  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.
