Water Transport on 3-Spiders: Unbounded Lengths of Shortest Optimal Strategies

Let $G$ be a finite simple graph. An averaging operation selects a nonempty connected vertex set and replaces the weights of its vertices by their arithmetic mean. For a nonnegative initial weight function $\omega :V(G)\to {\mathbb{R}}_{\ge 0}$ and a target vertex $x$, let $t(G,x,\omega )$ denote the minimum number of averaging operations needed to attain the maximum reachable weight at $x$. Gollin et al.[1] proved, using the Axiom of Choice, that a finite optimal strategy exists for every finite graph, every nonnegative initial weight function and every target vertex; see Tao [2] for a simplified proof. We prove that the lengths of shortest optimal strategies on $3$-spiders have no uniform finite bound. In particular, these lengths cannot be bounded in terms of the maximum degree of the graph.