Avdonin–Zhao's watershed conjecture for optimal graph control time

Let $\mathcal{G}$ be a metric graph, and let the free wave equation on $\mathcal{G}$ be observed or controlled from an open subset of the graph. A watershed is the decomposition used by Avdonin and Zhao to determine a control time from the longest river covering the uncontrolled part.

Avdonin–Zhao's watershed conjecture. The watershed description of Avdonin and Zhao provides the optimal time for the observation and control of the free wave equation on a graph.

Computing the optimal observation and control time on a general graph is described as an open problem. The watershed time is always shorter than the time obtained from the graph geometric control condition, but its optimality is not known in general.