On Pebbling Graphs by their Blocks

TL;DR

This paper investigates the pebbling numbers of graphs using their block structures, providing exact values for graphs with complete blocks and an upper bound for diameter-two graphs, advancing understanding of pebbling in complex graphs.

Contribution

It introduces a block-structure approach to determine pebbling numbers, including exact values for complete block graphs and bounds for diameter-two graphs.

Findings

Exact pebbling number for graphs with complete blocks.

Upper bound for k-pebbling number of diameter-two graphs.

Framework for analyzing pebbling numbers via graph blocks.

Abstract

Graph pebbling is a game played on a connected graph G. A player purchases pebbles at a dollar a piece, and hands them to an adversary who distributes them among the vertices of G (called a configuration) and chooses a target vertex r. The player may make a pebbling move by taking two pebbles off of one vertex and moving one pebble to a neighboring vertex. The player wins the game if he can move k pebbles to r. The value of the game (G,k), called the k-pebbling number of G, is the minimum cost to the player to guarantee a win. That is, it is the smallest positive integer m of pebbles so that, from every configuration of size m, one can move k pebbles to any target. In this paper, we use the block structure of graphs to investigate pebbling numbers, and we present the exact pebbling number of the graphs whose blocks are complete. We also provide an upper bound for the k-pebbling number of diameter-two graphs, which can be the basis for further investigation into the pebbling numbers of graphs with blocks that have diameter at most two.