Rubbling and Optimal Rubbling of Graphs

TL;DR

This paper explores a variant of graph pebbling called rubbling, introducing new moves and analyzing the minimum pebble counts needed for reachability and optimal distributions across various graph families.

Contribution

It defines rubbling and optimal rubbling numbers, and determines these values for specific graph families such as cycles, advancing understanding of pebbling variants.

Findings

Rubbling number for cycles determined

Optimal rubbling number for certain graph families established

New move in pebbling theory introduced

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.