A set of positive integers with is said to be equi-graphical if there exists a graph with exactly vertices of degree for each with . It is known that such a set is equi-graphical if and only if is even and . This concept is generalized to the following problem: Given a set of positive integers and a permutation on , determine when there exists a graph containing exactly vertices of degree for each (). If such a graph exists, then is called a graphical permutation. In this paper, the graphical permutations on sets of size four are characterized and using a criterion of Fulkerson, Hoffman, and McAndrew, we show that a permutation of , where and such that , is graphical if and only if is even and .