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 uu is Rk+-related to a vertex vv if there exists a 0-weighted walk from uu to vv whose every subwalk starting at uu has weight in the interval [0,k][0,k]. Similarly, a vertex uu is Rk−-related to a vertex vv if there exists a 0-weighted walk from uu to vv whose every subwalk starting at uu has weight in the interval [−k,0][−k,0]. For all positive integers kk, the relations Rk+ and Rk− 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 Z, the number of ends, growth conditions, and vertex degree.