Induced Subgraph Isomorphism on Interval and Proper Interval Graphs

The INDUCED SUBCRAPH IsomciRPHIsm problem On two input graphs G and H is to decide whether G has an induced subgraph isomorphic to H. Already for the restricted case where H is a complete graph the problem is NP-complete, as it is then equivalent to the CLIQUE problem. In a recent paper [7] Marx and Schlotter show that INDUCED SUBCRAPH ISOMORPHISM is NP-complete when G and H are restricted to be interval graphs. They also show that the problem is W[1]-hard with this restriction when pararnetrised by the number of vertices in H. In this paper we show that when G is an interval graph and H is a connected proper interval graph, the problem is solvable in polynomial time. As a more general result, we show that when G is an interval graph and H is an arbitrary proper interval graph, the problem is fixed parameter tractable when parametrised by the number of connected components of H. To complement our results, we prove that the problem remains NP-complete when G and H are both proper interval graphs and H is disconnected.