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{\bf Courtney R. Gibbons and Joshua D. Laison}
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{\bf Fixing Numbers of Graphs and Groups}
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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 automorphism
group $\Gamma$. Several authors have studied the distinguishing
number of a graph, the smallest number of labels needed to label
$G$ so that the automorphism group of the labeled graph is
trivial.  The fixing number can be thought of as a variation of
the distinguishing number in which every label may be used only
once, and not every vertex need be labeled.  We characterize the
fixing sets of finite abelian groups, and investigate the fixing
sets of symmetric groups.




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