# Partition into interval graphs

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.

• Isn't the recognition of a 2-track graph (in the West paper) exactly your problem ? Commented Feb 4, 2014 at 19:36
• I think recognizing 2-track graphs is the edge-version of the problem. Commented Feb 4, 2014 at 20:45

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).