The structure fault tolerance of burnt pancake networks

Huifen Ge; Chengfu Ye; Shumin Zhang

doi:10.1515/math-2023-0154

Article Open Access

The structure fault tolerance of burnt pancake networks

Huifen Ge , Chengfu Ye and Shumin Zhang

Published/Copyright: December 31, 2023

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$Open Mathematics$

From the journal Open Mathematics Volume 21 Issue 1

Abstract

One of the symbolic parameters to measure the fault tolerance of a network is its connectivity. The H -structure connectivity and H -substructure connectivity extend the classical connectivity and are more practical. For a graph G and its connected subgraph H , the H -structure connectivity κ ( G ; H ) (resp. H -substructure connectivity κ s ( G ; H ) ) of G is the cardinality of a minimum subgraph set such that every element of the set is isomorphic to H (resp. every element of the set is isomorphic to a connected subgraph of H ) in G , whose vertices removal disconnects G . In this article, we investigate the H -structure connectivity and H -substructure connectivity of the n -dimensional burnt pancake network BP n for each H ∈ { K 1 , K 1 , 1 , … , K 1 , n − 1 , P 4 , … , P 7 , C 8 } .

Keywords: interconnection networks; H-structure connectivity; H-substructure connectivity; burnt pancake networks

MSC 2010: 05C05; 05C25; 05C38; 05C40

1 Introduction

An interconnection network is often represented as a graph, in which a vertex corresponds to a processor, and an edge corresponds to a communication link. One fundamental consideration in the design of networks is fault tolerance, which can be measured by the connectivity of graphs. In general, if the connectivity is larger, then its fault tolerance is higher. For a connected graph G , its connectivity κ ( G ) is defined as the minimum cardinality of a vertex subset whose removal makes the remaining graph disconnected. In recent years, the conditional connectivity [1] and the restricted connectivity [2,3] were introduced in succession for more accurate assessment of the fault tolerance of an interconnection network.

However, the connectivity parameters mentioned above are still disadvantageous because they just take into account the influence of a private vertex failure on the networks rather than the influence of a vertex or the vertices around it. In reality, one failing vertex is bound to have some adverse effects on the surrounding vertices. Furthermore, stimulated by the current situation that networks and subnetworks of large scale are increasingly made into chips, people think that it is becoming more and more feasible to consider the fault situation of a structure. Lin et al. [4] came up with the structure connectivity and substructure connectivity, whose proposition perfectly accommodates this disadvantageous realistic environment. Instead of focusing on the effects of a single vertex failure, they started to pay attention to the influence caused by some structure failures.

Due to its advantages, there have been a number of results about the structure connectivity and substructure connectivity on some well-known networks, such as hypercube Q n [4,5], folded hypercube F Q n [6], balanced hypercube BH n [7], k -ary n -cube network Q n , k [8,9], twisted hypercube H n [10], crossed cube CQ n [11], bubble-sort star graph B S n [12], star graph S n [13], ( n , k ) -star graph S n , k [14], alternating group graph AG n [15,16], wheel network CW n [17], circulant graph Cir ( n , Ω ) [18], and divide-and-swap cube DSC n [19].

This article determines the H -structure connectivity and H -substructure connectivity of the n -dimensional burnt pancake network BP n , where H ∈ { K 1 , K 1 , r , P 4 , … , P 7 , C 8 } and 1 ≤ r ≤ n − 1 . For detailed results, see Table 1.

Table 1

The H -(sub)structure connectivity of BP n

H	K 1	K 1 , r	P 4	P 5	P 6	P 7	C 8
κ ( BP n ; H )	n	n	n	n	n ‒ 1	⌈ n 2 ⌉	∕
κ s ( BP n ; H )	n	n	n	n	n ‒ 1	⌈ n 2 ⌉	⌈ n 2 ⌉

The remaining of the article is organized as follows: Section 2 presents the basic notations and definitions and introduces burnt pancake networks and their relevant structural properties; Section 3–5 dedicate the H -structure connectivity and H -substructure connectivity of BP n such that H are K 1 , r ( 1 ≤ r ≤ n − 1 ) , P ℓ ( 4 ≤ ℓ ≤ 7 ) , and C 8 , respectively; Section 6 concludes the article.

2 Preliminaries

We simply describe some terminologies and notations of graph theory, give the definitions of the structure connectivity and substructure connectivity, and provide the topological structure and properties of BP n in this section.

2.1 Terminologies and notations

For notation and terminology not mentioned here, the reader can refer to the study by Bondy and Murty [20]. Given two graphs G and H , if V ( H ) ⊆ V ( G ) and E ( H ) ⊆ E ( G ) , then H is a subgraph of G , denoted by H ⊆ G ; if they have identical structure, then H is isomorphic to G , denoted by H ≅ G ; if H is isomorphic to a connected subgraph of G , denoted by H ≼ G . For S ⊆ V ( G ) , G [ S ] is the induced subgraph by S ; N G ( S ) = ⋃ v ∈ S N G ( v ) \ S and N G [ S ] = N G ( S ) ∪ S , where N G ( v ) denotes the neighbors of v in G (the subscript G can be omitted without ambiguity); G − S is the graph, which is obtained by removing all vertices of S and the edges incident with vertices of S in G . Note that K 1 , r is a star with r pendant vertices and P ℓ is a path with ℓ vertices. For any two subgraphs G 1 and G 2 in G , E ( G 1 , G 2 ) is the set of edges joining one vertex in G 1 to another vertex in G 2 .

Here are the definitions of the H -structure connectivity and H -substructure connectivity of G , where G and H are two connected graphs and V ( ℋ ) = ∪ H ∈ ℋ V ( H ) .

Definition 1

[4] Let H ⊆ G and ℋ = { H 1 , H 2 , … , H t } be a set of connected subgraph of G such that every H i ≅ H . If G − V ( ℋ ) is disconnected, then ℋ is an H-structure cut. The H-structure connectivity of G is defined as follows:

κ ( G ; H ) = min { ∣ ℋ ∣ : ℋ is an H -structure cut } .

Definition 2

[4] Let H ⊆ G and ℋ = { H 1 , H 2 , … , H t } be a set of connected subgraph of G such that every H i ≼ H . If G − V ( ℋ ) is disconnected, then ℋ is an H-substructure cut. The H-substructure connectivity of G is defined as follows:

κ s ( G ; H ) = min { ∣ ℋ ∣ : ℋ is an H -substructure cut } .

As a matter of fact, the structure connectivity and substructure connectivity are the natural generalizations of the classical connectivity with κ ( G ; K 1 ) = κ s ( G ; K 1 ) = κ ( G ) in this sense. It is obvious to obtain the following results from the definitions above.

Observation 2.1

κ s ( G ; H ) ≤ κ ( G ; H ) and κ s ( G ; H ) ≤ κ s ( G ; H ′ ) for H ′ ≼ H .

2.2 Burnt pancake networks

Burnt pancake networks originate from the Burnt Pancake Problem discussed in [21–24]. Incidentally, one of the authors of [22] is Microsoft co-founder Bill Gates. In the Burnt Pancake Problem, one is tasked with sorting a stack of burnt pancakes in the proper order and orientation. Similarly to pancake networks, burnt pancake networks are also Cayley graphs [25,26].

Given a positive integer n , let [ n ] and [ ± n ] be the set { 1 , 2 , … , n } and { − n , − ( n − 1 ) , … , − 1 } ∪ [ n ] , respectively. To simplify notation, it is usual to use i ¯ instead of − i , and we also replace i ¯ j ¯ with i j ¯ . A signed permutation on [ ± n ] is an n -permutation x 1 x 2 ⋯ x n of [ ± n ] such that after taking the absolute value of each element x i can make up a permutation of [ n ] , i.e., ∣ x 1 ∣ ∣ x 2 ∣ ⋯ ∣ x n ∣ forms a permutation of [ n ] . For example, all of the signed permutations on [ ± 2 ] are { 12 , 1 2 ¯ , 1 ¯ 2 , 12 ¯ , 21 , 2 1 ¯ , 2 ¯ 1 , 21 ¯ } .

Definition 3

[27] Let BP n be an n-dimensional burnt pancake network, whose vertex set and edge set are defined as follows:

V ( BP n ) = { all the signed permutations of [ ± n ] } .
E ( BP n ) = { ( u , u i ) : u = x 1 x 2 ⋯ x i ⋯ x n , u i = x i x i − 1 ⋯ x 2 x 1 ¯ x i + 1 ⋯ x n and i ∈ [ n ] } .

Moreover, u i is the unique i -neighbor of u for i ∈ [ n ] , u i is an in-neighbor of u if i ∈ [ n − 1 ] and u i is an out-neighbor of u if i = n . We label the edge ( u , u i ) (for short u u i ) by i , and the edge u u i is called an i-edge.

The n -dimensional burnt pancake networks for n = 1 , 2 , and 3 are shown in Figure 1. It is obvious that ∣ V ( BP n ) ∣ = 2 n × n ! and ∣ E ( BP n ) ∣ = n × 2 n − 1 × n ! . We denote a cycle or path by the edges it traverses. For example, an ( a , b , c , d , e , f , g , h ) -cycle traverses an a -edge, followed by a b -edge, and successively until the last edge traversed is an h -edge. In fact, BP 2 is a ( 1 , 2 , 1 , 2 , 1 , 2 , 1 , 2 ) -cycle.

$Figure 1 Burnt pancake networks BP 1 , BP 2 {{\rm{BP}}}_{1},{{\rm{BP}}}_{2} , and BP 3 {{\rm{BP}}}_{3} .$

Figure 1

Burnt pancake networks BP 1 , BP 2 , and BP 3 .

Lemma 2.2

Let BP n be an n-dimensional burnt pancake network, i , j ∈ [ ± n ] , and i ≠ j . Then, the following results hold:

[27, Theorem 3] BP n is n-regular and κ ( BP n ) = λ ( BP n ) = n .
[28, Lemma 2] BP n can be decomposed into 2 n vertex-disjoint subgraphs BP n i , which are induced by all of those signed permutations whose last position is i, where i ∈ [ ± n ] . Clearly, BP n i ≅ BP n − 1 , and every edge (if exists) between BP n i and BP n j is labeled by n .
[27, Lemma 1] For any two distinct subgraphs BP n i and BP n j , the number of edges between them are
∣ E ( BP n i , BP n j ) ∣ = ( n − 2 ) ! × 2 n − 2 , i f i ≠ j ¯ , 0 , o t h e r w i s e .

Lemma 2.3

Let n ≥ 2 be an integer, i , j ∈ [ ± n ] , and i ≠ j .

[29, Theorem 10] The girth (the length of a shortest cycle) of BP n equals to 8 for n ≥ 2 .
[25, Theorem 4.1] An eight-cycle in BP n is one of the following forms:
- ( k , j , i , j , k , k − j + i , i , k − j + i ) -cycle for 1 ≤ i < j ≤ k − 1 and 3 ≤ k ≤ n ;
- ( k , j , k , i , k , j , k , i ) -cycle for 2 ≤ i , j ≤ k − 2 , i + j ≤ k , and 4 ≤ k ≤ n ;
- ( k , i , k , 1 , k , i , k , 1 ) -cycle for 2 ≤ i ≤ k − 1 and 3 ≤ k ≤ n ;
- ( k , 1 , k , 1 , k , 1 , k , 1 ) -cycle for 2 ≤ k ≤ n .

Proposition 2.4

Let i , j ∈ [ ± n ] , i ≠ j , and u ∈ V ( BP n i ) .

Let U = N BP n i [ u ] . Then, the induced subgraph by U in BP n is isomorphic to K 1 , n − 1 . Furthermore, if the out-neighbors of vertices in U are denoted by U n , then they lie in different copies.
Each subgraph that is isomorphic to one of P 4 , P 5 , P 6 , and C 8 in BP n i is incident with at most two edges between BP n i and BP n j in BP n for all j ≠ i .

Proof

(1) Let u = x 1 x 2 ⋯ x n be a vertex of BP n i and U = N BP n i [ u ] . Then, the induced subgraph by U in BP n is isomorphic to K 1 , n − 1 because of

(1) U = { u , u 1 , u 2 , … , u n − 1 } = { x 1 x 2 ⋯ x n , x 1 ¯ x 2 ⋯ x n , x 2 x 1 ¯ ⋯ x n , … , x n − 1 ⋯ x 2 x 1 ¯ x n } . U n = { x n x n − 1 ⋯ x 1 ¯ , x n x n − 1 ⋯ x 2 ¯ x 1 , x n x n − 1 ⋯ x 3 ¯ x 1 x 2 , … , x n ¯ x 1 x 2 ⋯ x n − 1 } .

These vertices of U n belong to BP n x 1 ¯ , BP n x 1 , BP n x 2 , … , BP n x n − 1 , respectively. This implies that a subgraph isomorphic to K 1 , n − 1 in BP n i is incident with at most one edge between BP n i and BP n j in BP n .

(2) Let F be a subgraph in BP n i . If F is isomorphic to one of P 4 , P 5 , and P 6 , there are two subgraphs F 1 and F 2 such that F 1 ≼ K 1 , 2 , F 2 ≼ K 1 , 2 , and V ( F 1 ) ∪ V ( F 2 ) = V ( F ) . By equation (1), every F i for i ∈ { 1 , 2 } is incident with at most one edge between BP n i and BP n j . Therefore, F is incident with at most two edges between BP n i and BP n j . Suppose that F ≅ C 8 . If F is incident with three edges between BP n i and BP n j , then there exists a subgraph isomorphic to K 1 , 2 in F , and it is incident with two edges between BP n i and BP n j , a contradiction.□

Lemma 2.5

[26, Theorems 3.5, 3.7 and 3.9] κ 0 ( BP n ) = n , κ 1 ( BP n ) = 2 n − 2 , and κ 2 ( BP n ) = 3 n − 4 for n ≥ 4 , where κ h ( H ) , the h-extra connectivity of H, is defined the minimum number of vertices whose deletion yields the resulting graph disconnected and each remaining component has more than h vertices.

3 κ ( BP n ; K 1 , r ) and κ s ( BP n ; K 1 , r )

In light of the definition of structure connectivity, substructure connectivity, and κ ( BP n ) = n , the following result is immediate.

Theorem 3.1

κ ( BP n ; K 1 ) = κ s ( BP n ; K 1 ) = n .

Next, we determine K 1 , r -structure connectivity and K 1 , r -substructure connectivity of BP n for 1 ≤ r ≤ n − 1 by establishing the upper and lower bounds, respectively.

Lemma 3.2

κ ( BP n ; K 1 , r ) ≤ n for n ≥ 2 and 1 ≤ r ≤ n − 1 .

Proof

Let u ∈ V ( BP n ) and U i be a set of r vertices in N ( u i ) \ { u } for i = 1 , 2 , … , n . Using H i to represent the induced subgraph by U i ∪ { u i } in BP n , we have H i ≅ K 1 , r . Let ℋ = { H 1 , H 2 , … , H n } . Then, BP n − V ( ℋ ) is disconnected, and u becomes an isolated vertex. Therefore, κ ( BP n , K 1 , r ) ≤ ∣ ℋ ∣ = n .□

For convenience, we define some notations throughout the article.

ℋ = { H 1 , H 2 , … , H t } is a set of connected subgraphs of BP n ;
ℋ i = ℋ ∩ BP n i = { H 1 ∩ BP n i , H 2 ∩ BP n i , … , H t ∩ BP n i } for i ∈ [ ± n ] ;
BP n a 1 , BP n a 2 , … , BP n a 2 n are 2 n vertex-disjoint sub-burnt pancake networks of BP n such that ∣ ℋ a 1 ∣ ≥ ∣ ℋ a 2 ∣ ≥ ⋯ ≥ ∣ ℋ a 2 n ∣ ;
I = { i : ℋ ∩ BP n a i ≠ ∅ } and I 0 = { i : ℋ ∩ BP n a i = ∅ } ;
BP n I = BP n [ ⋃ i ∈ I V ( BP n a i ) ] and BP n I 0 = BP n [ ⋃ i ∈ I 0 V ( BP n a i ) ] ;
∣ E ( i , j ) ∣ = ∣ E ( BP n a i , BP n a j ) ∣ .

Lemma 3.3

κ s ( BP n ; K 1 , n − 1 ) ≥ n for n ≥ 2 .

Proof

Let ℋ and ℋ a i be defined as above. We will show that BP n − V ( ℋ ) is connected if ∣ ℋ ∣ ≤ n − 1 and every element of ℋ is isomorphic to a subgraph of K 1 , n − 1 by induction on n . For n = 2 , as BP 2 ≅ C 8 , it is clear that BP 2 − V ( K 1 , 1 ) and BP 2 − V ( K 1 ) are still connected. For 2 ≤ ℓ ≤ n − 1 , suppose that the statement holds for BP ℓ .

Since ∣ ℋ ∣ ≤ n − 1 and every element of ℋ distributes in at most two sub-burnt pancake networks, we have ∣ ℋ i ∣ ≤ n − 1 , ∑ j = 1 2 n ∣ ℋ a j ∣ ≤ 2 n − 2 , ∣ ℋ a 2 n − 1 ∣ = ∣ ℋ a 2 n ∣ = 0 , and ∣ ℋ a 2 n − 2 ∣ ≤ 1 .

Case 1 ∣ ℋ a 2 n − 2 ∣ = 1 .

In this case, ∣ ℋ a 1 ∣ = ∣ ℋ a 2 ∣ = ⋯ = ∣ ℋ a 2 n − 3 ∣ = 1 and every element of ℋ distributes in exactly two distinct sub-burnt pancake networks. It implies that every H i of ℋ contains exactly one n -edge. Notice that ∣ ℋ a k ∣ ≤ 1 , it is easy to see that BP n a k − V ( ℋ a k ) is connected for k ∈ [ 2 n ] by the induction hypothesis for n ≥ 3 . Without loss of generality, let ℋ a j ≼ K 1 , n − 2 , ℋ a j + 1 ≼ K 1 , and V ( H i ) = V ( ℋ a j ) ∪ V ( ℋ a j + 1 ) for j = 2 i − 1 and i ∈ [ n − 1 ] .

On the one hand, let G i = BP n [ V ( BP n a j ∪ BP n a j + 1 ) ] − V ( H i ) for i ∈ [ n − 1 ] . Then, G i is connected because every H i contains just one edge between BP n a j and BP n a j + 1 and ∣ E ( j , j + 1 ) ∣ = ( n − 2 ) ! × 2 n − 2 > 1 for n ≥ 3 . On the other hand, since every H i is incident with at most one edge between G i and BP n a 2 n − 1 (resp. BP n a 2 n ), there is at least one edge connecting G i and BP n a 2 n − 1 (resp. BP n a 2 n ). Therefore, BP n − V ( ℋ ) is connected.

Case 2 ∣ ℋ a 2 n − 2 ∣ = 0 .

In this case, { 2 n − 2 , 2 n − 1 , 2 n } ⊆ I 0 , ∣ I 0 ∣ ≥ 3 , and BP n I 0 is connected. We consider the following cases.

Case 2.1 ∣ ℋ a 1 ∣ ≤ n − 2 .

By the induction hypothesis, BP n a j − V ( ℋ a j ) is connected for j ∈ [ 2 n ] .

In light of Lemma 2.2(3) and Proposition 2.4(1), ∣ E ( i , i 0 ) ∣ = ( n − 2 ) ! × 2 n − 2 for a i ≠ a i 0 ¯ , and every element of ℋ a i ∪ ℋ a i 0 is incident with at most one edge between BP n a i and BP n a i 0 for i ∈ [ 2 n − 3 ] , i 0 ∈ I 0 . Clearly, ( n − 2 ) ! × 2 n − 2 > n − 2 for n ≥ 3 , and thus, there is at least one edge connecting BP n a i and BP n I 0 . Therefore, BP n − V ( ℋ ) is connected.

Case 2.2 ∣ ℋ a 1 ∣ = n − 1 and ∣ ℋ a j ∣ ≤ n − 2 for 2 ≤ j ≤ 2 n .

By the pigeonhole principle, we have ∣ ℋ a n ∣ ≤ 1 and ∣ ℋ a n + 1 ∣ = 0 .

Case 2.2.1 ∣ ℋ a n ∣ = 1 .

It is clear that BP n − V ( BP n a 1 ) − V ( ℋ ) is connected by Case 2.1. Since ∣ ℋ a n ∣ = 1 , ∣ ℋ a 2 ∣ = ∣ ℋ a 3 ∣ = ⋯ = ∣ ℋ a n − 1 ∣ = 1 and every element H i of ℋ distributes in exactly two distinct sub-burnt pancakes BP n a 1 and BP n a i for i = 2 , 3 , … , n .

Let D be a connected component of BP n a 1 − V ( ℋ a 1 ) , and so N BP n a 1 ( D ) ⊆ V ( ℋ a 1 ) .

Suppose that ∣ V ( D ) ∣ = 1 . Let D = { w } , as ∣ N BP n ( w ) ∣ = n and ∣ N BP n ( w ) ∩ V ( H i ) ∣ ≤ 1 for H i ∈ ℋ , then w n ∉ V ( ℋ ) . Now, we set ∣ V ( D ) ∣ ≥ 2 . Let u v ∈ E ( D ) such that N BP n a 1 ( v ) ∩ V ( ℋ a 1 ) ≠ ∅ . Then, v n ∉ V ( ℋ ) . Otherwise, there exists an i ∈ { 2 , 3 , … , n } such that ℋ a i ≼ K 1 , n − 2 and ℋ a i is incident with two n -edges between BP n a 1 and BP n a i , contradicts Proposition 2.4(1). Therefore, BP n − V ( ℋ ) is connected.

Case 2.2.2 ∣ ℋ a n ∣ = 0 .

In light of the discussion of Case 2.1, BP n − V ( BP n a 1 ) − V ( ℋ ) is connected. It suffices to show that an arbitrary vertex u of BP n a 1 − V ( ℋ a 1 ) can connect to BP n − V ( BP n a 1 ) − V ( ℋ ) .

To the contrary, suppose that there exists a vertex u = x 1 x 2 ⋯ x n − 1 a 1 in BP n a 1 − V ( ℋ a 1 ) , which cannot connect to BP n − V ( BP n a 1 ) − V ( ℋ ) . If x 1 ∈ { a n ¯ , a n + 1 ¯ , … , a 2 n ¯ } , then u n ∈ { a 1 x n − 1 ⋯ x 2 ¯ a n , a 1 x n − 1 ⋯ x 2 ¯ a n + 1 , … , a 1 x n − 1 ⋯ x 2 ¯ a 2 n } , and these vertices belong to BP n a n , BP n a n + 1 , … , BP n a 2 n , respectively. Thus, u u n is a path connecting u and BP n − V ( BP n a 1 ) − V ( ℋ ) , a contradiction. It implies that x 1 ∉ { a n ¯ , a n + 1 ¯ , … , a 2 n ¯ } , and there exists at least a pair of a i and a j such that a i = a j ¯ for n ≤ i , j ≤ 2 n . Without loss of generality, let a n = a n + 1 ¯ = 2 . It means that x 1 ≠ 2 and x 1 ≠ 2 ¯ , but 2 or 2 ¯ must occur at the vertex u , assuming x 2 = 2 .

If u n ∉ V ( ℋ ) , then u u n is a path connecting u and BP n − V ( BP n a 1 ) − V ( ℋ ) ; if u 1 ∉ V ( ℋ ) and ( u 1 ) 2 ∉ V ( ℋ ) , then u u 1 ( u 1 ) 2 ( ( u 1 ) 2 ) n is a path connecting u and BP n − V ( BP n a 1 ) − V ( ℋ ) ; if u 2 ∉ V ( ℋ ) , then u u 2 ( u 2 ) n is a path connecting u and BP n − V ( BP n a 1 ) − V ( ℋ ) ; if there exists an integer i ∈ { 3 , 4 , … , n − 1 } such that u i ∉ V ( ℋ ) and ( u i ) i − 1 ∉ V ( ℋ ) , then u u i ( u i ) i − 1 ( ( u i ) i − 1 ) n is a path connecting u and BP n − V ( BP n a 1 ) − V ( ℋ ) (Figure 2). All of these cases get contradictions. Summing up above, we know that u n ∈ V ( ℋ ) , at least one of u 1 and ( u 1 ) 2 is contained in V ( ℋ ) , u 2 ∈ V ( ℋ ) , and at least one of u i , ( u i ) i − 1 is contained in V ( ℋ ) for i = 3 , 4 , … , n − 1 .

Choose v 1 ∈ { u 1 , ( u 1 ) 2 } ∩ V ( ℋ ) and v i ∈ { u i , ( u i ) i − 1 } ∩ V ( ℋ ) for i = 3 , 4 , … , n − 1 and denote v n = u n and v 2 = u 2 . Let U = { v 1 , v 2 , v 3 , … , v n } . Then, U ⊆ V ( ℋ ) .

If we can show that every element of ℋ contains at most one vertex in U , then ∣ ℋ ∣ ≥ ∣ U ∣ = n , which contradicts ∣ ℋ ∣ ≤ n − 1 . If not, assume that there are two distinct vertices v i , v j ∈ U , which lie in an element of ℋ . By the definition of v i , v j , there exist a v i - u path R i and a v j - u path R j whose lengths are at most 2. Furthermore, since u ∉ V ( ℋ ) and v i , v j lie in an element of ℋ , there is a v i - v j path R whose length is at most 2. Consequently, R i R j R forms a cycle whose length is at most 6, which contradicts the girth of BP n .

Hence, any vertex u of BP n a 1 − V ( ℋ a 1 ) connects to BP n − V ( BP n a 1 ) − V ( ℋ ) , i.e., BP n − V ( ℋ ) is connected.

Case 2.3 ∣ ℋ a 1 ∣ = ∣ ℋ a 2 ∣ = n − 1 and ∣ ℋ a j ∣ = 0 for 3 ≤ j ≤ 2 n .

Assume that u = x 1 x 2 ⋯ x n is a vertex in BP n a 1 − V ( ℋ a 1 ) ; if u n ∈ B P a 3 ∪ B P a 4 ∪ ⋯ ∪ B P a 2 n , then the result is true. Next, let u n ∈ B P a 2 , and it is obvious that ( u 1 ) n , ( u 2 ) n , … , ( u n − 1 ) n ∉ B P a 2 . If one of u 1 , u 2 , … , u n − 1 is not in V ( ℋ ) , then it is true. Assume that u 1 , u 2 , … , u n − 1 belong to V ( ℋ ) . If u n ∈ V ( ℋ ) , then ∣ ℋ ∣ ≥ n , a contradiction. Assume that u n ∉ V ( ℋ ) , similarly, if ( ( u n ) 1 ) n , ( ( u n ) 2 ) n , … , ( ( u n ) n − 1 ) n ∉ B P a 1 and one of them is not in V ( ℋ ) , then it is true. Now, ( u n ) 1 , ( u n ) 2 , … , ( u n ) n − 1 ∈ V ( ℋ ) and ∣ ℋ ∣ = 2 n − 2 , a contradiction.

Therefore, any vertex u of BP n a 1 − V ( ℋ a 1 ) connects to BP n − V ( BP n a 1 ) − V ( ℋ ) , i.e., BP n − V ( ℋ ) is connected.□

$Figure 2 Illustration of Case 2.2.2 of Lemma 3.3.$

Figure 2

Illustration of Case 2.2.2 of Lemma 3.3.

Combining Lemmas 3.2 and 3.3, we can obtain the following result.

Theorem 3.4

κ ( BP n ; K 1 , r ) = κ s ( BP n ; K 1 , r ) = n for n ≥ 2 and 1 ≤ r ≤ n − 1 .

4 κ ( BP n ; P ℓ ) and κ s ( BP n ; P ℓ )

In this section, we investigate the P ℓ -structure connectivity and P ℓ -substructure connectivity of BP n for 4 ≤ ℓ ≤ 7 .

Lemma 4.1

κ ( BP n ; P 4 ) ≤ n and κ ( BP n ; P 5 ) ≤ n for n ≥ 3 .

Proof

Let u ∈ V ( BP n ) and H i (resp. F i ) be the ( i , j , i , j ) -path (resp. ( i , j , i , j , i ) -path) that starts at u , where i = 1 , 2 , … , n and j ≡ ( i + 1 ) mod n . Clearly, H i ≅ P 4 (resp. F i ≅ P 5 ). Let ℋ = { H 1 , H 2 , … , H n } (resp. ℱ = { F 1 , F 2 , … , F n } ). Then, BP n − V ( ℋ ) (resp. BP n − V ( ℱ ) ) is disconnected and u is an isolated vertex. Therefore, κ ( BP n ; P 4 ) ≤ ∣ ℋ ∣ = n (resp. κ ( BP n ; P 5 ) ≤ ∣ ℱ ∣ = n ).□

Lemma 4.2

κ s ( BP 3 ; P 5 ) ≥ 3 .

Proof

Let ℋ = { H 1 , … , H t } and H i ≼ P 5 for i ∈ [ t ] . It is easy to check that BP 3 − V ( ℋ ) is connected when t = 0 or 1. Suppose that t = 2 . If H 1 , H 2 ≼ P 3 , then BP 3 − V ( ℋ ) is connected by Theorem 3.4. Suppose that at least one of H 1 and H 2 is isomorphic to P 4 or P 5 . Since there are at most four n -edges in ℋ , we have the following cases.

Suppose that ℋ contains no n -edges. When H 1 ∪ H 2 ⊆ BP 3 a 1 , clearly, BP 3 − V ( ℋ ) is connected. When H 1 ⊆ BP 3 a 1 and H 2 ⊆ BP 3 a 2 , BP 3 a 1 − V ( H 1 ) is connected and ∣ V ( BP 3 a 1 − V ( H 1 ) ) ∣ ≥ 3 , and there is at least one n -edge connecting BP 3 a 1 − V ( H 1 ) and BP 3 I 0 . Similarly, BP 3 a 2 − V ( H 2 ) can connect to BP 3 I 0 , where I 0 = { 3 , 4 , 5 , 6 } . Therefore, BP 3 − V ( ℋ ) is connected.

First, assume that ℋ contains one n -edge. When ∣ ℋ a 1 ∣ = 2 and ∣ ℋ a 2 ∣ = 1 , clearly, BP 3 − V ( BP 3 a 1 ) − V ( ℋ ) is connected and BP 3 a 1 − V ( ℋ a 1 ) contains at most two components D 1 and D 2 . It is easy to check that N BP 3 ( D j ) ∩ V ( BP 3 − V ( BP 3 a 1 ) − V ( ℋ ) ) ≠ ∅ for j = 1 , 2 . Therefore, BP 3 − V ( ℋ ) is connected. When ∣ ℋ a 1 ∣ = ∣ ℋ a 2 ∣ = ∣ ℋ a 3 ∣ = 1 , BP 3 a i − V ( ℋ a i ) is connected and ∣ V ( BP 3 a i − V ( ℋ a i ) ) ∣ ≥ 3 for i = 1 , 2 , 3 , and there is at least one n -edge connecting BP 3 a i − V ( ℋ a i ) and BP 3 I 0 for I 0 = { 4 , 5 , 6 } . Therefore, BP 3 − V ( ℋ ) is connected.

Second, assume that ℋ contains two n -edges. Then, H 1 and H 2 , respectively, contain one n -edge, or H 1 has exactly two n -edges. When ∣ ℋ a 1 ∣ = ∣ ℋ a 2 ∣ = ∣ ℋ a 3 ∣ = ∣ ℋ a 4 ∣ = 1 , every BP n a i − V ( ℋ a i ) is connected for 1 ≤ i ≤ 4 , and there is at least one n -edge connecting BP 3 a i − V ( ℋ a i ) and ⋃ i = 5 6 BP 3 a i . When ∣ ℋ a 1 ∣ = 2 , and ∣ ℋ a 2 ∣ = ∣ ℋ a 3 ∣ = 1 , BP 3 − V ( BP 3 a 1 ) − V ( ℋ ) is connected and every component D of BP n a 1 − V ( BP n a 1 ) satisfies that N BP 3 ( D ) ∩ V ( BP 3 − V ( BP 3 a 1 ) − V ( ℋ ) ) ≠ ∅ . When ∣ ℋ a 1 ∣ = ∣ ℋ a 2 ∣ = 2 , BP 3 − V ( BP 3 a 1 ) − V ( BP 3 a 2 ) is connected and every component D i of BP n a i − V ( BP n a i ) satisfies that N BP 3 ( D ) ∩ V ( BP 3 − V ( BP 3 a 1 ) − V ( BP 3 a 2 ) ) ≠ ∅ for i = 1 , 2 . Summing up above, BP 3 − V ( ℋ ) is connected.

Third, assume that ℋ contains three n -edges. Then, H 1 contains exactly two n -edges and H 2 contains one n -edge. It easy to check that BP 3 − V ( ℋ ) is connected from the above discussion.

Fourth, assume that ℋ contains four n -edges. Both H 1 and H 2 contain two n -edges. When ∣ ℋ a 1 ∣ = ⋯ = ∣ ℋ a 6 ∣ = 1 , every ℋ a i ≼ K 1 , 1 or K 1 and BP n a i − V ( ℋ a i ) is connected for i = 1 , … , 6 . Without loss of generality, let H 1 ⊆ BP 3 [ ⋃ i = 1 3 V ( BP 3 a i ) ] = G 1 and H 2 ⊆ BP 3 [ ⋃ i = 4 6 V ( BP 3 a i ) ] = G 2 , and we have that G 1 − V ( H 1 ) is connected since there are two n -edges between any pair of BP 3 a i and BP 3 a j for a i ≠ a j ¯ and that every element of ℋ a i ∪ ℋ a j is incident with exactly one edge between BP 3 a i and BP 3 a j for i , j ∈ { 1 , 2 , 3 } . Similarly, G 2 − V ( H 2 ) is connected. Since the number of deleted n -edges between G 1 and G 2 are at most 4 and ∣ E ( G 1 , G 2 ) ∣ ≥ 12 , there exist edges connecting G 1 − V ( H 1 ) and G 2 − V ( H 2 ) . Hence, BP 3 − V ( ℋ ) is connected. Similarly, when ∣ ℋ a 1 ∣ = ∣ ℋ a 2 ∣ = ∣ ℋ a 3 ∣ = 2 or ∣ ℋ a 1 ∣ = ∣ ℋ a 2 ∣ = 2 , ∣ ℋ a 3 ∣ = ∣ ℋ a 4 ∣ = 1 or ∣ ℋ a 1 ∣ = 2 , ∣ ℋ a 2 ∣ = ⋯ = ∣ ℋ a 5 ∣ = 1 , we also can obtain that BP 3 − V ( ℋ ) is connected.□

Lemma 4.3

κ s ( BP n ; P 5 ) ≥ n for n ≥ 3 .

Proof

Let ℋ = { H 1 , H 2 , … , H t } be a set of subgraph of P 5 in BP n such that t ≤ n − 1 , and we show that BP n − V ( ℋ ) is connected by induction on n . By Lemma 4.2, this statement holds for BP 3 . Now, assume that the statement holds for BP ℓ ( 3 ≤ ℓ ≤ n − 1 ).

Since every element of ℋ lies in at most three different copies of BP n a i for i ∈ [ 2 n ] and contains at most two n -edges, we have ∣ ℋ a i ∣ ≤ n − 1 and ∑ i = 1 2 n ∣ ℋ a i ∣ ≤ 3 n − 3 .

Case 1 ∣ ℋ a 1 ∣ ≤ n − 2 .

By the inductive hypothesis, each BP n a i − V ( ℋ a i ) is connected for i ∈ [ 2 n ] . In light of Lemma 2.2(3) and Proposition 2.4(2), when a i ≠ a j ¯ , we know that ∣ E ( i , j ) ∣ = ( n − 2 ) ! × 2 n − 2 and the number of deleted n -edges between BP n a i and BP n a j is at most 4 ( n − 2 ) . Since ( n − 2 ) ! × 2 n − 2 > 4 ( n − 2 ) for n ≥ 5 , there exists at least one edge connecting BP n a i − V ( ℋ a i ) and BP n a j − V ( ℋ a j ) . For n = 4 , ( n − 2 ) ! × 2 n − 2 = 8 . Let J = { j : ∣ ℋ a j ∣ = 2 } . Then, 2 ≤ ∣ J ∣ ≤ 3 and ℋ a i ∪ ℋ a j is incident with at most four n -edges for i , j ∈ J . Thus, there exists at least one edge connecting BP 4 a i − V ( ℋ a i ) and BP 4 a j − V ( ℋ a j ) . Therefore, BP n − V ( ℋ ) is connected.

Case 2 ∣ ℋ a 1 ∣ = n − 1 .

By the pigeonhole principle, ∣ ℋ a 2 ∣ ≤ n − 1 , ∣ ℋ a 3 ∣ ≤ n − 2 , ∣ ℋ a 2 n − 1 ∣ ≤ 1 , and ∣ ℋ a 2 n ∣ = 0 .

By the inductive hypothesis, each BP n a i − V ( ℋ a i ) is connected for ∣ ℋ a i ∣ ≤ n − 2 . By Lemma 2.2(3) and Proposition 2.4(2), ∣ E ( i , 2 n ) ∣ = ( n − 2 ) ! × 2 n − 2 and every element of ℋ a i ∪ ℋ a 2 n is incident with at most two n -edges between BP n a i and BP n a 2 n for ∣ ℋ a i ∣ ≤ n − 2 and a i ≠ a 2 n ¯ . Since ( n − 2 ) ! × 2 n − 2 > 2 ( n − 2 ) for n ≥ 4 , there is at least one edge connecting BP n a i − V ( ℋ a i ) and BP n a 2 n .

Let J = { j : ∣ ℋ a j ∣ = n − 1 } . Then, every element of ℋ is intersecting with BP n a j .

Case 2.1 There exists j ∈ J such that a j = a 2 n ¯ .

Clearly, BP n − V ( BP n J ) − V ( ℋ ) is connected. By symmetry, assume that a 1 = a 2 n ¯ . Next, we will show that an arbitrary vertex u of BP n a 1 − V ( ℋ a 1 ) can connect to BP n − V ( BP n J ) − V ( ℋ ) . Note that BP n a 2 n ⊆ BP n − V ( BP n J ) − V ( ℋ ) .

On the contrary, assume that u is a vertex that belongs to BP n a 1 − V ( ℋ a 1 ) and disconnects BP n − V ( BP n J ) − V ( ℋ ) . Without loss of generality, let u = x 1 x 2 … x n − 1 1 . Then, a 2 n = 1 ¯ . If there exists an integer i ∈ { 1 , 2 , … , n − 1 } such that u i , ( u i ) n , ( ( u i ) n ) 1 , ( ( ( u i ) n ) 1 ) n ∉ V ( ℋ ) , then u u i ( u i ) n ( ( u i ) n ) 1 ( ( ( u i ) n ) 1 ) n is a path connecting u and BP n a 2 n ; if u n , ( u n ) 1 , ( ( u n ) 1 ) n ∉ V ( ℋ ) , then u u n ( u n ) 1 ( ( u n ) 1 ) n is a path connecting u and BP n a 2 n (see the left of Figure 3). All of these cases get contradictions.

Summing up above, at least one of u i , ( u i ) n , ( ( u i ) n ) 1 , and ( ( ( u i ) n ) 1 ) n is contained in V ( ℋ ) for i = 1 , 2 , … , n − 1 and at least one of u n , ( u n ) 1 , and ( ( u n ) 1 ) n is contained in V ( ℋ ) . Choose v i ∈ { u i , ( u i ) n , ( ( u i ) n ) 1 , ( ( ( u i ) n ) 1 ) n } ∩ V ( ℋ ) for i = 1 , 2 , … , n − 1 and v n ∈ { u n , ( u n ) 1 , ( ( u n ) 1 ) n } ∩ V ( ℋ ) . Let U = { v 1 , v 2 , … , v n } . Then, U ⊆ V ( ℋ ) . If we can show that every element of ℋ contains at most one vertex in U , then ∣ ℋ ∣ ≥ ∣ U ∣ = n , which contradicts ∣ ℋ ∣ ≤ n − 1 .

If not, assume that there are two distinct vertices v i , v j ∈ U , which lie in an element of ℋ . By the definition of v i and v j , there exist a v i - u path R i and a v j - u path R j whose lengths are at most 3. Recall that u ∉ V ( ℋ ) and V ( BP n a 2 n ) ⊈ V ( ℋ ) . Since v i and v j lie in an element of ℋ , there is a v i - v j path R whose length is at most 4. Consequently, R i R j R forms a cycle C whose length is at most 10. When C is a C 8 , we can check that either the edge of R i R j does not match with a C 8 of BP n (Figure 4 shows four kinds of C 8 containing n -edges) or R ∩ V ( BP n a 1 ) = ∅ , a contradiction. When C is a C 9 or C 10 , R is an element of ℋ but R ∩ V ( BP n a 1 ) = ∅ , also a contradiction.

Therefore, any vertex u of BP n a 1 − V ( ℋ a 1 ) connects to BP n − V ( BP n J ) − V ( ℋ ) .

Case 2.2 a j ≠ a 2 n ¯ for every j ∈ J .

On the one hand, BP n a 2 n ¯ − V ( ℋ a 2 n ¯ ) can connect to any BP n a i − V ( ℋ a i ) for ∣ ℋ a i ∣ ≤ n − 2 when n ≥ 5 because of ( n − 2 ) ! × 2 n − 2 > 4 ( n − 2 ) . On the other hand, there exist i ∉ J and ∣ ℋ a i ∣ ≤ n − 3 such that BP n a i − V ( ℋ a i ) can connect to BP n a 2 n ¯ − V ( ℋ a 2 n ¯ ) when n = 4 . Hence, BP n − V ( BP n J ) − V ( ℋ ) is connected.

It suffices to show that an arbitrary vertex w of BP n a 1 − V ( ℋ a 1 ) can connect to BP n − V ( BP n J ) − V ( ℋ ) for a 1 ≠ a 2 n ¯ . Recall that BP n a 2 n ⊆ BP n − V ( BP n J ) − V ( ℋ ) .

On the contrary, assume that w ∈ BP n a 1 − V ( ℋ a 1 ) and w disconnects to BP n − V ( BP n J ) − V ( ℋ ) . Without loss of generality, let a 1 = 1 and w = x 1 x 2 … x n − 1 1 . Then, a 2 n ≠ 1 ¯ . Let a 2 n = 2 ¯ . If x 1 = 2 , then w w n is an edge connecting w and BP n a 2 n , a contradiction. It means that x 1 ≠ 2 , assuming x 1 = 2 ¯ .

If w 1 ∉ V ( ℋ ) , then w w 1 ( w 1 ) n is a path connecting w and BP n a 2 n ; if there exists an integer i ∈ { 2 , 3 , … , n − 1 } such that w i , ( w i ) n , ( ( w i ) n ) n + 1 − i , ( ( ( w i ) n ) n + 1 − i ) n ∉ V ( ℋ ) , then w w i ( w i ) n ( ( w i ) n ) n + 1 − i ( ( ( w i ) n ) n + 1 − i ) n is a path connecting w and BP n a 2 n ; if w n , ( w n ) 1 , ( ( w n ) 1 ) n , ( ( ( w n ) 1 ) n ) 1 ∉ V ( ℋ ) , then w w n ( w n ) 1 ( ( w n ) 1 ) n ( ( ( w n ) 1 ) n ) 1 ( ( ( ( w n ) 1 ) n ) 1 ) n is a path connecting w and BP n a 2 n (see the right of Figure 3). All of these cases get contradictions. Thus, w 1 ∈ V ( ℋ ) , at least one of w i , ( w i ) n , ( ( w i ) n ) n + 1 − i , ( ( ( w i ) n ) n + 1 − i ) n belongs to V ( ℋ ) for i = 2 , 3 , … , n − 1 , and at least one of w n , ( w n ) 1 , ( ( w n ) 1 ) n , ( ( ( w n ) 1 ) n ) 1 belongs to V ( ℋ ) .

Let v 1 = w 1 , and choose v i ∈ { w i , ( w i ) n , ( ( w i ) n ) n + 1 − i , ( ( ( w i ) n ) n + 1 − i ) n } ∩ V ( ℋ ) for i = 2 , 3 , … , n − 1 and v n ∈ { w n , ( w n ) 1 , ( ( w n ) 1 ) n , ( ( ( w n ) 1 ) n ) 1 } ∩ V ( ℋ ) . Let W = { v 1 , v 2 , … , v n } . Then, W ⊆ V ( ℋ ) . We will show that every element of ℋ contains at most one vertex in W .

If not, assume that there are two distinct vertices v i , v j ∈ U ( i > j ) , which lie in an element of ℋ . By the definition of v i and v j , there is a v i - w path Q i whose length is at most 4 and a v j - w path Q j whose length is at most 3. Recall that w ∉ V ( ℋ ) and V ( BP n a 2 n ) ⊈ V ( ℋ ) . Since v i and v j lie in an element of ℋ , there is a v i - v j path Q whose length is at most 4. Consequently, Q i Q j Q forms a cycle C whose length is at most 11. When C is a C 8 , we can check that either the edge of Q i Q j does not match with a C 8 of BP n or Q ∩ V ( BP n a 1 ) = ∅ , a contradiction. When C is a C 9 , C 10 , or C 11 , Q is an element of ℋ but Q ∩ V ( BP n a 1 ) = ∅ , also a contradiction.

Thus, every element of ℋ contains at most one vertex in U . Hence, ∣ ℋ ∣ ≥ ∣ U ∣ = n , a contradiction. Therefore, any vertex w of BP n a 1 − V ( ℋ a 1 ) connects to BP n − V ( BP n J ) − V ( ℋ ) .

By symmetry, any vertex of BP n a j − V ( ℋ a j ) for j ∈ J can connect to BP n − V ( BP n J ) − V ( ℋ ) . Therefore, BP n − V ( ℋ ) is connected.□

$Figure 3 Illustration of Case 2 of Lemma 4.3.$

Figure 3

Illustration of Case 2 of Lemma 4.3.

$Figure 4 Four kinds C 8 {C}_{8} with n n -edge in BP n {{\rm{BP}}}_{n} .$

Figure 4

Four kinds C 8 with n -edge in BP n .

Theorem 4.4

κ ( BP n ; P 5 ) = κ s ( BP n ; P 5 ) = κ s ( BP n ; P 4 ) = κ ( BP n ; P 4 ) = n for n ≥ 3 .

Proof

By Observation 2.1 and Lemmas 4.1 and 4.3, κ ( BP n ; P 5 ) = n = κ s ( BP n ; P 5 ) ≤ κ s ( BP n ; P 4 ) ≤ κ ( BP n ; P 4 ) = n .□

Lemma 4.5

κ ( BP n ; P 6 ) ≤ n − 1 for n ≥ 3 .

Proof

For any vertex u ∈ V ( BP n ) , let C i be ( i , 1 , i , 1 , i , 1 , i , 1 ) -cycle containing u and H i = C i − { u , u 1 } for i ∈ { 2 , 3 , … , n } . Denote ℋ = { H 2 , H 3 , … , H n } . It is easy to see that H i ≅ P 6 , N ( { u , u 1 } ) = ⋃ i = 2 n { u i , ( u 1 ) i } ⊆ V ( ℋ ) and u , u 1 ∉ V ( ℋ ) . Hence, BP n − V ( ℋ ) is disconnected, u u 1 is an isolated edge, and κ ( BP n , P 6 ) ≤ ∣ ℋ ∣ = n − 1 .□

Lemma 4.6

κ s ( BP n ; P 6 ) ≥ n − 1 for n ≥ 3 .

Proof

Let ℋ = { H 1 , … , H t } and H i ≼ P 6 for i ∈ [ t ] . For n = 3 , it suffices to show that BP 3 − V ( ℋ ) is connected with ∣ ℋ ∣ ≤ 1 . It is clear that BP 3 − V ( ℋ ) is connected if ∣ ℋ ∣ = 0 . Assume that ∣ ℋ ∣ = 1 and ℋ = { H 1 } , either H 1 is isomorphic to a connected subgraph of P 5 or there are H 1 , 1 and H 1 , 2 such that H 1 , 1 and H 1 , 2 are isomorphic to a connected subgraph of P 5 and V ( H 1 , 1 ) ∪ V ( H 1 , 2 ) = V ( H 1 ) . By Theorem 4.4, κ s ( BP 3 ; P 5 ) = 3 , then BP 3 − V ( ℋ ) is connected.

For n ≥ 4 , let ℋ be a minimum P 6 -substructure cut set of BP n . It suffices to show that ∣ ℋ ∣ ≥ n − 1 .

Case 1 There exists an isolated vertex u in BP n − V ( ℋ ) .

Since ∣ N ( u ) ∣ = n and ∣ N ( u ) ∩ H i ∣ ≤ 1 , we have ∣ ℋ ∣ ≥ n . Hence, ∣ ℋ ∣ > n − 1 .

Case 2 There is no isolated vertex in BP n − V ( ℋ ) .

Let D be a minimum connected component of BP n − V ( ℋ ) . Then, ∣ V ( D ) ∣ ≥ 2 .

By Lemma 2.5, if n ≥ 4 and ∣ V ( D ) ∣ = 2 , then ∣ N ( D ) ∣ = 2 n − 2 . Since ∣ N ( D ) ∩ H i ∣ ≤ 2 , we have ∣ ℋ ∣ ≥ n − 1 .

By Lemma 2.5, if n ≥ 4 and ∣ V ( D ) ∣ ≥ 3 , then ∣ N ( D ) ∣ ≥ 3 n − 4 , and N ( D ) contains at least 3 n − 4 mutually nonadjacent vertices. However, there are at most 3 n − 6 mutually nonadjacent vertices in V ( ℋ ) if ∣ ℋ ∣ ≤ n − 2 . Therefore, ∣ ℋ ∣ ≥ n − 1 .□

Applying Observation 2.1 and Lemmas 4.5 and 4.6, the following results are obtained immediately.

Theorem 4.7

κ ( BP n ; P 6 ) = κ s ( BP n ; P 6 ) = n − 1 for n ≥ 3 .

Lemma 4.8

κ ( BP n ; P 7 ) ≤ n 2 for n ≥ 3 .

Proof

For any u ∈ V ( BP n ) , let C k be ( k − 2 , k − 1 , k , k − 1 , k − 2 , k − 1 , k , k − 1 ) -cycle starting at u for k = 2 i + 1 and i = { 1 , 2 , … , n − 1 2 } .

If n is odd, let C n be ( n , 1 , n , 2 , n , 1 , n ) -path starting at u .

If n is even, let C n be ( n − 1 , 1 , n − 1 , n , 2 , 1 , 2 , n ) -cycle starting at u .

Let H k = C k − { u } for k = 2 i + 1 and i = { 1 , 2 , … , n 2 } . Denote ℋ = { H 1 , H 2 , … , H n 2 } . It is not hard to see that H k ≅ P 7 , N ( u ) = { u 1 , u 2 , … , u n } ⊆ V ( ℋ ) , and u ∉ V ( ℋ ) . Hence, BP n − V ( ℋ ) is disconnected, u is an isolated vertex, and κ ( BP n , P 7 ) ≤ ∣ ℋ ∣ = n 2 .□

5 κ ( BP n ; C 8 ) and κ s ( BP n ; C 8 )

We study the C 8 -substructure connectivity and C 8 -structure connectivity of BP n as follows.

Lemma 5.1

κ s ( BP n ; C 8 ) ≥ ⌈ n 2 ⌉ for n ≥ 3 .

Proof

It suffices to show that BP n − V ( ℋ ) is connected for ℋ = { H 1 , … , H s , H s + 1 , … , H t } with t ≤ ⌈ n 2 ⌉ − 1 and every H i ≼ C 8 . Suppose that every H i ( s + 1 ≤ i ≤ t ) is just isomorphic to a subgraph of P 4 , then every H i ( 1 ≤ i ≤ s ) can be divided into two subgraphs of P 4 , set V ( H i ) = V ( H i , 1 ) ∪ V ( H i , 2 ) . Let ℋ ′ = { H 1 , 1 , H 1 , 2 , … , H s , 1 , H s , 2 , H s + 1 , … , H t } . Then, V ( ℋ ) = V ( ℋ ′ ) . Moreover, every element of ℋ ′ is isomorphic to a subgraph of P 4 and ∣ ℋ ′ ∣ = 2 s + ( t − s ) = t + s ≤ 2 t ≤ 2 ( ⌈ n 2 ⌉ − 1 ) ≤ n − 1 . But κ s ( BP n ; P 4 ) = n by Theorem 4.4, hence, BP n − V ( ℋ ) is connected.□

By Observation 2.1 and Lemmas 4.8 and 5.1, we obtain n 2 ≤ κ s ( BP n ; C 8 ) ≤ κ s ( BP n ; P 7 ) ≤ κ ( BP n ; P 7 ) ≤ n 2 . Then, the following theorem is obvious.

Theorem 5.2

κ s ( BP n ; C 8 ) = κ s ( BP n ; P 7 ) = κ ( BP n ; P 7 ) = n 2 for n ≥ 3 .

Lemma 5.3

κ ( BP n ; C 8 ) ≤ n for n ≥ 3 .

Proof

For u ∈ V ( BP n ) , let H i be ( 1 , 2 , 1 , 2 , 1 , 2 , 1 , 2 ) -cycle going through u i for i = 3 , … , n . In particular, let H 1 be the ( 2 , 3 , 1 , 3 , 2 , 3 , 1 , 3 ) -cycle going through u 1 and H 2 be the ( 1 , 3 , 1 , 3 , 1 , 3 , 1 , 3 ) -cycle going through u 2 . Denote ℋ = { H 1 , H 2 , … , H n } . Clearly, H i ≅ C 8 , N ( u ) = { u 1 , u 2 , … , u n } ⊆ V ( ℋ ) , and u ∉ V ( ℋ ) . Hence, BP n − V ( ℋ ) is disconnected, u is an isolated vertex, and κ ( BP n , C 8 ) ≤ ∣ ℋ ∣ = n . (Figure 5 shows a C 8 -structure cut in BP 4 , where u = 1243 .)□

$Figure 5 A C 8 {C}_{8} -structure cut in BP 4 {{\rm{BP}}}_{4} .$

Figure 5

A C 8 -structure cut in BP 4 .

Combining Observation 2.1, Theorem 5.2, and Lemma 5.3, we have the following result.

Theorem 5.4

n 2 ≤ κ ( BP n ; C 8 ) ≤ n for n ≥ 3 .

6 Conclusion

In this article, we investigate the H -structure and H -substructure connectivity of the n -dimensional burnt pancake network BP n when H is isomorphic to K 1 , t ( 1 ≤ t ≤ n − 1 ) , P ℓ ( 4 ≤ ℓ ≤ 7 ) , and C 8 . More details,

for H = K 1 , t ( 1 ≤ t ≤ n − 1 ) , κ s ( BP n ; K 1 , t ) = κ ( BP n ; K 1 , t ) = n ;
for H = P ℓ ( 4 ≤ ℓ ≤ 7 ) , κ s ( BP n ; P ℓ ) = κ ( BP n ; P ℓ ) = n when ℓ = 4 , 5 , κ s ( BP n ; P 6 ) = κ ( BP n ; P 6 ) = n − 1 and κ s ( BP n ; P 7 ) = κ ( BP n ; P 7 ) = n 2 ;
for H = C 8 , κ s ( BP n ; C 8 ) = n 2 and n 2 ≤ κ ( BP n ; C 8 ) ≤ n .

This work provides constructive ideas for other networks in the process of showing the structure and substructure connectivity. In addition, the exact value of κ ( BP n ; C 8 ) remains open, and we have the following conjecture.

Conjecture 6.1

κ ( BP n ; C 8 ) = n for n ≥ 3 .

Acknowledgements

The author thanks the referees and editors for their valuable comments and constructive suggestions that helped improving this article.

Funding information: This work was supported by the National Natural Science Foundation of China (Nos. 11661068, 12261074, and 12201335) and the Science Found of Qinghai Province (No. 2021-ZJ-703).
Author contributions: All authors contributed equally to the writing of this article. All authors read and approved the final manuscript.
Conflict of interest: The authors state that there is no conflict of interest.

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Received: 2022-08-07

Revised: 2023-08-30

Accepted: 2023-10-30

Published Online: 2023-12-31

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/math-2023-0154

Keywords for this article

interconnection networks; H-structure connectivity; H-substructure connectivity; burnt pancake networks

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