ARC dimension of a digraph
Publication Type:Journal Article
Source:Proceedings of the Jangjeon Mathematical Society, Jangjeon Research Institute for Mathematical Sciences and Physics, Volume 19, Number 1, p.107-114 (2016)
The results of Harary, Norman, and Cartwright on point-bases in finite digraphs to point- and arc-bases in infinite digraphs was extended by Acharya et.al,  by introducing the notion of arc bases of digraphs as follows: in a digraph D = (X,U), not necessarily finite, an arc (x,y) εU is reachable from a vertex u if there exists a directed walk W that originates from u and contains (x,y). A subset S ⊆ X is an arc-reaching set of D if for every arc (x,y) there exists a diwalk W originating at a vertex u ε S and containing (x,y) and an arc-basis as a minimal arc-reaching set. One of the main results reported in  is that all the arc bases of any finite digraph D have the same cardinality which led to the introduction of the notion of arc-dimension of D, denoted σ(D), as the cardinality of an arc basis of D. In this article we establish the upper and lower bounds on σ(D)+σ(D→), σ(D)+σ(Dc) and establish some related results.
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