Reachability relations in digraphsSeifter, Norbert (Avtor)
Zgrablić, Boris (Avtor)
Malnič, Aleksander (Avtor)
Šparl, Primož (Avtor)
Marušič, Dragan (Avtor)
graph theorydigraphreachability relationsend of a graphproperty ▫$\mathbb{Z}$▫growthIn 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. We prove that, for transitive digraphs, properties of these relations are closely related to other properties such as having property ▫$\mathbb{Z}$▫, the number of ends, growth conditions, and vertex degree.20082016-04-08 16:46:15Delo ni kategorizirano7717ISSN: 0195-6698UDK: 519.17OceCobissID: 25427968COBISS_ID: 2017509DOI: 10.1016/j.ejc.2007.11.003sl