Distance-preserving Subgraphs of Interval Graphs

We consider the problem of finding small distance-preserving subgraphs of undirected, unweighted interval graphs with $k$ terminal vertices. To start with, we show that finding an optimal distance-preserving subgraph is $\mathsf{NP}$-hard for general graphs. Then, we show that every interval graph admits a subgraph with $O(k)$ branching vertices that approximates pairwise terminal distances up to an additive term of $+1$. We also present an interval graph $G_{\mathrm{int}}$ for which the $+1$ approximation is necessary to obtain the $O(k)$ upper bound on the number of branching vertices. In particular, any distance-preserving subgraph of $G_{\mathrm{int}}$ has $Ω(k\log k)$ branching vertices. Furthermore, we prove that every interval graph admits a distance-preserving subgraph with $O(k\log k)$ branching vertices, implying that the $Ω(k\log k)$ lower bound for interval graphs is tight. To conclude, we show that there exists an interval graph such that every optimal distance-preserving subgraph of it has $O(k)$ branching vertices and $Ω(k\log k)$ branching edges, thereby providing a separation between branching vertices and branching edges. The $O(k)$ bound for distance-approximating subgraphs follows from a naïve analysis of shortest paths in interval graphs. $G_{\mathrm{int}}$ is constructed using bit-reversal permutation matrices. The $O(k\log k)$ bound for distance-preserving subgraphs uses a divide-and-conquer approach. Finally, the separation between branching vertices and branching edges employs Hansel's lemma for graph covering.