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On automorphisms of digraphs without symmetric cycles. (English) Zbl 0881.05051
Summary: 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 $$D$$.

##### MSC:
 05C20 Directed graphs (digraphs), tournaments
##### Keywords:
asymmetric digraphs
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