Feedback numbers of de Bruijn digraphs

Authors:
Xirong Xu;Yongchang Cao;Jun-Ming Xu;Yezhou Wu
Affiliations:
Department of Computer Science, Dalian University of Technology, Dalian, 116024, China and Department of Mathematics, University of Science and Technology of China, Hefei, Anhui, 230026, China;Department of Mathematics, University of Science and Technology of China, Hefei, Anhui, 230026, China;Department of Mathematics, University of Science and Technology of China, Hefei, Anhui, 230026, China;Department of Mathematics, University of Science and Technology of China, Hefei, Anhui, 230026, China
Venue:
Computers & Mathematics with Applications
Year:
2010

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Abstract

A subset of vertices of a graph G is called a feedback vertex set of G if its removal results in an acyclic subgraph. Let f(d,n) denote the minimum cardinality over all feedback vertex sets of the de Bruijn digraph B(d,n). This paper proves that for any integers d=2 and n=2f(d,n)={1n@?i|nd^i@f(ni)for 2@?n@?4;d^nn+O(nd^n^-^4)for n=5, where i|n means i divides n, and @f(i) is the Euler totient function.