a graph "l" is the line graph of some other graph or multigraph if and only if it is possible to find a collection of cliques in "l" (allowing some of the cliques to be single vertices) that partition the edges of "l", such that each vertex of "l" belongs to exactly two of the cliques. it is the line graph of a graph (rather than a multigraph) if this set of cliques satisfies the additional condition that no two vertices of "l" are both in the same two cliques. given such a family of cliques, the underlying graph "g" for which "l" is the line graph can be recovered by making one vertex in "g" for each clique, and an edge in "g" for each vertex in "l" with its endpoints being the two cliques containing the vertex in "l". by the strong version of whitney's isomorphism theorem, if the underlying graph "g" has more than four vertices, there can be only one partition of this type.