Optimal pebbling and rubbling of graphs with given diameter

A pebbling move on a graph removes two pebbles from a vertex and adds one pebble to an adjacent vertex. A vertex is reachable from a pebble distribution if it is possible to move a pebble to that vertex using pebbling moves. The optimal pebbling number pi(opt) is the smallest number m needed to guarantee a pebble distribution of m pebbles from which any vertex is reachable. A rubbling move is similar to a pebbling move, but it can remove the two pebbles from two different vertices. The optimal rubbling number rho(opt) is defined analogously to the optimal pebbling number. In this paper we give lower bounds on both the optimal pebbling and rubbling numbers using the distance k domination number. With this bound we prove that for each k there is a graph G with diameter k such that rho(opt)(G) = pi(opt)(G) = 2(k). (C) 2018 Elsevier B.V. All rights reserved.