Throttling processes equivalent to full throttling on trees

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