Bondage Numbers of 𝐶₄ Bundles over a Cycle 𝐶_𝑛

Abstract

Graph bundles generalize the notion of covering graphs and graph products. Graph bundles have been applied in computer architecture and communication networks. The bondage number is an important parameter for measuring the vulnerability and stability of the network domination under link failure. The bondage number 𝑏(𝐺) of a graph 𝐺 is the minimum number of edges whose removal enlarges the domination number. In this paper, we show that the bondage number of every 𝐶₄ bundles over a cycle 𝐶_𝑛(𝑛≥4) is equal to 4.

1. Introduction

For notation and graph theoretical terminology not defined here, we follow [1]. Specifically, let 𝐺=(𝑉,𝐸) be an undirected graph without loops and multiedges, where 𝑉=𝑉(𝐺) is the vertex set and 𝐸=𝐸(𝐺) is the edge set, which is a subset of the set of all unordered pair of 𝑉. A graph 𝐺 is nonempty if 𝐸(𝐺)≠∅. Two vertices 𝑥 and 𝑦 are adjacent if {𝑥,𝑦}∈𝐸(𝐺). We will denote an edge {𝑥,𝑦} by 𝑥𝑦 if it does not make any confusion. For any disjoint subset 𝑆,𝑇⊆𝑉, we denote [𝑆,𝑇] by the edge set between 𝑆 and 𝑇 of 𝐺. For any subset 𝑆 of 𝑉, let ⟨𝑆⟩ be the subgraph induced by 𝑆. Throughout this paper, we use 𝑃_𝑛 and 𝐶_𝑛 to denote a path and a cycle of order 𝑛, respectively. A subset 𝐷⊆𝑉(𝐺) is called a dominating set of 𝐺 if every vertex not in 𝐷 has at least one neighbor in 𝐷. The domination number of 𝐺, denoted by 𝛾(𝐺), is the minimum cardinality among all dominating sets.

The domination is very important and classic conception that it has become one of the most widely studied topics in graph theory and also is frequently used to study properties of interconnection networks. The early results on this subject have been, in detail, surveyed in the two excellent books written by Haynes et al. [2, 3]. In the recent decade, a large number of research papers on domination as well as related topics appear in many scientific journals because of their applications in many fields such as networks, and wireless communication.

In 1990, Fink et al. [4] introduced the bondage number as a parameter for measuring the vulnerability of the interconnection network under link failure. The minimum dominating set of sites plays an important role in the network because it dominates the whole network with the minimum cost. So we must consider whether its function remains good under attack. Suppose that someone such as a saboteur does not know which sites in the network take part in the dominating role but does know that the set of these special sites corresponds to a minimum dominating set in the related graph. Then how many links does he have to attack so that the cost can not remain the same in order to dominate the whole network? That minimum number of links is just the bondage number.

A subset 𝐵 of 𝐸(𝐺) is called a bondage set of 𝐺 if its removal from 𝐺 results in a graph with larger domination number than 𝛾(𝐺). The bondage number 𝑏(𝐺) of a nonempty graph 𝐺 is the minimum number of edges in a bondage set of 𝐺. Since the domination number of every spanning subgraph of a nonempty graph 𝐺 is at least as great as 𝛾(𝐺), the bondage number of a nonempty graph is well defined. Many results on this topic are obtained in the literature. Fink et al. [4] computed the exact value of the bondage number for only a few classes of graphs, such as complete graphs, paths, cycles, and complete 𝑡-partition graphs. In 2012, Hu and Xu [5] have showed that the problem of determining bondage number for general graphs is NP hard. Recently, Xu [6] gave a review on bondage numbers in 2013.

For the Cartesian product 𝐺₁□𝐺₂ of two graphs 𝐺₁ and 𝐺₂, Hu and Xu [7] determined 𝑏(𝑃_𝑛□𝑃_𝑚) for 𝑚≤4, Dunbar et al. [8] determined 𝑏(𝐶_𝑛□𝑃₂) for 𝑛⩾3, Sohn et al. [9] determined 𝑏(𝐶_𝑛□𝐶₃) for 𝑛⩾4, Kang et al. [10] determined 𝑏(𝐶_𝑛□𝐶₄) for 𝑛⩾4, Huang and Xu [11] determined 𝑏(𝐶_5𝑖□𝐶_5𝑗) for any positive integers 𝑖 and 𝑗, and Xiang et al. [12] determined 𝑏(𝐶_𝑛□𝐶₅) for 𝑛⩾5 and 𝑛≢3(mod5). Sohn and Cao [13] determined bondage number of the graph bundle 𝐶_𝑛□_𝜑𝐶₅ having reflection voltage assignment.

Let 𝐵 and 𝐹 be graphs. A graph 𝐺 is a Cartesian graph bundle with fibre 𝐹 over the base graph 𝐵 if there is a graph map 𝑝∶𝐺→𝐵 such that for each vertex 𝑣∈𝑉(𝐵), 𝑝⁻¹(𝑣)≃𝐹, and for each edge 𝑒∈𝐸(𝐵), 𝑝⁻¹(𝑒)≃𝐾₂□𝐹. The triple (𝐺,𝑝,𝐵) is called a presentation of 𝐺 as a Cartesian graph bundle. We can also understand the Cartesian graph bundle as a graph, which is obtained from the base graph by replacing each of its vertices with a copy of the fibre graph and each of its edges by matching between the copies of the fibre, corresponding to the endpoints of the edge. The edges of the matching define an isomorphism between the copies of the fibre. For the completeness, let 𝐵 be the digraph obtained from 𝐵 by replacing each edge of 𝐵 with a pair of oppositely directed edges. The set of directed edges of 𝐵 is denoted by 𝐸(𝐵). By 𝑒⁻¹, we mean the reverse edge to an edge 𝑒∈𝐸(𝐵). We denote the directed edge 𝑒 of 𝐵 by 𝑢𝑣 if the initial and terminal vertices of 𝑒 are 𝑢 and 𝑣, respectively. For a finite group Γ, a Γ-voltage assignment of 𝐺 is a function 𝜑∶𝐸(𝐵)→Γ such that 𝜑(𝑒⁻¹)=𝜙(𝑒)⁻¹ for all 𝑒∈𝐸(𝐵). We denote the set of all Γ-voltage assignments of 𝐵 by 𝐶¹(𝐵;Γ). Let Aut(𝐹) be the set of all automorphisms of 𝐹 and let 𝜑∈𝐶¹(𝐵;Aut(𝐹)). The bundle 𝐺 is determined by 𝐹, 𝐵 and 𝜑 is the graph 𝐵□_𝜑𝐹 with the vertex set 𝑉(𝐵□_𝜙𝐹)=𝑉(𝐵)□𝑉(𝐹), and two vertices (𝑢₁,𝑣₁) and (𝑢₂,𝑣₂) are adjacent in 𝐵□_𝜙𝐹 if either 𝑢₁𝑢₂∈𝐸(𝐵) and 𝑣₂=𝑣^{𝜙(𝑢1𝑢2)}₁ or 𝑢₁=𝑢₂ and 𝑣₁𝑣₂∈𝐸(𝐹).

It is well known [14, 15] and easy to see that for any spanning tree 𝑇 of 𝐵, there is a 𝜓∶𝐸(𝐵)→Aut(𝐹) such that 𝜓(𝑒)=identity for 𝑒∈𝐸(𝑇) and 𝐺=𝐵□_𝜑𝐹≃𝐵□_𝜓𝐹. From the definition, it follows that Cartesian graph bundles over paths and trees are exactly Cartesian product graphs and that we can represent Cartesian graph bundles over a cycle with a set of isomorphisms over the edges of the cycle, with at most one nonidentical isomorphism (if all isomorphisms are identities, the bundle is the Cartesian product of two graphs 𝐵 and 𝐹).

The domination numbers of the Cartesian product of two graphs have been given great attention in domination theory. The most interesting problem is Vizing’s conjecture [16]: For every pair of finite graphs 𝐺 and 𝐻, 𝛾(𝐺□𝐻)≥𝛾(𝐺)𝛾(𝐻). Vizing’s conjecture is arguably the main open problem in the area of domination theory. There are many related articles about Vizing conjecture and see the recent survey [17]. There are many variants about Vizing’s conjecture with different dominations and product. The domination number of the graph bundle is an interesting problem. But the conjecture on graph bundle corresponding to Vizing’s conjecture on the Cartesian product of two graphs is not true [18].

In this paper, we compute the bondage number of the graph bundle 𝐶_𝑛□_𝜑𝐶₄ (𝑛≥4) as follows.

Main Theorem. For any   𝑛≥4  and   𝜑∈𝐶¹(𝐶_𝑛;𝐴𝑢𝑡(𝐶₄)), 𝑏(𝐶_𝑛□_𝜑𝐶₄)=4.

2. Preliminary Results

Throughout this paper, we assume that a path 𝑃_𝑛 and a cycle 𝐶_𝑛 both have the vertex set {1,2,…,𝑛}. Use 𝐺_𝑛 to denote the Cartesian graph bundle product 𝐶_𝑛□_𝜑𝐶₄ of two cycles 𝐶_𝑛 and 𝐶₄. Use 𝐻_𝑛 to denote the Cartesian product 𝑃_𝑛□𝐶₄ of two paths 𝑃_𝑛 and 𝐶₄. Let both 𝐺_𝑛 and 𝐻_𝑛 have vertex set {𝑢_𝑖𝑗∣1≤𝑖≤𝑛,1≤𝑗≤4}. Clearly, for any Cartesian graph bundle, 𝑃_𝑛□_𝜑𝐶₄ is isomorphic to 𝐻_𝑛. For a bundle 𝐶_𝑛□_𝜑𝐶₄, we define 𝜓∶𝐸(𝐵)→Aut(𝐶₄) by 𝜓(𝑒)= identity for 𝑒≠(𝑛−1)𝑛 or 𝑒≠𝑛(𝑛−1) and 𝜙(𝑛−1𝑛)=𝜑(12)𝜑(23)⋯𝜑((𝑛−1)𝑛)𝜑(𝑛1). Then 𝐶_𝑛□_𝜑𝐶₄ and 𝐶_𝑛□_𝜙𝐶₄ are isomorphic (see Figure 1). In the rest of the paper, we will denote Cartesian graph bundle over cycle 𝐶_𝑛 by 𝐶_𝑛□_𝜑𝐹, where 𝜑(𝑒)={identity} for 𝑒≠(𝑛−1)𝑛 or 𝑒≠𝑛(𝑛−1).

Figure 1 

Size representation of Cartesian graph bundles over cycle 𝐶4.

In the following we always use this representation to represent 𝐺_𝑛 for simplicity. For given 𝐺_𝑛 and 𝐻_𝑛, the 𝑖th column of them is the vertex set 𝑋_𝑖={𝑢_𝑖1,𝑢_𝑖2,𝑢_𝑖3,𝑢_𝑖4} for 1≤𝑗≤𝑛 and an element in it is said to be vertical and the 𝑗th raw 𝑌_𝑗={𝑢_1𝑗,𝑢_2𝑗,…,𝑢_𝑛𝑗} for 1≤𝑗≤4 and an element in it is said to be horizontal.

Let 𝐺_𝑛−𝐻_𝑖 and 𝐻_𝑛−𝐻_𝑖 be the subgraph of 𝐺_𝑛 and 𝐻_𝑛 induced by {𝑢_𝑘𝑗∣𝑖+1≤𝑘≤𝑛,1≤𝑗≤4} for each 𝑖=1,2,…,𝑛−1, respectively. Clearly, 𝐺_𝑛−𝐻_𝑖≅𝐻_𝑛−𝐻_𝑖≅𝐻_𝑛−𝑖. In this paper, when we write a vertex 𝑢_𝑖(𝑗+1) for 1≤𝑗≤4 and if 𝑗=4, then 𝑢_𝑖(𝑗+1) means 𝑢_𝑖1. We state some useful results on 𝛾(𝐺_𝑛) and 𝛾(𝐻_𝑛), which will be used in our proofs.

Lemma 1 (see [19]). Consider 𝛾(𝐶_𝑛□𝐶₄)=𝑛, 𝑛≥2.

Lemma 2 (see [18]). Consider 𝛾(𝐺_𝑛)=𝑛, 𝑛≥2.

Lemma 3 (see [10]). Consider 𝛾(𝐻_𝑛)=𝑛, 𝑛≥2.

Lemma 4 (see [10]). If 𝑒₁,𝑒₂∈𝐸(𝐻_𝑛), 𝑛≥2 are adjacent that is, they have a common end-vertex, then 𝛾(𝐻_𝑛−{𝑒₁,𝑒₂})=𝑛.

Lemma 5 (see [10]). If 𝑒₁ is a horizontal edge and 𝑒₂ is a vertical edge of 𝐻_𝑛 for 𝑛≥2, then 𝛾(𝐻_𝑛−{𝑒₁,𝑒₂})=𝑛.

Lemma 6. Let 𝐷 be a dominating set of 𝐻_𝑛. Then |𝐷∩𝑉(𝐻_𝑛−𝐻_𝑖)|≥𝛾(𝐻_𝑛−𝐻_𝑖+1) and |𝐷∩𝑉(𝐻_𝑖+1)|≥𝛾(𝐻_𝑖) for each 𝑖=0,1,…,𝑛−1, and 𝑛≥2.

Proof. We only need to prove that |𝐷∩𝑉(𝐻_𝑖+1)|≥𝛾(𝐻_𝑖) since 𝐻_𝑛−𝐻_𝑖≅𝐻_𝑛−𝑖. Let 𝐷^′=𝐷∩𝑉(𝐻_𝑖+1).
If 𝐷^′∩𝑋_𝑖+1=∅, then 𝐷^′ is a dominating set of 𝐻_𝑖, and hence |𝐷∩𝑉(𝐻_𝑖+1)|≥𝛾(𝐻_𝑖).
Assume 𝐷^′∩𝑋_𝑖+1≠∅. Let 𝐵_𝑖={𝑗∣𝑢_(𝑖+1)𝑗∈𝐷^′}. Then 𝐷^′′=(𝐷^′⧵𝑋_𝑖+1)∪{𝑢_𝑖𝑗∣𝑗∈𝐵_𝑖} is a dominating set of 𝐻_𝑖 and |𝐷^′′|⩽|𝐷^′|. Thus, we have |𝐷∩𝑉(𝐻_𝑖+1)|≥|𝐷^′′|≥𝛾(𝐻_𝑖).

Lemma 7. Let 𝐷 be a minimum dominating set of 𝐻_𝑛. Then |𝑋₁∩𝐷|=1 and |𝑋_𝑛∩𝐷|=1 for 𝑛≥2.

Proof. By Lemma 3, |𝐷|=𝑛. By the symmetry of 𝐻_𝑛, we only need to prove |𝑋₁∩𝐷|=1. It is easy to see that the conclusion is true for 𝑛=2,3. Assume now 𝑛≥4. By contradiction.
Assume |𝑋₁∩𝐷|=0 and then |𝑋₂∩𝐷|=4. By Lemmas 3 and 6, |𝐷∩𝑉(𝐻_𝑛−𝐻₂)|≥𝛾(𝐻_𝑛−𝐻₃)≥𝑛−3. Thus |𝐷|=|𝑋₂∩𝐷|+|𝐷∩𝑉(𝐻_𝑛−𝐻₂)|≥𝑛+1, a contradiction with |𝐷|=𝑛. So |𝑋₁∩𝐷|≥1.
Assume now |𝑋₁∩𝐷|≥3. By Lemmas 3 and 6, |𝐷∩𝑉(𝐻_𝑛−𝐻₁)|≥𝛾(𝐻_𝑛−𝐻₂)=𝑛−2. Thus |𝐷|=|𝑋₁∩𝐷|+|𝐷∩𝑉(𝐻_𝑛−𝐻₁)|≥𝑛+1, a contradiction with |𝐷|=𝑛. So |𝑋₁∩𝐷|≤2.
Finally, assume |𝑋₁∩𝐷|=2.
If 𝑋₂∩𝐷=∅, then 𝐷^′=𝐷⧵𝑋₁ is a dominating set of 𝐻_𝑛−𝐻₂ and |𝑋₃∩𝐷^′|⩾2. By the induction hypothesis, 𝐷^′ is not a minimum dominating set of 𝐻_𝑛−𝐻₂, and hence |𝐷^′|⩾𝛾(𝐻_𝑛−𝐻₂)+1=𝑛−1 by Lemma 3. Then |𝐷|=2+|𝐷^′|⩾𝑛+1, a contradiction with |𝐷|=𝑛.
If 𝑋₂∩𝐷≠∅, there exists a vertex 𝑢_2𝑗 such that 𝐷^′′=(𝐷⧵𝑋₁)∪{𝑢_2𝑗} is a dominating set of 𝐻_𝑛−𝐻₁ and |𝑋₂∩𝐷^′′|⩾2. By the induction hypothesis, 𝐷^′′ is not a minimum dominating set of 𝐻_𝑛−𝐻₁, and hence |𝐷^′′|⩾𝛾(𝐻_𝑛−𝐻₁)+1=𝑛 by Lemma 3. Then |𝐷|⩾|𝑋₁∩𝐷|+|𝐷^′′|−1⩾𝑛+1, a contradiction with |𝐷|=𝑛. The lemma follows.

Lemma 8. If both 𝑒₁ and 𝑒₂ are horizontal edges of 𝐻_𝑛 for 𝑛≥3, then 𝛾(𝐻_𝑛−{𝑒₁,𝑒₂})=𝑛.

Proof. Without loss of generality, assume 𝑒₁=𝑢_𝑖1𝑢_(𝑖+1)1 for some integer 𝑖 with 1≤𝑖≤⌊𝑛/2⌋. Let 𝑢𝐷=^𝑘2∣𝑘isodd∪𝑢_ℓ4∣ℓisevenif𝑒₂∈𝐸𝑌₁∪𝑌₃;𝑢_𝑘1∣𝑘isodd∪𝑢_ℓ3∣ℓisevenif𝑒₂∈𝐸𝑌₂∪𝑌₄. (1) Then 𝐷 is a dominating set of 𝐻_𝑛−{𝑒₁,𝑒₂} and |𝐷|=𝑛.

Lemma 9. Let 𝑒₁=𝑢_𝑖𝑗𝑢_𝑖(𝑗+1) and 𝑒₂=𝑢_𝑘𝑗′𝑢_{𝑘(𝑗′+1)}(𝑖≤𝑘) be any two vertical edges of 𝐻_𝑛 for 𝑛≥4 and let 𝑒₁ and 𝑒₂ be adjacent for 𝑖=𝑘=1 or 𝑖=𝑘=𝑛. Then, one has 𝛾(𝐻_𝑛−{𝑒₁,𝑒₂})=𝑛.

Proof. If 𝑒₁ and 𝑒₂ are adjacent, then the conclusion is true by Lemma 4. Next, assume 𝑒₁ and 𝑒₂ are nonadjacent. This imply that |𝑗^′−𝑗|=2 when 𝑖=𝑘.
Assume 1<𝑖<𝑘<𝑛. We consider 𝐻_𝑖 and 𝐻_𝑛−𝐻_𝑖. By Lemma 4, 𝛾(𝐻_𝑖−𝑒₁)=𝑖 and 𝛾(𝐻_𝑛−𝐻_𝑖−𝑒₂)=𝑛−𝑖 and hence 𝛾(𝐻_𝑛−{𝑒₁,𝑒₂})=𝑛.
Assume 1<𝑖=𝑘<𝑛. Without loss of generality, suppose 𝑗=1, 𝑗^′=3 and 𝑖=𝑘≤⌈𝑛/2⌉≤𝑛−2. Let 𝐷₁={𝑢_𝑝3∣𝑝isodd}∪{𝑢_𝑞1|𝑞iseven} where 1≤𝑝, 𝑞≤𝑖 and 𝐷₂={𝑢_𝑝2∣𝑝isodd}∪{𝑢_𝑞4|𝑞iseven} where 𝑖<𝑝, 𝑞≤𝑛. Then 𝐷=𝐷₁∪𝐷₂ is a dominating set of 𝐻_𝑛−{𝑒₁,𝑒₂} and |𝐷|=𝑛.
Assume now 𝑖=1 or 𝑘=𝑛. Without loss of generality, suppose 𝑖=1.
If 𝑘=1, then by the assumption, 𝑒₁ and 𝑒₂ are adjacent. By Lemma 4, 𝛾(𝐻_𝑛−{𝑒₁,𝑒₂})=𝑛.
If 𝑘=2, then without loss of generality, suppose 𝑗=1. Let 𝑢𝐷=¹³∪𝑢₂₁∪𝑢_𝑝2∣𝑝isodd∪𝑢,𝑝≥3_𝑞4∣𝑞iseven,𝑞≥3,if𝑗^′𝑢=1;_𝑝3∣𝑝isodd∪𝑢_𝑞1∣𝑞iseven,if𝑗^′=2or𝑢3;_𝑝4∣𝑝isodd,𝑝≥3∪𝑢_𝑞2∣𝑞iseven,𝑞≥3,if𝑗^′=4. (2) Then 𝐷 is a dominating set of 𝐻_𝑛−{𝑒₁,𝑒₂} and |𝐷|=𝑛.
If 𝑘>2, then we consider 𝐻₂ and 𝐻_𝑛−𝐻₂. By Lemma 4, 𝛾(𝐻₂−𝑒₁)=𝑖+1 and 𝛾(𝐻_𝑛−𝐻₂−𝑒₂)=𝑛−𝑖−1 and hence 𝛾(𝐻_𝑛−{𝑒₁,𝑒₂})=𝑛.

Proposition 10. Let 𝑥 be a vertex that belongs to 𝑋₁ or 𝑋_𝑛. Then there exists a minimum dominating set 𝐷 of 𝐻_𝑛 such that 𝑥∈𝐷 for 𝑛≥2.

3. Main Results

Theorem 11. 𝑏(𝐻_𝑛)=2 for 𝑛≥2.

Proof. For any edge 𝑒∈𝐸(𝐻_𝑛), by Lemmas 3 and 4, 𝛾(𝐻_𝑛−{𝑒})=𝑛=𝛾(𝐻_𝑛). Thus 𝑏(𝐻_𝑛)≥2. Next, we prove 𝑏(𝐻_𝑛)≤2.
Let 𝑒₁=𝑢₁₁𝑢₁₂ and 𝑒₂=𝑢₁₃𝑢₁₄. Let 𝐷 be a minimum dominating set of 𝐻_𝑛−{𝑒₁,𝑒₂}. We only need to prove |𝐷|>𝛾(𝐻_𝑛)=𝑛. It is easy to verify that |𝐷|=3>𝛾(𝐻_𝑛) for 𝑛=2. Next assume 𝑛≥3.
Notice that 𝐷 is also a dominating set of 𝐻_𝑛. If |𝐷∩𝑋₁|≠1, then by Lemma 7, 𝐷 is not a minimum dominating set of 𝐻_𝑛 and hence |𝐷|>𝛾(𝐻_𝑛). Now assume |𝐷∩𝑋₁|=1. Then |𝐷∩𝑋₂|≥2. Notice that 𝐷^′=𝐷⧵𝑋₁ is a dominating set of 𝐻_𝑛−𝐻₁≅𝐻_𝑛−1. By Lemma 7, 𝐷^′ is not a minimum dominating set of 𝐻_𝑛−𝐻₁; hence, |𝐷^′|>𝛾(𝐻_𝑛−𝐻₁)=𝑛−1, and hence, |𝐷|=|𝐷^′|+1>𝑛.

Lemma 12. Consider 𝑏(𝐺_𝑛)≤4 for 𝑛≥4.

Proof. Let 𝐷 be a minimum dominating set of 𝐺_𝑛−𝑢₁₄. Then 𝛾(𝐺_𝑛−𝐸_𝐺𝑛(𝑢₁₄))=|{𝑢₁₄}|+𝛾(𝐺_𝑛−𝑢₁₄)=1+|𝐷| where 𝐸_𝐺𝑛(𝑢₁₄) is all the edges incidents with 𝑢₁₄. Therefore, we only need to prove |𝐷|≥𝛾(𝐺_𝑛)=𝑛. If one of {𝑢₁₁,𝑢₁₃,𝑢₂₄,𝑢_𝑛4} belongs to 𝐷, then 𝐷 is also a dominating set of 𝐺_𝑛. By Lemma 2, |𝐷|≥𝛾(𝐺_𝑛)=𝑛. Assume {𝑢₁₁,𝑢₁₃,𝑢₂₄,𝑢_𝑛4}∩𝐷=∅ in the following.
If 𝑋₁∩𝐷=∅, then 𝐷 is also a dominating set of 𝐺_𝑛−𝐻₁≅𝐻_𝑛−1, |𝑋₂∩𝐷|≥2 or |𝑋_𝑛∩𝐷|≥2. By Lemma 7, 𝐷 is not a minimum dominating set of 𝐺_𝑛−𝐻₁, and hence |𝐷|≥𝛾(𝐻_𝑛−1)+1=𝑛.
Assume now 𝑋₁∩𝐷≠∅. Then 𝑋₁∩𝐷={𝑢₁₂} since {𝑢₁₁,𝑢₁₃,𝑢₂₄,𝑢_𝑛4}∩𝐷=∅. If |(𝑋₂∪𝑋_𝑛)∩𝐷|≥2, then there exists a vertex 𝑥∈𝑋₂∪𝑋_𝑛 such that 𝐷^′=(𝐷⧵{𝑢₁₂})∪{𝑥} is a dominating set of 𝐺_𝑛−𝐻₁≅𝐻_𝑛−1, and |𝑋₂∩𝐷^′|≥2 or |𝑋_𝑛∩𝐷^′|≥2. By Lemma 7, 𝐷^′ is not a minimum dominating set of 𝐺_𝑛−𝐻₁ and hence |𝐷|≥|𝐷^′|≥𝛾(𝐻_𝑛−1)+1=𝑛.
If |(𝑋₂∪𝑋_𝑛)∩𝐷|≤1, then 𝑋₂∩𝐷=∅ or 𝑋_𝑛∩𝐷=∅. Without loss of generality, assume 𝑋₂∩𝐷=∅. So {𝑢₃₁,𝑢₃₃,𝑢₃₄}⊆𝐷. Let 𝐷^′′=(𝐷⧵{𝑢₃₁,𝑢₃₃,𝑢₃₄})∪{𝑢₂₄,𝑢₃₂,𝑢₄₄}. Then 𝐷^′′ is a dominating set of 𝐺_𝑛, and hence |𝐷|≥|𝐷^′′|≥𝛾(𝐺_𝑛)=𝑛.

Lemma 13. Consider 𝑏(𝐺_𝑛)≥4 for 𝑛≥4.

Proof. It suffices to show that 𝛾(𝐺_𝑛−{𝑒₁,𝑒₂,𝑒₃})=𝑛 for any three edges 𝑒₁,𝑒₂,𝑒₃∈𝐸(𝐺_𝑛).
Note. Since 𝑛≥4, by the isomorphism of 𝐺_𝑛, we can assume that none of these three edges is in [𝑋_𝑛−1,𝑋_𝑛]. That is, 𝑒_𝑖 is either horizontal or vertical for each 𝑖=1,2,3.
According to the number of horizontal and vertical edges, it divides into two cases.
Case 1. At least one of 𝑒₁, 𝑒₂, and 𝑒₃ is horizontal. Without loss of generality, assume 𝑒₁=𝑢₁₁𝑢_𝑛1.
Assume at least one of 𝑒₂ and 𝑒₃ is horizontal, or both 𝑒₂ and 𝑒₃ are vertical and they satisfy the condition in Lemma 9. It is easy to see that 𝐺_𝑛−[𝑋₁,𝑋_𝑛]≅𝐻_𝑛. By Lemmas 5, 8, and 9,  𝛾(𝐺_𝑛−[𝑋₁,𝑋_𝑛]−{𝑒₂,𝑒₃})=𝑛. Thus 𝛾(𝐺_𝑛−{𝑒₁,𝑒₂,𝑒₃})≤𝛾(𝐺_𝑛−[𝑋₁,𝑋_𝑛]−{𝑒₂,𝑒₃})=𝑛.
The remaining case is that 𝑒₂ and 𝑒₃ do not satisfy the condition in Lemma 9. Without loss of generality, assume 𝑒₂=𝑢₁₁𝑢₁₂. Then 𝑒₃=𝑢₁₃𝑢₁₄.
By Proposition 10, let 𝐷^′ be a minimum dominating set of 𝐺_𝑛−𝐻₂≅𝐻_𝑛−2 that contains 𝑢_𝑛4. Let 𝐷={𝑢₁₃,𝑢₂₁}∪𝐷^′. Then 𝐷 is a dominating set of 𝐺_𝑛−{𝑒₁,𝑒₂,𝑒₃} with cardinality 𝑛.
Case 2. All of 𝑒₁, 𝑒₂, and 𝑒₃ are vertical. Without loss of generality, assume 𝑒₁=𝑢₁₁𝑢₁₂. Let 𝑒₂=𝑢_𝑘𝑗𝑢_𝑘(𝑗+1), 𝑒₃=𝑢_𝑙𝑗′𝑢_{𝑙(𝑗′+1)} and 𝑘≤ℓ.
If 𝑘=ℓ=1, then without loss of generality, assume 𝑗=2 and 𝑗^′=3. By Proposition 10, let 𝐷^′ be a minimum dominating set of 𝐺_𝑛−𝐻₂≅𝐻_𝑛−2 such that 𝑢_𝑛2∈𝐷^′. Then 𝐷^′∪{𝑢₁₁,𝑢₂₃} is a dominating set of 𝐺_𝑛−{𝑒₁,𝑒₂,𝑒₃} with cardinality 𝑛.
Assume 𝑘=ℓ>1 or 1=𝑘<ℓ. Without loss of generality, suppose 𝑘=ℓ>1. If 𝑘≥3 or 𝑘=2 and |𝑗−𝑗^′|≠2 (see Figure 2), then 𝑒₂ and 𝑒₃ are adjacent. We consider 𝐻_𝑛−1−𝐻₁≅𝐻_𝑛−2 and ⟨𝑋₁∪𝑋_𝑛⟩≅𝐻₂. By Lemma 4, 𝛾(𝐻_𝑛−1−𝐻₁−{𝑒₂,𝑒₃})=𝑛−2 and 𝛾(⟨𝑋₁∪𝑋_𝑛⟩−{𝑒₁})=2. Thus 𝛾(𝐺_𝑛−{𝑒₁,𝑒₂,𝑒₃})≤𝑛. If 𝑘=2 and |𝑗−𝑗^′|=2, then 𝑒₂ and 𝑒₃ are nonadjacent. Without loss of generality, assume 𝑗=1 and 𝑗^′=3. By Proposition 10, let 𝐷^′′ be a minimum dominating set of 𝐺_𝑛−𝐻₂≅𝐻_𝑛−2 such that 𝑢₃₂∈𝐷^′′. Then 𝐷^′′∪{𝑢₁₃,𝑢₂₁} is a dominating set of 𝐺_𝑛−{𝑒₁,𝑒₂,𝑒₃} with cardinality 𝑛.
Assume now 1<𝑘<ℓ. We consider 𝐻_𝑛−1−𝐻₁ and ⟨𝑋₁∪𝑋_𝑛⟩≅𝐻₂. By Lemmas 4 and 9, 𝛾(⟨𝑋₁∪𝑋_𝑛⟩−{𝑒₁})=2 and 𝛾(𝐻_𝑛−1−𝐻₁−{𝑒₁,𝑒₂})=𝑛−2. Therefore, 𝛾(𝐺_𝑛−{𝑒₁,𝑒₂,𝑒₃})≤𝛾(⟨𝑋₁∪𝑋_𝑛⟩−{𝑒₁})+𝛾(𝐻_𝑛−1−𝐻₁−{𝑒₁,𝑒₂})=𝑛.

Figure 2 

Dominating sets (bold vertices) of 𝐺9−{𝑒1,𝑒2,𝑒3} in Case 2 of Lemma 13.

From Lemmas 12 and 13, we have the main theorem.

Theorem 14. For any 𝑛≥4 and 𝜑∈𝐶¹(𝐶_𝑛;Aut (𝐶₄)), 𝑏(𝐶_𝑛□_𝜑𝐶₄)=4.

Since Cartesian product graph 𝐶_𝑛□𝐶₄ is the bundle graph 𝐶_𝑛□_𝜑𝐶₄ corresponding to the 𝜑 such that 𝜑(𝑒)= identity for each edge 𝑒∈𝐸(𝐶_𝑛), we have the following.

Corollary 15 (see [10]). For any 𝑛≥4, 𝑏(𝐶_𝑛□𝐶₄)=4.

Acknowledgments

This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2012R1A1A20 05115). The third author was supported by the National Research Foundation of Korea (NRF) Grant funded by the Korea government (2013000416).