Improved Feedback Vertex Sets in Kautz Digraphs K (d, n)

This paper considers the feedback number f (d, n) of Kautz digraph K (d, n), which is the minimum number of vertices in K (d, n) whose removal results in an acyclic digraph. Xu et al. have determined the exact values of f (d, n) for 1 ≤ n ≤ 7 and obtained an asymptotic formula f (d, n)=d
ⁿ
/n + d
^n-1
/n-1 + O (nd
^n-4
) for n ≥ 8. In this paper, we construct a feedback vertex set of K (d, n) and obtain f (d, n)= d
ⁿ
/n +dn-1/n-1 + O (d
^n/2
) for n ≥ 8 asymptotically.