Suppose there is a graph $G=(V,E)$. I want to test if $V$ can be partitioned into two disjoint sets $V_1$ and $V_2$ such that the subgraphs induced by $V_1$ and $V_2$ are unit interval graphs.
I know about the NP-completeness of determining interval numbers but the above problem is different. Now, in the literature I found this work by A. Gyárfás and D. West on multitrack interval graphs but I'm not sure if it is relevant to above problem.
Any citation to existing literature on the above or similar problem would be helpful. Also please let me know if there is a formal name for the above problem.