Reachability relations in digraphs

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 R k + -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 R k − -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 R k + and R k − are equivalence relations on the vertex set of a given digraph. ve that, for transitive digraphs, properties of these relations are closely related to other properties such as having property Z , the number of ends, growth conditions, and vertex degree.