Critical Pebbling Numbers of Graphs

TL;DR

This paper introduces three new pebbling parameters for connected graphs, explores their relationships with existing parameters, and characterizes certain classes of graphs based on these parameters.

Contribution

It defines the $r$-, $g$-, and $u$-critical pebbling numbers, analyzes their relationships with other graph parameters, and classifies graphs according to these new parameters.

Findings

Determined relationships between seven graph parameters including pebbling numbers.

Investigated properties of the $r$-critical pebbling number.

Characterized classes of graphs such as greedy and thrifty graphs.

Abstract

We define three new pebbling parameters of a connected graph $G$, the $r$-, $g$-, and $u$-critical pebbling numbers. Together with the pebbling number, the optimal pebbling number, the number of vertices $n$ and the diameter $d$ of the graph, this yields 7 graph parameters. We determine the relationships between these parameters. We investigate properties of the $r$-critical pebbling number, and distinguish between greedy graphs, thrifty graphs, and graphs for which the $r$-critical pebbling number is $2^d$.