### Abstract

A pebbling move on a graph removes two pebbles at a vertex and adds one pebble at an adjacent vertex. Rubbling is a version of pebbling where an additional move is allowed. In this new move, one pebble each is removed at vertices v and w adjacent to a vertex u, and an extra pebble is added at vertex u. A vertex is reachable from a pebble distribution if it is possible to move a pebble to that vertex using rubbling moves. The rubbling number is the smallest number m needed to guarantee that any vertex is reachable from any pebble distribution of m pebbles. The optimal rubbling number is the smallest number m needed to guarantee a pebble distribution of m pebbles from which any vertex is reachable. We give bounds for rubbling and optimal rubbling numbers. In particular, we find an upper bound for the rubbling number of n-vertex, diameter d graphs, and estimates for the maximum rubbling number of diameter 2 graphs. We also give a sharp upper bound for the optimal rubbling number, and sharp upper and lower bounds in terms of the diameter.

Original language | English (US) |
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Pages (from-to) | 535-551 |

Number of pages | 17 |

Journal | Graphs and Combinatorics |

Volume | 29 |

Issue number | 3 |

DOIs | |

State | Published - May 2013 |

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### Keywords

- Pebbling
- Rubbling

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Theoretical Computer Science

### Cite this

*Graphs and Combinatorics*,

*29*(3), 535-551. https://doi.org/10.1007/s00373-012-1146-2