Geodesic
1-Restricted Optimal Rubbling on Graphs
Discussiones Mathematicae. Graph Theory, Tome 39 (2019) no. 2, pp. 575-588

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Let G be a graph with vertex set V and a distribution of pebbles on the vertices of V. A pebbling move consists of removing two pebbles from a vertex and placing one pebble on a neighboring vertex, and a rubbling move consists of removing a pebble from each of two neighbors of a vertex v and placing a pebble on v. We seek an initial placement of a minimum total number of pebbles on the vertices in V, so that no vertex receives more than one pebble and for any given vertex v ∈ V, it is possible, by a sequence of pebbling and rubbling moves, to move at least one pebble to v. This minimum number of pebbles is the 1-restricted optimal rubbling number. We determine the 1-restricted optimal rubbling numbers for Cartesian products. We also present bounds on the 1-restricted optimal rubbling number.
Keywords: graph pebbling, graph rubbling, optimal rubbling, t -restricted optimal pebbling 
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     title = {1-Restricted {Optimal} {Rubbling} on {Graphs}},
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Beeler, Robert A.; Haynes, Teresa W.; Murphy, Kyle. 1-Restricted Optimal Rubbling on Graphs. Discussiones Mathematicae. Graph Theory, Tome 39 (2019) no. 2, pp. 575-588. http://geodesic.mathdoc.fr/item/DMGT_2019_39_2_a19/