The INDUCED SUBTREE ISOMORPHISM problem takes as input a graph G and a tree T, and the task is to decide whether G has an induced subgraph that is isomorphic to T. This problem is known to be NP-complete on bipartite graphs, but it can be solved in polynomial time when G is a forest. We show that INDUCED SUBTREE ISOMORPHISM can be solved in polynomial time when G is an interval graph. In contrast to this positive result, we show that the closely related SUBTREE ISOMORPHISM problem is NP-complete even when G is restricted to the class of proper interval graphs, a well-known subclass of interval graphs.
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