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 â€¦

The fixing number of a graph $G$ is the smallest cardinality of a set of vertices $S$ such that only the trivial automorphism of $G$ fixes every vertex in $S$. The fixing set of a group $\Gamma$ is the set of all fixing numbers of finite graphs with â€¦

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