  This paper revisits two classical distributed problems in anonymous networks,
namely spanning tree construction and topology recognition, from the point of
view of graph covering theory. For both problems, we characterize necessary and
sufficient conditions on the communication graph in terms of directed symmetric
coverings. These characterizations answer along-standing open question posed by
Yamashita and Kameda [YK96], and shed new light on the connection between
coverings and the concepts of views and quotient graphs developed by the same
authors. Characterizing conditions in terms of coverings is significant because
it connects the field with a vast body of classical literature in graph theory
and algebraic topology. In particular, it gives access to powerful tools such
as Reidemeister's theorem and Mazurkiewicz's algorithm. Combined together,
these tools allow us to present elegant proofs of otherwise intricate results,
and their constructive nature makes them effectively usable in the algorithms.
This paper also gives us the opportunity to present the field of covering
theory in a pedagogical way, with a focus on the two aforementioned tools,
whose potential impact goes beyond the specific problems considered in this
work.
