It is NP -hard to determine the minimum number of branching vertices needed in a single-source distance-preserving subgraph of an undirected graph. We show that this problem can be solved in polynomial time if the input graph is an interval graph. In earlier work, it was shown that every interval graph with k terminal vertices admits an all-pairs distance-preserving subgraph with O(klog k) branching vertices [13]. We extend this result to bi-interval graphs; these are graphs that can be expressed as the strong product of two interval graphs. We present a polynomial time algorithm that takes a bi-interval graph with k terminal vertices as input, and outputs an all-pairs distance-preserving subgraph of it with O(k2) branching vertices. This bound is tight. © Springer Nature Switzerland AG 2019.