Tree spanners of bounded degree graphs

TL;DR

This paper investigates the computational complexity of finding tree t-spanners in graphs, providing efficient algorithms for graphs with bounded degree and specific t-values, especially highlighting the case t=3.

Contribution

It introduces a dynamic programming algorithm for deciding tree t-spanner admissibility in bounded degree graphs, with special efficiency for t=3.

Findings

Efficient algorithms for t<3 and bounded degree graphs.

NP-completeness for t>3.

Special case algorithm for t=3 with degree constraints.

Abstract

A tree $t$-spanner of a graph $G$ is a spanning tree of $G$ such that the distance between pairs of vertices in the tree is at most $t$ times their distance in $G$. Deciding tree $t$-spanner admissible graphs has been proved to be tractable for $t<3$ and NP-complete for $t>3$, while the complexity status of this problem is unresolved when $t=3$. For every $t>2$ and $b>0$, an efficient dynamic programming algorithm to decide tree $t$-spanner admissibility of graphs with vertex degrees less than $b$ is presented. Only for $t=3$, the algorithm remains efficient, when graphs $G$ with degrees less than $b\log |V(G)|$ are examined.