Pebbling on Directed Graphs with Fixed Diameter

TL;DR

This paper explores the pebbling number in directed graphs with fixed diameter, revealing that orientation and structure significantly influence pebbling complexity, with new bounds established for various graph classes.

Contribution

It extends pebbling theory to directed graphs, providing bounds on the pebbling number based on diameter and orientation, including sharp bounds for diameter 2 graphs.

Findings

Pebbling number for oriented diameter-2 graphs is n or n+1.

Directed graphs with diameter 2 can have pebbling number up to 1.5n+1.

General directed graphs have pebbling number bounded by 2^d n / d + f(d).

Abstract

Pebbling is a game played on a graph. The single player is given a graph and a configuration of pebbles and may make pebbling moves by removing 2 pebbles from one vertex and placing one at an adjacent vertex to eventually have one pebble reach a predetermined vertex. The pebbling number, $\pi(G)$, is the minimum number of pebbles such that regardless of their exact configuration, the player can use pebbling moves to have a pebble reach any predetermined vertex. Previous work has related $\pi(G)$ to the diameter of $G$. Clarke, Hochberg, and Hurlbert demonstrated that every connected undirected graph on $n$ vertices with diameter 2 has $\pi(G) = n$ unless it belongs to an exceptional family of graphs, consisting of those that can be constructed in a specific manner; in which case $\pi(G) = n +1$. By generalizing a result of Chan and Godbole, Postle showed that for a graph with diameter $d$, $\pi(G) \le n 2^{\lceil \frac{d}{2} \rceil} (1+o_n(1))$. In this article, we continue this study relating pebbling and diameter with a focus on directed graphs. This leads to some surprising results. First, we show that in an oriented directed graph $G$ (in the sense that if $i \to j$ then we cannot have $j \to i$), it is indeed the case that if $G$ has diameter 2, $\pi(G) = n$ or $n + 1$, and if $\pi(G) = n+1$, the directed graph has a very particular structure. In the case of general directed graphs (that is, if $i \to j$, we may or may not have an arc $j \to i$) with diameter 2, we show that $\pi(G)$ can be as large as $\frac32 n + 1$, and further, this bound is sharp. More generally, we show that for general directed graphs, $\pi(G) \le 2^d n / d + f(d)$ where $f(d)$ is some function of only $d$.