We consider the problem of finding small distance-preserving subgraphs of undirected, unweighted interval graphs that have k terminal vertices. We show that every interval graph admits a distancepreserving subgraph with O(k log k) branching vertices. We also prove a matching Ω (k log k) lower bound by exhibiting an interval graph based on bit-reversal permutation matrices. In addition, we show that interval graphs admit subgraphs with O(k) branching vertices that approximate distances up to an additive term of +1.