# Reachability relations in digraphs

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Primož Šparl

University of Ljubljana and University of Primorska, Slovenia

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Aleksander Malnič

University of Ljubljana and University of Primorska, Slovenia

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Primož Potočnik

University of Ljubljana and University of Primorska, Slovenia

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Norbert Seifter

Montanuniversitaet Leoben, Austria

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**Minisymposium:**
STRUCTURE AND PROPERTIES OF VERTEX-TRANSITIVE GRAPHS

**Content:**
In 2002 Maru\v si\v c and Poto\v cnik introduced an infinite family of
relations (now called {\em reachability relations}) on (finite) digraphs and proved
various interesting results regarding them. In 2008 these relations were studied in
the context of infinite digraphs by Malni\v c, Maru\v si\v c, Seifter, \v Sparl and
Zgrabli\'c, again producing some nice results. The interplay of various (di)graph
theoretic properties and properties of these relations on infinite
(vertex-transitive) digraphs was further investigated in a paper by Seifter and
Trofimov from 2009.
In this talk we present a selection of the results on reachability relations from
the above mentioned papers, together with a recent result by Malni\v c, Poto\v cnik,
Seifter and \v Sparl, linking the properties of reachability relations of a given
vertex-transitive digraph admitting a nilpotent normal subgroup of automorphisms to
the nilpotency class of this subgroup. We also present some preliminary results from
a work in progress regarding the potential use of these relations in the
polycirculant conjecture problem for infinite digraphs.
This is joint work with Aleksander Malni\v c, Primo\v z Poto\v cnik and Norbert Seifter.