Stacking and Clearing in Directed Graph Pebbling

Abstract

Suppose that pebbles are distributed on the vertices of a directed graph D. A directed pebbling step u -> v along an arc u -> v removes two pebbles from u and places one pebble on v. We study the stacking number stack(D), the least integer t >= 2 such that every configuration with t pebbles can be transformed by a finite sequence of pebbling steps into a configuration with all pebbles on a single vertex, and the clearing number clear(D), defined analogously by requiring a final configuration with one pebble. Our main result is the formula stack(C_n) = n(2^{n-1}-1)+1 for the directed n-cycle C_n, for n >= 2. We also prove that, for finite simple digraphs with at least two vertices, stack(D) is defined precisely for strongly connected digraphs, and clear(D) is defined precisely for strongly connected digraphs whose directed cycle lengths have greatest common divisor 1.