Optimal pebbling and rubbling of graphs with given diameter

TL;DR

This paper establishes lower bounds on optimal pebbling and rubbling numbers of graphs based on their diameter and constructs graphs where these numbers are exactly exponential in the diameter.

Contribution

It introduces bounds relating optimal pebbling and rubbling numbers to the distance domination number and constructs graphs achieving these bounds.

Findings

Lower bounds on pebbling and rubbling numbers using distance domination.

Existence of graphs with diameter k where optimal pebbling and rubbling numbers are 2^k.

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 $\pi_{opt}$ 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 vertex. 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 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 $\rho_{opt}(G)=\pi_{opt}(G)=2^k$.