Abstract
Given a finite loopless graph G (resp. digraph D), let σ( G), ϕ( G) and ψ( D) denote the minimal cardinalities of a completely separating system of G, a separating system of G and a separating system of D, respectively. The main results of this paper are: δ(G) = min m m ⌊m/2⌋ ⩾γ(G) and ϕ(G) = ⌉ log 2 γ(G)⌉ where γ(G) denotes the chromatic number of G. (ii) All the problems of determining σ( G), ϕ( G) and ψ( D) are NP-complete.
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