Abstract
For a digraph , the feedback vertex number , (resp. the feedback arc number ) is the minimum number of vertices, (resp. arcs) whose removal leaves the resultant digraph free of directed cycles. In this note, we determine and for the Cartesian product of directed cycles . Actually, it is shown that , and if then .
1. Introduction
Let be an undirected graph. A set is called a feedback vertex set of if contains no cycle. The feedback vertex number of , denoted by , is the cardinality of a minimum feedback vertex set of . In general, it is NP-hard to determine the feedback vertex number of a graph [1]. However, it becomes polynomial for specific families of graphs such as interval graphs [2], permutation graphs [3], graphs with maximum degree 3 [4], and -trees. The readers are referred to [5, 6] for a review of some earlier results and open problems, and [7–9] for some recent results on the feedback vertex number of graphs. Some bounds or exact values are established for various families of graph, for instance, outerplanar graphs [10], grids and butterflies [11], cubic graphs [12, 13], bipartite graphs [14], generalized Petersen graphs [15], regular graphs [16, 17]. Bau et al. [18] investigated the feedback number of grid graphs.
Apart from its graph-theoretical importance, the feedback vertex problem has many applications, such as operating system [19, 20], artificial intelligence [21], synchronous distributed systems [22, 23], optical networks [24]. The feedback vertex set and the feedback vertex number are also known as decycling set and the decycling number, respectively, see [25].
In 2005, Pike and Zou [26] determined the feedback vertex number of the Cartesian products of two cycles as follows:
Our main concern in this note is the directed version of the feedback vertex number. A directed graph is said to be acyclic if it does not contain any directed cycle. A feedback vertex set in a digraph is a set of vertices such that is acyclic, and the feedback vertex number of is the minimum size of such a set is denoted by . We denote by the number of vertex-disjoint cycles of . Clearly, for any digraph . A feedback arc set of a digraph is a set of arcs such that is acyclic. The feedback arc number of , denoted by , is the cardinality of a minimum feedback arc set of . We denote by the number of arc-disjoint cycles of . Clearly, for any digraph .
Not much works were known for the feedback vertex number or the feedback arc number of directed graphs. Lien et al. [27] gave an upper bound for the feedback vertex number of generalized Kautz digraphs. Figueroa et al. [28] investigated the relation for the relationship between the minimum feedback arc set and the acyclic disconnection of a digraph. Even et al. [29] gave a -approximation algorithm for the feedback vertex problem for a digraph of order . For planar digraphs, the approximation ratio is not greater than [30], and for tournament, it is 2.5 [31]. We refer to [32–34] for more results on feedback vertex set problems for tournaments and bipartite tournaments.
The Cartesian product of directed digraph is the digraph with the vertex set , in which there is an arc directed from to if and only if there exists an integer such that and for any other . For any integer , denotes the directed cycle of order . Various kinds of properties of are investigated. Trotter and Erdös [35] give a necessary and sufficient condition for being hamiltonian. Keating [36] gave a necessary and sufficient condition for being decomposed into directed Hamilton cycles. Recently, the previous result is extended by Bogdanowicz [37] with the decomposition into directed cycles of equal length. The domination number [38–41], respectively, the total domination number [42] of the Cartesian product of two directed cycles are investigated.
We shall determine the exact values of and for the Cartesian product of directed cycles .
2. Main Results
In this section, we denote by or, simply by . For convenience, label the vertices of as , where for each . For an integer , let be the subgraph of induced by the set of vertices
It is clear that for each .
Theorem 1. For any integers with for each ,
Proof. First, we show that by showing that.
We proceed with induction on . Let . By our notation, for each . Note that and are vertex-disjoint (and thus arc-disjoint). Moreover, since , . Now assume that . Since for each , by the induction hypothesis,
for each . After removing these cycles from , it results in exactly arc-disjoint directed cycles. This gives
Next we show that by finding a feedback arc set of with cardinality . Such a set feedback arc set for is constructed recursively as follows. For convenience, let . Note that is a feedback arc set of . For , . By the above construction and the induction hypothesis,
Moreover, since is acyclic, we conclude that is a feedback arc set of . This proves .
For an illustration, for the case when , we have , and , that is, the set of arcs colored in red as shown in Figure 1.
Theorem 2. For any integers with ,
Proof. Let .
First, we show that by showing that
We proceed with induction on . For the case when , for each . It follows that contains vertex-disjoint copies of , and thus . Now assume that . For every integer , , and hence, by the induction hypothesis,
Moreover, since and are vertex-disjoint for any , we have
As an example for the case when and , we have and , see Figure 2 for an illustration. Clearly, is a feedback vertex set of with .
For any , and , let
Since for any given value of with , there exists unique value of with satisfying , implying that .
To show that is a feedback vertex set of , we consider any directed cycle of , where for each . Since for each , , we have
Moreover, since , there exists an integer such that
implying that , and thus is a feedback vertex set of . This proves
3. Conclusion
In this note, we determined the two important parameters and for the Cartesian product of directed cycles . Actually, it is shown that and if , then .
Data Availability
Data sharing not applicable to this article as no datasets were generated or analysed during the current study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
All authors contributed equally and significantly in conducting this research work and writing this paper.
Acknowledgments
Research supported by NSFC (No. 11571294), the Key Laboratory Project of Xinjiang (2018D04017). The authors are grateful to the referees for their helpful suggestions.