The t-admissibility problem aims to decide whether a graph G has a spanning tree T in which the distance between any two adjacent vertices of G is at most t. Regarding its optimization version, the smallest t for which G is t-admissible is the stretch index of G, denoted by ${\sigma }_T(G)$. The problem of deciding whether ${\sigma }_T(G)\le t$, $t\ge 4$ is $\mathsf{NP}$-complete and polynomial-time solvable for $t=2$. However, deciding if $t=3$ is an open problem. We determine 3-admissible graph classes by studying graphs with few $P_4$'s, and we partially classify the $\mathsf{P}$ vs $\mathsf{NP}$-complete dichotomy of the t-admissibility problem for $(k,\ell )$-graphs. These graph classes generalize some others in which the computational complexity of the t-admissibility problem was already determined. Moreover, we determine the stretch index for cycle-power graphs and for $(2,1)$-chordal graphs, which are subclasses of $(k,\ell )$-graphs and the t-admissibility problem is $\mathsf{NP}$-complete.

所述吨-admissibility问题的目的，以决定一个图是否ģ具有生成树Ť其中的任意两个相邻的顶点之间的距离G ^为至多吨。关于它的优化版本，最小吨为其ģ是吨-admissible是的拉伸指数ģ，记${\sigma }_{Ť}（G）$。决定是否${\sigma }_{Ť}（G）\le Ť$， $Ť\ge 4$ 是 $\mathsf{NP}$完全和多项式时间可解 $Ť=2$。但是，决定是否$Ť=3$是一个开放的问题。我们通过研究很少的图来确定3容许图类$P_4$，我们将其部分分类 $\mathsf{P}$ 与 $\mathsf{NP}$的t-可容许性问题的完全二分法$（ķ，\ell ）$-图。这些图类概括了其中一些已经确定了t容许性问题的计算复杂度的其他图类。此外，我们确定了循环功率图和$（2，\mathrm{1个}）$-chordal图，它们是 $（ķ，\ell ）$图和t的可容许性问题是$\mathsf{NP}$-完成。