Critical Pebbling Numbers of Graphs

We define three new pebbling parameters of a connected graph \( G \), the \( r \)-, \( g \)-, and \( u \)-\emph{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 \( 2^d \).