Restricted optimal pebbling is NP-hard
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Consider a distribution of pebbles on a graph. A pebbling move removes two pebbles from a vertex and place one at an adjacent vertex. A vertex is reachable under a pebble distribution if it has a pebble after the application of a sequence of pebbling moves. A pebble distribution is solvable if each vertex is reachable under it. The size of a pebble distribution is the total number of pebbles. The optimal pebbling number π^*(G) is the size of the smallest solvable distribution. A t-restricted pebble distribution places at most t pebbles at each vertex. The t-restricted optimal pebbling number π_t^*(G) is the size of the smallest solvable t-restricted pebble distribution. We show that deciding whether π^*_2(G)≤ k is NP-complete. We prove that π_t^*(G)=π^*(G) if δ(G)≥2|V(G)|/3-1 and we show infinitely many graphs which satisfies δ(H)≈1/2|V(H)| but π_t^*(H)≠π^*(H), where δ denotes the minimum degree.
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