The critical Almost Stacked Hypothesis fails on a nine-vertex graph
In graph pebbling, a move removes two pebbles from a vertex and adds one to a neighbour; a configuration is stackable if moves can bring all pebbles onto one vertex and clearable if moves can leave exactly one pebble. The stacking (clearing) number of a connected (connected non-bipartite) graph is the least t >= 2 such that every configuration of t pebbles is stackable (clearable). The Almost Stacked Hypothesis, in its critical form, says that this number equals the least t0 >= 2 such that every almost stacked configuration (one vertex arbitrary, every other vertex at most one pebble) of…
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