Feedback numbers of Kautz digraphs

A subset of vertices (resp. arcs) of a graph G is called a feedback vertex (resp. arc) set of G if its removal results in an acyclic subgraph. Let f (d, n) (f_a (d, n)) denote the minimum cardinality over all feedback vertex (resp. arc) sets of the Kautz digraph K (d, n). This paper proves that for any integers d ≥ 2 and n ≥ 1{Mathematical expression}where (φ{symbol} ȯ θ) (n) = ∑_{i | n} φ{symbol} (i) θ (n / i), i | n means i divides n, θ (i) = dⁱ + (- 1)ⁱ d, φ{symbol} (1) = 1 and φ{symbol} (i) = i · ∏_{j = 1}^r (1 - 1 / p_j) for i ≥ 2, where p₁, ..., p_r are the distinct prime factors of i, not equal to 1.