On the characterization of graphs with tree 3-spanners

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 $\ddot{I}ƒ(G)$. It is well known in the literature that determining $\ddot{I}ƒ(G)\le 2$ is polynomially solvable, while determining $\ddot{I}ƒ(G)\le k$ for $k\ge 4$ is NP-complete. A long-standing open problem is to characterize graphs with $\ddot{I}ƒ(G)=3$. This paper settles this open problem by proving that it is polynomially solvable.

19 pages, 5 figures