Two-Player Pebbling on Diameter 2 Graphs

TL;DR

This paper introduces a two-player variant of graph pebbling on diameter 2 graphs, analyzing strategies and characterizing winning conditions for each player in this competitive setting.

Contribution

It presents a novel two-player pebbling game, providing strategies and characterizations for diameter two graphs, expanding the understanding of pebbling dynamics.

Findings

Identified configurations where each player has a winning strategy

Characterized the winning player for a specific class of diameter two graphs

Extended pebbling theory to a competitive two-player context

Abstract

A pebbling move refers to the act of removing two pebbles from one vertex and placing one pebble on an adjacent vertex. The goal of graph pebbling is: Given an initial distribution of pebbles, use pebbling moves to reach a specified goal vertex called the root. The pebbling number of a graph $\pi(G)$ is the minimum number of pebbles needed so every distribution of $\pi(G)$ pebbles can reach every choice of the root. We introduce a new variant of graph pebbling, a game between two players. One player aims to move a pebble to the root and the other player aims to prevent this. We show configurations of various classes of graphs for which each player has a winning strategy. We will characterize the winning player for a specific class of diameter two graphs.