  In this note we prove an optimal volume growth condition for stochastic
completeness of graphs under very mild assumptions. This is realized by proving
a uniqueness class criterion for the heat equation which is an analogue to a
corresponding result of Grigor'yan on manifolds. This uniqueness class
criterion is shown to hold for graphs that we call globally local, i.e., graphs
where we control the jump size far outside. The transfer from general graphs to
globally local graphs is then carried out via so called refinements.
