On the characterization of graphs with tree 3-spanners

TL;DR

This work settles the open problem of characterizing graphs with σ(G)=3 by establishing a precise, testable condition (Theorem 3.2) based on inner star-sets and vertex cuts, and by designing a polynomial-time recognition algorithm. The core idea is that σ(G)≤3 is achievable if either the optimal spanning tree has diameter at most 3 or inner star-sets decompose G into outer components in a controlled way, enabling a constructive assembly via fixed subtrees. The proposed Stretch-three Recognition Algorithm combines a neighbor-cut decomposition with solved ICS and CFE subproblems, achieving a total time bound of O(n^5) for 2-connected graphs. By proving polynomial solvability for σ(G)≤3, the paper delineates a sharp boundary between tractable and intractable cases for tree k-spanners and broadens the class of graphs (e.g., star-separating, interval, split, and permutation graphs) for which efficient tree-spanner recognition is possible.

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