an:07168972
Zbl 1453.05088
Chen, Xiaohong; Wu, Baoyindureng
Feedback arc number and feedback vertex number of Cartesian product of directed cycles
EN
Discrete Dyn. Nat. Soc. 2019, Article ID 7028573, 4 p. (2019).
00446588
2019
j
05C69 05C20 05C70 05C76
Summary: For a digraph \(D\), the feedback vertex number \(\tau \left(D\right)\), (resp. the feedback arc number \(\tau' \left(D\right))\) is the minimum number of vertices, (resp. arcs) whose removal leaves the resultant digraph free of directed cycles. In this note, we determine \(\tau \left(D\right)\) and \(\tau' \left(D\right)\) for the Cartesian product of directed cycles \(D = \overrightarrow{C_{n_1}} \square \overrightarrow{C_{n_2}} \square \cdots \overrightarrow{C_{n_k}} \). Actually, it is shown that \(\tau' \left(D\right) = n_1 n_2 \cdots n_k \sum_{i = 1}^k 1 / n_i\), and if \(n_k \geq \dots \geq n_1 \geq 3\) then \(\tau \left(D\right) = n_2 \ldots n_k\).