Geodesic
On automorphisms of digraphs without symmetric cycles
Commentationes Mathematicae Universitatis Carolinae, Tome 37 (1996) no. 3, pp. 457-467.

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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 $D$.
Classification : 05C20, 05C25
Keywords: asymmetric diagraphs 
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     title = {On automorphisms of digraphs without symmetric cycles},
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Wójcik, Piotr. On automorphisms of digraphs without symmetric cycles. Commentationes Mathematicae Universitatis Carolinae, Tome 37 (1996) no. 3, pp. 457-467. http://geodesic.mathdoc.fr/item/CMUC_1996__37_3_a2/