Consider a discrete-time process on a graph G where a set B of initial vertices are chosen to be colored blue (the remainder being white) and then a time step consists of every currently blue vertex forcing all of its neighbors to become blue; this process stops when every vertex of the graph is blue, and the process is called full forcing. The full throttling number of G is then defined to be the minimum sum of the cardinality of B and the number of time steps needed to complete the forcing process. On trees, the full throttling number is equivalent to the throttling numbers of several other graph processes, such as positive-semidefinite zero forcing, the game of cops and robbers, and the distance domination number (alternately, the k-radius) of a graph. For all of these, it is known that maximum possible throttling number for a tree on n vertices is somewhere between $1.4502\sqrt{n}$ and $\frac{\sqrt{14}}{2}\sqrt{n}$, with the former exhibited by a family of spiders. After introducing some new ideas and methods for working with throttling on trees, this paper determines the exact full throttling number of all balanced spiders (trees with equal-length paths extending from a center vertex), and proves that their full throttling numbers are bounded above by that of paths of the same order n, which are known to have full throttling number $\lceil \sqrt{2n}-\frac{1}{2}\rceil$.

考虑图G上的离散时间过程，在该过程中，一组初始顶点B被选择为蓝色（其余为白色），然后一个时间步长由每个当前蓝色的顶点组成，迫使其所有邻居变为蓝色。当图形的每个顶点都是蓝色时，此过程停止，该过程称为“完全强制”。然后将G的全部节流数定义为B的基数的最小总和以及完成强制过程所需的时间步数。在树上，完整的节流数等于其他几个图形过程的节流数，例如正半定零强迫，警察和强盗的博弈以及图形的距离支配数（或者，k半径） 。对于所有这些，已知在n个顶点上一棵树的最大可能节流数是介于$1.4502\sqrt{ñ}$ 和 $\frac{\sqrt{14}}{\mathrm{2个}}\sqrt{ñ}$，前者由一个蜘蛛家族展出。在介绍了一些在树上进行节流的新思想和方法之后，本文确定了所有平衡蜘蛛（具有从中心顶点延伸的等长路径的树）的确切全部节流数，并证明了它们的全部节流数都在上面由已知的具有完整节流数的相同n阶路径的$\lceil \sqrt{\mathrm{2个}ñ}-\frac{\mathrm{1个}}{\mathrm{2个}}\rceil$。