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 |
Keywords
- Pebbling
- Rubbling
ASJC Scopus subject areas
- Theoretical Computer Science
- Discrete Mathematics and Combinatorics