Minimizing Branching Vertices in Distance-preserving Subgraphs

TL;DR

This paper studies the complexity of minimizing branching vertices in distance-preserving subgraphs, showing polynomial solutions for special graph classes and establishing tight bounds for these cases.

Contribution

It provides polynomial-time algorithms for constructing minimal distance-preserving subgraphs in interval graphs and their strong products, with tight bounds on branching vertices.

Findings

Polynomial-time algorithm for interval graphs.

All-pairs distance-preserving subgraph with O(k^2) branching vertices.

Bound is tight for the considered graph classes.

Abstract

It is $\mathsf{NP}$-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 $O(k\log k)$ branching vertices. We consider graphs that can be expressed as the strong product of two interval graphs, and present a polynomial time algorithm that takes such a graph with $k$ terminals as input, and outputs an all-pairs distance-preserving subgraph of it with $O(k^2)$ branching vertices. This bound is tight.