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) (fa(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 f (d, n)= ⎪⎪⎪⎨ ⎪⎪⎪⎩ d for n= 1, ( )(n) n + ( )(n− 1) n− 1 for 2 n 7, dn n + d n−1 n− 1 + O(nd n−4) for n 8, fa(d, n)= f (d, n+ 1) for n 1, where ( )(n)=∑i|n (i) (n/i), i|nmeans i divides n, (i)= di + (−1)id, (1)= 1 and (i)= i ·∏rj=1(1− 1/pj ) for i 2, where p1, . . . , pr are the distinct prime factors of i, not equal to 1. © 2006 Elsevier B.V. All rights reserved. MSC: 05C85; 68R10