Abstract
A vertex subset is a cyclic vertex-cut of a connected graph if is disconnected and at least two of its components contain cycles. The cyclic vertex-connectivity is denoted as the cardinality of a minimum cyclic vertex-cut. In this paper, we show that the cyclic vertex-connectivity of the -star network is for any integer and .
1. Introduction
Let be a simple connected graph, where and are the vertex set and the edge set, respectively. is an induced subgraph by , whose vertex set is and whose edge set consists of all the edges of with both ends in . For any vertex , define the neighborhood . Let and the set is denoted by . We use to replace , to replace , and to replace . A graph is said to be -regular if for any vertex . For any subset , or denotes the graph obtained by removing all vertices in from . If there exists a nonempty subset such that is disconnected, then is called a vertex-cut of . The connectivity is the minimum number of vertices whose removal results in a disconnected graph or only one vertex left. Let and denote the minimum degree and the girth of , respectively. As usual, we use and to denote the complete graph and the cycle of order , respectively.
In this work, we study a kind of restricted vertex-connectivity known as the cyclic vertex-connectivity. A vertex subset is a cyclic vertex-cut of if has at least two components containing cycles. Not all connected graphs have a cyclic vertex-cut. The cyclic vertex-connectivity of a graph is the cardinality of the minimum cyclic vertex-cut of . When has no cyclic vertex-cut, the definition of can be found in [1] using Betti number. A graph is said to be -connected if has a cyclic vertex-cut. Similarly, changing “edge” to “vertex,” the cyclic edge-connectivity of graph can be defined.
The definition of the cyclic vertex- (edge-) connectivity dates to Tait in attacking four color conjecture [2] and the graph colouring [2, 3]. It is used in many classic fields, such as integer flow conjectures [4] and n-extendable graphs [5, 6]. In many works, the cyclic vertex-connectivity has been studied. Cheng et al. [7] studied the cyclic vertex-connectivity of Cayley graphs generated by transposition trees. Yu et al. [8] obtained the cyclic vertex-connectivity of star graphs. For more research studies on the cyclic vertex-connectivity, see [7, 9–11] for references.
This paper focuses on the cyclic vertex-connectivity of the -star network . We will show that for any integer and find out the minimum circle vertex-cut structure of the -star network .
2. Some Preliminaries
We provide the definition of the -star graph and its structural properties, which are useful for the following discussion.
For convenience, let and for any integers and with . Clearly, .
Definition 1 (see [12]). The -star graph, denoted by (see Figure 1), is a graph with the vertex-set and the edge set defined as follows:(1)A vertex is adjacent to the vertex through an edge of dimension , where (i.e., exchange with )(2)A vertex is adjacent to the vertex through an edge of dimension 1, where (i.e., replace by )
The edges of type (1) are referred to as -edges, and the corresponding neighboring vertices are called -neighbors. The edges of type (2) are referred to as 1-edges, and the corresponding neighboring vertices are called 1-neighbors. Let be induced by all the vertices having the symbol in one of the rightmost positions of . Clearly, can be decomposed into subgraphs and , where and .
Lemma 1 (see [13]). The -star graph has the following properties:(1) is a graph of degree with vertices and edges.(2), , and , where is a complete graph, is a -dimensional star graph, and is a -dimensional alternating group network.(3) is the set of all cross edges between any two subgraphs and , and .(4)For any two vertices and in ,
Lemma 2 (see [13]). is a -regular -connected graph.
Theorem 1 (see [14]). Let be a faulty vertex set of with . Then, the survival graph satisfies one of the following conditions:(1) is connected(2) has two components, one of which has exactly one vertex or two vertices with one 1-edge(3) has three components, two of which are singletons
Lemma 3 (see [13]). If a cycle has a length at least 6 in an , then it contains one -edge, .
Theorem 2 (see [8]). For any integer , .
3. Main Result
By Lemma 1 and Theorem 2, we know if and . Thus, we determine the value of with for .
Lemma 4. For , the girth of is 3 and the edges of every 3-cycle are 1-edges.
Proof. Choose any vertex from and make it as . Since , there exist and . From the definition of , we have two vertices and in and . Clearly, is one 3-cycle, and all the edges of it are 1-edges. Hence, the lemma holds.
Lemma 5. Let be any cycle of length 3 in . Then, .
Proof. From Lemma 4, we can suppose . Let , then . By the definition of , we haveSo,
Lemma 6. Let be a 3-cycle of . Then, is a cyclic vertex-cut of .
Proof. Clearly, is disconnected and contains as a connected component. In order to prove the lemma, it suffices to show that the subgraph has a cycle. In fact, we can prove a stronger property as follows.
Suppose . By Lemma 5, and . If , and then there exists a vertex satisfying and . Since is -regular, has at least neighbors in . Let be two distinct vertices in .
If , without loss of generality, let , , where and . By Definition 1, all the edges in are -edges or 1-edges. Since , , and . It means , contradicting .
If and , without loss of generality, then let and , where and . Since , , and , it means , contradicting .
If , clearly, if , then . If , then , similarly, contradicting .
From the above discussion, we know that . Furthermore, is a cyclic vertex-cut of .
Combining Lemma 5 and 6, we have the following theorem.
Theorem 3. For any integer , .
Proof. Let be a 3-cycle in and . By Lemmas 5 and 6, is a cyclic vertex-cut of . Hence, . By Lemma 2, . We have , and then .
Theorem 4. For any integer , and .
Proof. Let be a faulty vertex set of with . By Theorem 1, is connected or is disconnected, and at most one of its component contains cycles. Hence, . To prove the converse, we need to find a cyclic vertex-cut of with .
Suppose , we have two vertices , such that and are 1-edges by . Without loss of generality, we can assume and . From the definition of , we have . Hence, in . Let , then and . Since have another -neighbors in , respectively, . Clearly, is disconnected and contains as a connected component. In order to find a cyclic vertex-cut, it suffices to show that has a cycle. In fact, we can prove . Suppose that there exists one vertex of with . Since is -regular , .
If is adjacent to one neighbor vertex of , , and , respectively, it means there are three vertices , , and such that , , , and . Then, all of are -edges. Let , , and . By the definition of , we know is adjacent to at most one of , , and , a contradiction.
If is adjacent to two neighbor vertices of and one neighbor vertex of , it means there are three vertices , , and such that , , and . From the definition of , both are 1-edges and , . Furthermore, , a contradiction with .
If is adjacent to three neighbor vertices of , it means there are three vertices , , and such that and . From the definition of , all of are 1-edges and and . Furthermore, , a contradiction with .
From the above discussion, we know that . Then, contains a cycle, and is a cyclic vertex-cut of with . Hence, . The theorem holds.
Combining Theorems 3 and 4, we have the following theorem.
Theorem 5. For any integer .
4. Conclusion
In this paper, we determine the cyclic vertex-connectivity of the -star network . We can consider the cyclic vertex-connectivity of other graphs and the cyclic edge-connectivity of the -star network in our future research.
Data Availability
No data were used in this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by the Natural Science Foundation of Qinghai Province, China (nos. 2021-ZJ-703 and 2019-ZJ-7012).