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## Graph Invariants and Their Applications

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Research Article | Open Access

Volume 2021 |Article ID 5570761 | https://doi.org/10.1155/2021/5570761

Yalan Li, Shumin Zhang, Chengfu Ye, "On Cyclic-Vertex Connectivity of -Star Graphs", Mathematical Problems in Engineering, vol. 2021, Article ID 5570761, 4 pages, 2021. https://doi.org/10.1155/2021/5570761

# On Cyclic-Vertex Connectivity of -Star Graphs

Revised31 Mar 2021
Accepted03 Apr 2021
Published13 Apr 2021

#### 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  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  and the graph colouring [2, 3]. It is used in many classic fields, such as integer flow conjectures  and n-extendable graphs [5, 6]. In many works, the cyclic vertex-connectivity has been studied. Cheng et al.  studied the cyclic vertex-connectivity of Cayley graphs generated by transposition trees. Yu et al.  obtained the cyclic vertex-connectivity of star graphs. For more research studies on the cyclic vertex-connectivity, see [7, 911] 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 ). 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 ). 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 ). is a -regular -connected graph.

Theorem 1 (see ). 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 ). If a cycle has a length at least 6 in an , then it contains one -edge, .

Theorem 2 (see ). 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).

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