Generalized 4-connectivity of hierarchical star networks

1 Introduction

The graphs considered in this article are simple undirected finite graphs. For graph theory symbols and terms that are covered but not mentioned in this article, please refer to [1]. Let G = ( V , E ) be a connected graph with vertex set V ( G ) and edge set E ( G ) . Let u ∈ V ( G ) , and N G ( u ) = { v ∈ V ( G ) \ u ∣ u v ∈ E ( G ) } be the neighbour set of u in the graph G , and N G [ u ] = N G ( u ) ∪ { u } . Let d G ( u ) = ∣ N G ( u ) ∣ be the degree of u in G . If d G ( v ) = k for any vertex v ∈ V ( G ) , then the graph G is k -regular. For any vertices u , v ∈ V ( G ) , an ( u , v ) -path starts at u and ends at v . Any two ( u , v ) -paths P and Q are internally disjoint if and only if V ( P ) ∩ V ( Q ) = { u , v } .

The classic connectivity of graph G , denoted as κ ( G ) , is an important parameter to measure the reliability and fault-tolerance of the network. The connectivity κ ( G ) is larger, the reliability of the network is higher. There are two versions to define κ ( G ) . The version definition of “cut” is that deleting the minimum number of vertices disconnects the graph G . The version of “path” is defined as follows: for any vertex set S = { u , v } , κ G ( S ) represents the maximum number of internally disjoint paths joining u and v in G , and κ ( G ) = min { κ G ( S ) ∣ S ⊆ V ( G ) and ∣ S ∣ = 2 } .

Although there are fruitful research results in the study of classical connectivity, classical connectivity itself has certain limitations, which lead to large defects in evaluating the reliability of the network. For example, in the actual application of an interconnection network, all processors connected to the same processor are less likely to fail at the same time, so this parameter is not accurate enough to measure network reliability and fault tolerance. In view of this, Chartrand et al. [2] generalized classical connectivity and proposed the concept of generalized connectivity. Let G be a connected graph with S ⊆ V ( G ) . An S -tree of graph G is a tree T = ( V ′ , E ′ ) that contains all the vertices in S subject to S ⊆ V ( G ) . Two S -trees T and T ′ are internally disjoint if and only if E ( T ) ∩ E ( T ′ ) = ∅ and V ( T ) ∩ V ( T ′ ) = S . Denote κ ( S ) as the maximum number of internally disjoint S -trees in graph G . The generalized k-connectivity is defined as κ k ( G ) = min { κ ( S ) ∣ S ⊆ V ( G ) and ∣ S ∣ = k } , clearly, κ 2 ( G ) = κ ( G ) .

The bounds of generalized connectivity and the relationship between connectivity and generalized connectivity have been extensively investigated [3,4,5]. In addition, the generalized 3-connectivity of many networks has been studied, for example, complete graphs [6], card product graphs [7], product graphs [8], complete bipartite graph [9], star graphs, and alternating group graphs [10]. However, there are few results about generalized 4-connectivity, such as hypercubes [11], exchanged hypercubes [12], dual cube [13], and hierarchical cubic networks [14]. For more works and results on generalized connectivity, please refer to [15,16, 17,18].

In this article, we study the generalized 4-connectivity of the hierarchical star networks.

2 Hierarchical star networks and their properties

In this section, we give the definition, structure of the hierarchical star networks, and some lemmas.

The star graph S 4 is shown in Figure 1.

Figure 1

The graph of S 4 .

Let Γ n be a permutation group on the set [ n ] and S = { ( 1 , 2 ) , ( 1 , 3 ) , … , ( 1 , n ) } , where ( 1 , i ) is a transposition of Γ n . Then S n is the undirected Cayley graph Cay ( Γ n , S ) . For a permutation x ∈ Γ n , the permutation by interchanging the first element with i th element of x is denoted as x ( 1 , i ) for i ∈ [ n ] ⧹ { 1 } .

For two integers i , j ∈ [ n ] , denoted by S n j : i the subgraph of S n induced by all the vertices with the j th element being i . For a fixed dimension j ∈ [ n ] \ 1 , S n can be partitioned into n subgraphs S n j : i , which is isomorphic to S n − 1 for each i ∈ [ n ] .

The hierarchical star networks H S 2 and H S 3 are shown in Figure 2.

Figure 2

The graph of H S 2 and H S 3 .

Let ⟨ a , b ⟩ be a vertex of H S n . The neighbor set of ⟨ a , b ⟩ is exactly { h i ( ⟨ a , b ⟩ ) ∣ i ∈ [ n ] } . Furthermore, h 1 ( ⟨ a , b ⟩ ) is called the external neighbor of ⟨ a , b ⟩ and h i ( ⟨ a , b ⟩ ) is called the internal neighbor of ⟨ a , b ⟩ for i ∈ [ n ] ⧹ { 1 } . We denote by H S n a the subgraph induced by the vertex set { ⟨ a , b ⟩ ∈ V ( H S n ) ∣ b ∈ V ( S n ) } , which is isomorphic to an n -dimensional star graph S n identified by a . Moreover, we define H S n a , b as a subgraph of H S n induced by the vertex set V ( H S n a ) ∪ V ( H S n b ) and H S n Γ n − { a } as a subgraph of H S n induced by the vertex set V ( H S n ) \ V ( H S n a ) . For a vertex x ∈ V ( H S n ) , we use x ′ to denote the external neighbor of x .

3 Generalized 4-connectivity of H S n

In this section, we will study the generalized 4-connectivity of hierarchical star networks. To prove the main result, the following results are useful.

By Lemma 3.1, κ 4 ( H S n ) ≤ δ ( H S n ) − 1 = n − 1 . To obtain Theorem 3.3, it suffices to establish the following claim.

Claim. For any given vertex set S = { x , y , z , w } ⊆ V ( H S n ) , there exist n − 1 internally disjoint S -trees in H S n .

Recall that H S n consists of n ! copies isomorphic to S n , denoted by H S n ℓ , ℓ ∈ Γ n . We consider the following cases.

Case 1. The vertices of S are distributed in one copy of H S n .

Without loss of generality, suppose x , y , z , w ∈ V ( H S n α ) , where α ∈ Γ n . By Theorem 3.2, there exist n − 2 internally disjoint S -trees connecting x , y , z , w in H S n α , say T 1 , T 2 , … , T n − 2 . Let H = H S n Γ n − { α } , by Lemma 2.3, H is connected. Clearly, x ′ , y ′ , z ′ , w ′ ∈ V ( H ) , then there exists a tree T ^ n − 1 connecting x ′ , y ′ , z ′ , w ′ in H . Let T n − 1 = T ^ n − 1 ∪ x x ′ ∪ y y ′ ∪ z z ′ ∪ w w ′ . Then T 1 , T 2 , … , T n − 1 are n − 1 internally disjoint S -trees.

Case 2. The vertices of S are distributed in two distinct copies of H S n .

Without loss of generality, suppose x , y , z ∈ V ( H S n α ) and w ∈ V ( H S n β ) , where α , β ∈ Γ n , α ≠ β . By Lemma 3.5, there are n − 2 internally disjoint trees connecting x , y , z , say T ^ i , and u i ∈ V ( T ^ i ) is distinct with x , y , z for i ∈ { 1 , … , n − 2 } .

Case 2.1. There is no cross edge between { T ^ 1 , … , T ^ n − 2 } and H S n β .

Without loss of generality, suppose u i ′ ∈ V ( H S n j i ) , where j i ∈ Γ n − { α , β } , i ∈ { 1 , … , n − 2 } . First, we suppose that different vertices of { u 1 ′ , … , u n − 2 ′ } belong to different copies.

Case 2.1.1. x ′ , y ′ , z ′ ∈ V ( H S n Γ n − { α , β , ⋃ i = 1 n − 2 j i } ) for i ∈ { 1 , … , n − 2 } .

Figure 3

The illustration of Case 2.1.1.

Clearly, there exists a path P i joining u i ′ and m i in H S n j i such that m i ′ ∈ V ( H S n β ) . Let W = { m 1 ′ , … , m n − 2 ′ , m } . By Lemma 3.3, there are n − 1 internally disjoint paths joining w and W , say P 1 ′ , … , P n − 1 ′ , m i ′ ∈ V ( P i ′ ) , m ∈ V ( P n − 1 ′ ) , m ′ ∈ V ( H S n Γ n − { α , β , ⋃ i = 1 n − 2 j i } ) , for i ∈ { 1 , … , n − 2 } . Clearly, x ′ , y ′ , z ′ , m ′ ∈ V ( H S n Γ n − { α , β , ⋃ i = 1 n − 2 j i } ) , and so there is a tree T ^ n − 1 connecting x ′ , y ′ , z ′ , m ′ . Let T i = T ^ i ∪ u i u i ′ ∪ P i ∪ m i m i ′ ∪ P i ′ for i ∈ { 1 , … , n − 2 } and T n − 1 = x x ′ ∪ y y ′ ∪ z z ′ ∪ m m ′ ∪ P n − 1 ′ ∪ T ^ n − 1 . Then T 1 , T 2 , … , T n − 1 are n − 1 internally disjoint S -trees (Figure 3).

Case 2.1.2. One of { x ′ , y ′ , z ′ } belongs to some H S n j i for i ∈ { 1 , … , n − 2 } .

Suppose x ′ belongs to the copy H S n j n − 2 , clearly u n − 2 ′ also belongs to H S n j n − 2 . Let m n − 2 , h ∈ V ( H S n j n − 2 ) , X = { u n − 2 ′ , x ′ } , and Y = { m n − 2 , h } such that m n − 2 ′ ∈ V ( H S n β ) , h ′ ∈ V ( H S n Γ n − { α , β , ⋃ i = 1 n − 2 j i } ) . By Lemma 3.4, there are two internally disjoint paths between X and Y , say P n − 2 and P n − 1 , where P n − 2 is the path joining u n − 2 ′ and m n − 2 and P n − 1 is the path joining x ′ and h . Let W = { m 1 ′ , … , m n − 2 ′ , m } , by Lemma 3.3, there are n − 1 internally disjoint paths between w and W , say P 1 ′ , … , P n − 1 ′ , and m i ′ ∈ V ( P i ′ ) , m ∈ V ( P n − 1 ′ ) , such that m ′ ∈ V ( H S n Γ n − { α , β , ⋃ i = 1 n − 2 j i } ) . In view of y ′ , z ′ , m ′ , h ′ ∈ V ( H S n Γ n − { α , β , ⋃ i = 1 n − 2 j i } ) , there exists a tree connecting y ′ , z ′ , m ′ , h ′ , say T ^ n − 1 . Let T i = T ^ i ∪ u i u i ′ ∪ P i ∪ m i m i ′ ∪ P i ′ for i ∈ { 1 , … , n − 2 } and T n − 1 = x x ′ ∪ y y ′ ∪ z z ′ ∪ P n − 1 ∪ h h ′ ∪ m m ′ ∪ P n − 1 ′ ∪ T ^ n − 1 . Then T 1 , T 2 , … , T n − 1 are n − 1 internally disjoint S -trees (Figure 4).

Figure 4

The illustration of Case 2.1.2.

Next, we suppose two of { u 1 ′ , … , u n − 2 ′ } belong to the same copy. Let X ′ = ( X \ { x ′ } ) ∪ { u n − 3 ′ } , Y ′ = ( Y \ { h } ) ∪ { m n − 3 } , and m n − 3 ′ ∈ V ( H S n β ) . We obtain n − 3 internally disjoint S -tees T 1 , T 2 , … , T n − 4 , T n − 2 , similar to that of Case 2.1.2 by replacing X with X ′ and Y with Y ′ . In view of x ′ , y ′ , z ′ , m ′ ∈ V ( H S n Γ n − { α , β , ⋃ i = 1 n − 3 j i } ) , there is a tree connecting x ′ , y ′ , z ′ , m ′ say T ^ n − 1 . Let T n − 3 = T ^ n − 3 ∪ u n − 3 u n − 3 ′ ∪ P n − 3 ∪ m n − 3 m n − 3 ′ ∪ P n − 3 ′ , T n − 1 = x x ′ ∪ y y ′ ∪ z z ′ ∪ m m ′ ∪ P n − 1 ′ ∪ T ^ n − 1 . Then T 1 , T 2 , … , T n − 1 are n − 1 internally disjoint S -trees.

Case 2.2. There is one cross edge between { T ^ 1 , … , T ^ n − 2 } and H S n β .

Consequently, we just consider the case different vertices of { u 1 ′ , … , u n − 2 ′ } belong to different copies and the case that two of { u 1 ′ , … , u n − 2 ′ } in the same copy is similar.

Case 2.2.1. For some i ∈ { 1 , … , n − 2 } , u i ′ ∈ V ( H S n β ) .

Without loss of generality, suppose u n − 2 ′ ∈ V ( H S n β ) , u i ′ ∈ V ( H S n j i ) for i ∈ { 1 , … , n − 3 } .

When x ′ , y ′ , z ′ ∈ V ( H S n Γ n − { α , β , ⋃ i = 1 n − 3 j i } ) , we set W ′ = ( W \ { m n − 2 ′ } ) ∪ { u n − 2 ′ } . We obtain n − 2 internally disjoint S -trees T 1 , T 2 , … , T n − 3 , T n − 1 , similar to that of Case 2.1.1 by replacing W with W ′ . And T n − 2 = T ^ n − 2 ∪ P n − 2 ′ ∪ u n − 2 u n − 2 ′ (Figure 5[a]).

Figure 5

The illustrations of Case 2.2.1: (a) x ′ , y ′ , z ′ ∈ V ( H S n Γ n − { α , β , ⋃ i = 1 n − 3 j i } ) ; and (b) x ′ , u n − 3 ′ ∈ V ( H S n j n − 3 ) .

When one of { x ′ , y ′ , z ′ } and u i ′ belong to the same copy, for some i ∈ { 1 , … , n − 3 } , we suppose that x ′ , u n − 3 ′ belong to the same copy H S n j n − 3 . Let X ′ = ( X \ { u n − 2 ′ } ) ∪ { u n − 3 ′ } , Y ′ = ( Y \ { m n − 2 } ) ∪ { m n − 3 } , and W ′ = ( W \ { m n − 2 ′ } ) ∪ { u n − 2 ′ } . We obtain n − 2 internally disjoint S -trees T 1 , T 2 , … , T n − 3 , T n − 1 , similar to that of Case 2.1.2 by replacing X with X ′ , Y with Y ′ , W with W ′ . And T n − 2 = T ^ n − 2 ∪ P n − 2 ′ ∪ u n − 2 u n − 2 ′ (Figure 5[b]).

Case 2.2.2. One of { x ′ , y ′ , z ′ } belongs to H S n β .

Without loss of generality, suppose z ′ ∈ V ( H S n β ) , u i ′ ∈ V ( H S n j i ) , where j i ∈ Γ n − { α , β } , i ∈ { 1 , … , n − 2 } . There is a path joining u i ′ and m i in H S n j i , say P i , and m i ′ ∈ V ( H S n β ) , for i ∈ { 1 , … , n − 2 } .

Case 2.2.2.1. x ′ , y ′ ∈ V ( H S n Γ n − { α , β , ⋃ i = 1 n − 2 j i } ) .