The optimal rubbling number of ladders, prisms and Möbius-ladders

Katona, Gyula Y. and Papp, László F. (2016) The optimal rubbling number of ladders, prisms and Möbius-ladders. DISCRETE APPLIED MATHEMATICS, 209. pp. 227-246. ISSN 0166-218X

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Abstract

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 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 optimal rubbling number of ladders ( P n □ P 2 ), prisms ( C n □ P 2 ) and Möbius-ladders.

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