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arXiv · 0707.4256

Rubbling and Optimal Rubbling of Graphs

Christopher Belford·Nandor Sieben
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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 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 of a graph 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 determine the rubbling and optimal rubbling number of some families of graphs including cycles.

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Christopher Belford, Nandor Sieben. 2007-07-28. Rubbling and Optimal Rubbling of Graphs. https://arxiv.org/abs/0707.4256

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