Reachability relations in digraphs
Dragan Marušič, Aleksander Malnič, Norbert Seifter, Primož Šparl and Boris Zgrablić
Eur. j. comb. Volume 29, Number 7, pp. 1566-1581, 2008. ISSN 0195-6698

Abstract

In this paper we study reachability relations on vertices of digraphs, informally defined as follows. First, the weight of a walk is equal to the number of edges traversed in the direction coinciding with their orientation, minus the number of edges traversed in the direction opposite to their orientation. Then, a vertex u is View the MathML source-related to a vertex v if there exists a 0-weighted walk from u to v whose every subwalk starting at u has weight in the interval [0,k]. Similarly, a vertex u is View the MathML source-related to a vertex v if there exists a 0-weighted walk from u to v whose every subwalk starting at u has weight in the interval [−k,0]. For all positive integers k, the relations View the MathML source and View the MathML source are equivalence relations on the vertex set of a given digraph. We prove that, for transitive digraphs, properties of these relations are closely related to other properties such as having property View the MathML source, the number of ends, growth conditions, and vertex degree.

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