A digraph is a symmetric cycle if it is symmetric and its underlying graph is a cycle. It is proved that if $D$ is an asymmetric digraph not containing a symmetric cycle, then $D$ remains asymmetric after removing some vertex. It is also showed that each digraph $D$ without a symmetric cycle, whose underlying graph is connected, contains a vertex which is a common fixed point of all automorphisms of $D$.
@article{118852,
author = {Piotr W\'ojcik},
title = {On automorphisms of digraphs without symmetric cycles},
journal = {Commentationes Mathematicae Universitatis Carolinae},
volume = {37},
year = {1996},
pages = {457-467},
zbl = {0881.05051},
mrnumber = {1426910},
language = {en},
url = {http://dml.mathdoc.fr/item/118852}
}
Wójcik, Piotr. On automorphisms of digraphs without symmetric cycles. Commentationes Mathematicae Universitatis Carolinae, Tome 37 (1996) pp. 457-467. http://gdmltest.u-ga.fr/item/118852/
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