The Optimal Pebbling Number of Staircase Graphs

TL;DR

This paper determines the minimum number of pebbles needed to ensure reachability of any vertex in staircase graphs, a class of subgraphs of the square grid, through pebbling moves.

Contribution

It introduces the optimal pebbling number for staircase graphs and provides exact values for several classes of these induced subgraphs.

Findings

Exact optimal pebbling numbers for certain staircase graphs.

Methodology for calculating pebbling numbers in grid-based graphs.

Insights into pebbling strategies on staircase graph structures.

Abstract

Let G be a graph with a distribution of pebbles on its vertices. A pebbling move consists of removing two pebbles from one vertex and placing one pebble on an adjacent vertex. The optimal pebbling number of G is the smallest number of pebbles which can placed on the vertices of G such that, for any vertex v of G, there is a sequence of pebbling moves resulting in at least one pebble on v. We determine the optimal pebbling number for several classes of induced subgraphs of the square grid, which we call staircase graphs.