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arXiv · 1708.09177

Optimal pebbling and rubbling of graphs with given diameter

Ervin Győri·Gyula Y. Katona·László F. Papp
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Abstract

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 $π_{opt}$ 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 vertex. The optimal rubbling number $ρ_{opt}$ is defined analogously to the optimal pebbling number. In this paper we give lower bounds on both the optimal pebbling and rubbling numbers by the distance $k$ domination number. With this bound we prove that for each $k$ there is a graph $G$ with diameter $k$ such that $ρ_{opt}(G)=π_{opt}(G)=2^k$.

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Ervin Győri, Gyula Y. Katona, László F. Papp. 2017-08-30. Optimal pebbling and rubbling of graphs with given diameter. https://arxiv.org/abs/1708.09177

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