Partition into interval graphs
I think, your problem is NP-complete. It is a special case of a theorem 
by Farrugia, 
stating that it is NP-hard to test if the vertex set a graph can be partitioned into 
two subsets $V_1,$ and $V_2$ such that $G(V_1)$ belongs to the graph class $\mathcal{P}$ and 
$G(V_2)$ belongs to the graph class $\mathcal{Q}$, provided $\mathcal{P}$ and $\mathcal{Q}$ 
are closed under taking vertex-disjoint unions and talking induced subgraphs, and at least 
one of $\mathcal{P}$ and $\mathcal{Q}$ is non-trivial (meaning not all graphs in the class are 
edgeless).