The Optimal Rubbling Number of Ladders, Prisms and M\"obius-ladders

TL;DR

This paper determines the minimum number of pebbles needed to ensure any vertex in ladders, prisms, and Möbius-ladders can be reached through rubbling moves, extending pebbling concepts.

Contribution

It introduces the optimal rubbling number for specific graph families and provides exact values for ladders, prisms, and Möbius-ladders.

Findings

Optimal rubbling number for ladders ($P_n\square P_2$)

Optimal rubbling number for prisms ($C_n\square P_2$)

Optimal rubbling number for Möbius-ladders

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 ($P_n\square P_2$), prisms ($C_n\square P_2$) and M\"oblus-ladders.