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. 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.
2008
2016-04-08 16:46:15
1033
graph theory, digraph, reachability relations, end of a graph, property ▫$\mathbb{Z}$▫, growth
teorija grafov, usmerjeni grafi, rast
r6
Norbert
Seifter
70
Boris
Zgrablić
70
Aleksander
Malnič
70
Primož
Šparl
70
Dragan
Marušič
70
ISSN
2
0195-6698
UDK
4
519.17
OceCobissID
13
25427968
COBISS_ID
3
2017509
DOI
15
10.1016/j.ejc.2007.11.003
0
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2016-04-08 16:46:16