The stability of a communication network, composed of processing nodes and communication links, is of prime importance to network designers. Many graph theoretical parameters have been used in measuring the vulnerability of networks, such as connectivity  , scattering number , tenacity , toughness   and the rupture degree .
A hypergraph is a generalization of a graph in which an edge can join any number of vertices. For a hypergraph , is a set of elements called vertices, and is a set of nonempty subset of V called edges. If for , we call H a k-uniform hypergraph. Clearly, ordinary graphs are referred as 2-uniform hypergraphs. A vertex is said to be incident to an edge if . Two vertices are said to be adjacent if they are contained in one edge. And two edges are said to be adjacent if their intersection is not empty.
A k-uniform linear hypergraph is a k-uniform hypergraph such that every two edges have at most one common vertex. In this paper, all k-uniform hypergraphs are referred as k-uniform linear hypergraph that each edge has at most two non-1-degree vertices. For a vertex , its degree is the number of edges that contain .
For an ordinary graph and an integer , the k-uniform hypergraph underlies graph G, denoted by , whose vertex set
and edge set
Hypergraphs are often used as a model of communication networks, paper index data, etc. The vulnerability of hypergraph is an important topic in graph theory research. Here we employ the rupture degree to measure the vulnerability of hypergraph . The rupture degree is similarly defined as
where is a cut set (or destruction strategy) of , and denote the number of components and the order of a largest component in , respectively. A vertex cut set (or a destruction strategy) X of hypergraph is called r-set (or the optimal destruction strategy) if .
Although this vulnerability parameter of is similar in form as the rupture degree of graph G, the optimal destruction strategy is different between G and . For a tree T and hypergraph in Figure 1, the cut sets and are the optimal destruction strategy of T and , respectively.
Figure 1. The optimal destruction strategies for trees T and T4.
In this paper, we determine the rupture degree for some k-uniform hypergraphs such as k-uniform hypercomet , k-uniform hypercycles , k-uniform hypercomplete and complete t-partite k-uniform hypergraph .
Throughout this paper, we use for the largest integer not large than x and for the smallest integer not smaller than x. A comet is a graph obtained by identifying one end of the path with the center of the star and the center is called the center of the comet. Any undefined terminology and notations can be found in  .
2. The Rupture Degree of Some k-Uniform Linear Hypergraphs
In this section, we will determine the rupture degree of some k-uniform hypergraphs such as k-uniform hypercomet , k-uniform hypercycles , k-uniform hyper complete and complete t-partite k-uniform hypergraph .
Lemma 2.1. Let X be an r-set of k-uniform hypergraph . Then .
If not, assume that , then . Suppose for are connected components of satisfying . Now select and let for . Clearly, and . Thus we get
This contradicts to the choice of X. So .n
Theorem 2.2. Let be k-uniform hypergraph that underlies comet graph . Then
Proof. We distinguish two cases by the parity of t to complete the proof.
Case 1 t is even.
Suppose X be an r-set of , by Lemma 2.1, we have . Thus . Now we discuss the value of by .
If , then . If , then , this contradicts to the choice of X. If , then we get
Based on the above argument and the definition of the rupture degree, we have
Case 2. t is odd.
Similar to case 1, we directly get
Let in Theorem 2.2, we get the rupture degree of k-uniform hyperpath .
Corollary 2.3. .
Let in Theorem 2.2, we get the rupture degree of k-uniform hyperstar .
Corollary 2.4. .
Let in Theorem 2.2, we get the rupture degree of comet , which is shown in .
Theorem 2.6. Let be k-uniform hypergraph that underlies complete t-partite graph . Then
Proof. Suppose are the partite sets of with for . Without loss of generality, let , then is a vertex cut of . And then , and . Thus we get
Therefore, we have
On the other hand, based on the fact that for any vertex cut of , there exists a partite set such that , we have
Thus, by the definition of rupture degree we get
So we get
Let and in Theorem 2.6, we get the rupture degree of and for , respectively.
Corollary 2.7. .
Corollary 2.8. .
Theorem 2.9 Let be k-uniform hypergraph underlies cycle . Then
Proof. Let X be an r-set of , by Lemma 2.1, we know that . Note that for any we have . So is an r-set of . Combine this with the fact , we get
Let in Theorem 2.9 get the rupture degree of , which is obtained in .
Theorem 2.11. Let be a k-uniform hypergraph of complete graph . Then
Proof. Let X be a cut set of and suppose . It is not difficult to get and . Thus we have
Let with . Note that , the function is a monotonically increasing in interval . So the maximum value of in interval is . Thus by the definition of rupture degree we get .
The author would like to thank anonymous reviewers for their valuable comments and suggests to improve the quality of the article. This work was supported by QHFRP 2020-ZJ-920 and the National Science Foundation of China (No. 11761056).