On the characterization of graphs with tree 3-spanners

TL;DR

This paper resolves a long-standing open problem by providing a polynomial-time characterization of graphs that admit a tree 3-spanner, advancing understanding of graph spanner properties.

Contribution

It proves that determining whether a graph has a tree 3-spanner is polynomially solvable, settling a major open question in graph theory.

Findings

Characterization of graphs with tree 3-spanners is polynomially computable.

The paper establishes the complexity boundary for tree spanner problems.

It completes the classification of the complexity for tree k-spanner problems.

Abstract

The tree spanner problem for a graph $G$ is as follows: For a given integer $k$, is there a spanning tree $T$ of $G$ (called a tree $k$-spanner) such that the distance in $T$ between every pair of vertices is at most $k$ times their distance in $G$? The minimum $k$ that $G$ admits a tree $k$-spanner is denoted by $\sigma(G)$. It is well known in the literature that determining $\sigma(G)\leq 2$ is polynomially solvable, while determining $\sigma(G)\leq k$ for $k\geq 4$ is NP-complete. A long-standing open problem is to characterize graphs with $\sigma(G)=3$. This paper settles this open problem by proving that it is polynomially solvable.