Graphs of interval count two with a given partition

We describe a polynomial time algorithm to decide for a given connected graph G and a given partition of its vertex set into two sets A and B  , whether it is possible to assign a closed interval I(u)I(u) to each vertex u of G such that two distinct vertices u and v of G   are adjacent if and only if I(u)I(u) and I(v)I(v) intersect, all intervals assigned to vertices in A   have some length LALA, and all intervals assigned to vertices in B   have some length LBLB where LA