On the Domination Number of Cartesian Product of Two Directed Cycles

Abstract

Denote by 𝛾(𝐺) the domination number of a digraph 𝐺 and 𝐶_𝑚□𝐶_𝑛 the Cartesian product of 𝐶_𝑚 and 𝐶_𝑛, the directed cycles of length 𝑚,𝑛≥2. In 2010, Liu et al. determined the exact values of 𝛾(𝐶_𝑚□𝐶_𝑛) for 𝑚=2,3,4,5,6. In 2013, Mollard determined the exact values of 𝛾(𝐶_𝑚□𝐶_𝑛) for 𝑚=3𝑘+2. In this paper, we give lower and upper bounds of 𝛾(𝐶_𝑚□𝐶_𝑛) with 𝑚=3𝑘+1 for different cases. In particular, ⌈(2𝑘+1)𝑛/2⌉≤𝛾(𝐶_3𝑘+1□𝐶_𝑛)≤⌊(2𝑘+1)𝑛/2⌋+𝑘. Based on the established result, the exact values of 𝛾(𝐶_𝑚□𝐶_𝑛) are determined for 𝑚=7 and 10 by the combination of the dynamic algorithm, and an upper bound for 𝛾(𝐶₁₃□𝐶_𝑛) is provided.

1. Introduction

All of the digraphs considered in this paper are finite and simple, that is, without multiple edges or loops. For a digraph 𝐺=(𝑉,𝐸) and a vertex 𝑣∈𝑉, 𝑁⁺_𝐺(𝑣) and 𝑁⁻_𝐺(𝑣) denote the set of out-neighbors and in-neighbors of 𝑣. Given two vertices 𝑢 and 𝑣 in 𝐺, we say 𝑢 dominates 𝑣 if 𝑢=𝑣 or 𝑢𝑣∈𝐸. Let 𝑁⁺_𝐺[𝑣]=𝑁⁺_𝐺(𝑣)∪{𝑣}. A vertex 𝑣 dominates all vertices in 𝑁⁺_𝐺[𝑣]. A set 𝐷⊆𝑉 is a dominating set of 𝐺 if 𝐷 dominates 𝑉(𝐺). The domination number of 𝐺, denoted by 𝛾(𝐺), is the minimum cardinality of a dominating set of 𝐺.

The Cartesian product of graphs 𝐺 and 𝐻 is the graph 𝐺□𝐻 with the vertex set 𝐺×𝐻, and (𝑔,ℎ)(𝑔^′,ℎ^′)∈𝐸(𝐺□𝐻) if either 𝑔𝑔^′∈𝐸(𝐺) and ℎ=ℎ^′ or ℎℎ^′∈𝐸(𝐻) and 𝑔=𝑔^′. The Cartesian product is commutative and associative, having the one-vertex graph as a unit. The subgraph of 𝐺□𝐻 induced by 𝑉(𝐺)×ℎ, where ℎ∈𝑉(𝐻), is isomorphic to 𝐺, called a 𝐺-layer (over ℎ) and denoted by  𝐺^ℎ. For more information on the Cartesian product of graphs see [1].

We use a 0-1 matrix pattern 𝒜 with 𝑚 rows and 𝑛 columns to represent a dominating set 𝐷 of 𝐶_𝑚□𝐶_𝑛, where (𝑖,𝑗)∈𝐷 if and only if the value at the entry (𝑖,𝑗) of 𝒜 equals 1. For example, the pattern in Figure 2 represents the dominating set (the set of black circles) shown in Figure 1. Let 𝑃 be a pattern; we denote by 𝑐(𝑃) the column number of 𝑃. Let 𝐴 and 𝐵 be patterns; then the pattern 𝐴𝐵 denotes the concatenation of patterns 𝐴 and 𝐵, and 𝐴^𝑘 denotes the concatenation of patterns 𝐴 for 𝑘 times.

Figure 1 

A dominating set of 𝐶7□𝐶18.

Figure 2 

Pattern of the dominating set shown in Figure 1.

Let [𝑛] denote the set {0,1,…,𝑛−1}. For an integer 𝑛≥3, the directed cycle of length 𝑛 is the graph 𝐶_𝑛 whose vertices are 0,1,…,𝑛−1 and whose edges are the pairs 𝑖, 𝑖+1, where the arithmetic is done modulo 𝑛. We denote 𝑉(𝐶_𝑛) by [𝑛]. In this paper, when we consider the Cartesian product graph 𝐶_𝑚□𝐶_𝑛 of two cycles 𝐶_𝑚 and 𝐶_𝑛, the edge set of 𝐶_𝑚 is {(𝑖,𝑖+1)∶𝑖∈[𝑚]} and 𝐶_𝑛 is {(𝑖+1,𝑖)∶𝑖∈[𝑛]}. Throughout the paper, when considering the vertex (𝑖,𝑗) in the graph 𝐶_𝑚□𝐶_𝑛, we use the arithmetic operations of the index 𝑖 over modulo 𝑚 and 𝑗 modulo 𝑛.

The dominating set problem requires determining the domination number of a given graph. It has natural applications in numerous facility location problems. In such problems, the vertices of a graph correspond to locations, adjacency represents some notion of accessability, and the goal is to find a subset of locations accessible from all other locations at which to install fire stations, bus stops, post offices, or similar facilities [2]. Dominating sets have also been applied in coding theory [2] and social networks [3]. For more information on the history and applications of the dominating set problem, see [4].

In 1990, Faudree and Schelp [5] initially discussed the domination number of the Cartesian product of two undirected graphs. The domination number for Cartesian product of two graphs has attracted lots of attention, and there are many works on undirected graphs (see, e.g., [6–12]). Recently, there are also some works on the domination number of Cartesian product of two directed graphs, in particular, the Cartesian product of directed paths and cycles [13–15].

In [13, 15], Liu et al. determined the exact values of 𝛾(𝐶_𝑚□𝐶_𝑛) for 𝑚=2,3,4,5,6 and showed the following.

Theorem 1. 𝛾(𝐶_𝑚□𝐶_𝑛)=𝑚𝑛/3 if 𝑚≡0  (mod3) and 𝑛≡0  (mod3).

Mollard [14] determined exact values of 𝛾(𝐶_𝑚□𝐶_𝑛) for 𝑚=3𝑘+2. This paper investigates properties on the domination of Cartesian product of 𝐶_𝑚 and 𝐶_𝑛 with 𝑚=3𝑘+1. We give lower and upper bounds of 𝛾(𝐶_𝑚□𝐶_𝑛) with 𝑚=3𝑘+1 for different cases. In particular, ⌈(2𝑘+1)𝑛/2⌉≤𝛾(𝐶_3𝑘+1□𝐶_𝑛)≤⌊(2𝑘+1)𝑛/2⌋+𝑘. Based on the established result, the exact values of 𝛾(𝐶_𝑚□𝐶_𝑛) are determined for 𝑚=7 and 10 by the combination of the dynamic algorithm, and an upper bound for 𝛾(𝐶₁₃□𝐶_𝑛) is provided.

2. Main Results

2.1. Bounds on 𝐶_3𝑘+1□𝐶_𝑛

Proposition 2 (Mollard [14]). Let 𝑆 be a dominating set of 𝐶_𝑚□𝐶_𝑛; then for all 𝑖∈[𝑛] considered modulo 𝑛 one has |𝑆∩𝐶_𝑚^𝑖−1|+2|𝑆∩𝐶^𝑖_𝑚|≥𝑚.

Lemma 3. Let 𝑛≥3, 𝑘≥1. Then there exists a minimum dominating set 𝐷 of 𝐶_3𝑘+1□𝐶_𝑛 such that |𝐶^𝑖_3𝑘+1∩𝐷|≥𝑘 for every 𝑖∈𝑉(𝐶_𝑛).

Proof. For a dominating set 𝐷 of 𝐶_3𝑘+1□𝐶_𝑛, we define a function 𝑓 as follows: 𝑓(𝐷)=^𝑛−1_𝑡=0𝑥_𝑡, (1) where 𝑥_𝑡=0,if𝐷∩𝐶^𝑡_3𝑘+1≥𝑘+1,𝐷∩𝐶^𝑡_3𝑘+1,if𝐷∩𝐶^𝑡_3𝑘+1≤𝑘. (2) We now assume that 𝐷 is a minimum dominating set of 𝐶_3𝑘+1□𝐶_𝑛 and suppose to the contrary that |𝐶^𝑖+1_3𝑘+1∩𝐷|<𝑘 for some 𝑖 such that 𝑓(𝐷) is maximized. Then there exists a set 𝑇={(𝑗,𝑖+1),(𝑗+1,𝑖+1),(𝑗+2,𝑖+1)} such that 𝑢∉𝐷 for any 𝑢∈𝑇. Thus, {(𝑗,𝑖),(𝑗+1,𝑖)}⊆𝐷. By Proposition 2, we have 𝑥_𝑡=0 if 0<𝑥_𝑡+1≤𝑘. Thus 𝐷^′=𝐷⧵{(𝑗+1,𝑖)}∪{(𝑗+1,𝑖+1)} is a dominating set with |𝐷|=|𝐷^′| such that 𝑓(𝐷^′)>𝑓(𝐷), a contradiction with the fact that 𝑓(𝐷) is maximized.

Lemma 4. Let 𝑛≥3, 𝑘≥1. Then there exists a minimum dominating set 𝐷 of 𝐶_3𝑘+1□𝐶_𝑛 such that |(𝐶^𝑖_3𝑘+1∪𝐶^𝑖+1_3𝑘+1)∩𝐷|≥2𝑘+1 for every 𝑖∈𝑉(𝐶_𝑛).

Proof. By Lemma 3, it is sufficient to consider a minimum dominating set 𝐷 of 𝐶_3𝑘+1□𝐶_𝑛 such that |𝐶^𝑖_3𝑘+1∩𝐷|≥𝑘 for every 𝑖∈𝑉(𝐶_𝑛). If |𝐶^𝑖_3𝑘+1∩𝐷|=𝑘, together with Proposition 2, we have that |𝐶^𝑖−1_3𝑘+1∩𝐷|≥𝑘+1, which completes the proof.

By Lemmas 3 and 4, there is an obvious lower bound in terms of 𝛾(𝐶_3𝑘+1□𝐶_𝑛) as follows.

Lemma 5. Let 𝑘≥1 and 𝑛≥3 be two integers; then 𝛾𝐶_3𝑘+1□𝐶_𝑛≥(2𝑘+1)𝑛2. (3)

Lemma 6. For two integers  𝑘≥1 and 𝑛≥3, then for any ℓ∈[𝑘], one has(1)𝛾(𝐶_3𝑘+1□𝐶_𝑛)≤⌊(2𝑘+1)𝑛/2⌋+ℓ if 𝑛≡6ℓ(mod6𝑘+2),(2)𝛾(𝐶_3𝑘+1□𝐶_𝑛)≤⌊(2𝑘+1)𝑛/2⌋+𝑘−ℓ if 𝑛≡2+6ℓ(mod6𝑘+2),(3)𝛾(𝐶_3𝑘+1□𝐶_𝑛)≤⌊(2𝑘+1)𝑛/2⌋+𝑘 if 𝑛≡4+6ℓ(mod6𝑘+2),(4)𝛾(𝐶_3𝑘+1□𝐶_𝑛)≤⌊(2𝑘+1)𝑛/2⌋+ℓ+1 if 𝑛≡3+6ℓ(mod6𝑘+2),(5)𝛾(𝐶_3𝑘+1□𝐶_𝑛)≤⌊(2𝑘+1)𝑛/2⌋+𝑘−ℓ if 𝑛≡5+6ℓ(mod6𝑘+2),(6)𝛾(𝐶_3𝑘+1□𝐶_𝑛)≤⌊(2𝑘+1)𝑛/2⌋+𝑘 if 𝑛≡7+6ℓ(mod6𝑘+2),(7)𝛾(𝐶_3𝑘+1□𝐶_𝑛)≤⌊(2𝑘+1)𝑛/2⌋+𝑘 for 𝑛≡1(mod6𝑘+2).

Proof. For any 𝑖∈[𝑛], let 𝐴_𝑖=𝑖3𝑞+2𝑖,𝑖∶0≤𝑞≤𝑘,if𝑛iseven,3𝑞+2+2,𝑖∶0≤𝑞≤𝑘−1,if𝑛isodd. (4) It can be seen that if 𝑖≡0(mod2), |𝐴_𝑖|=𝑘+1; otherwise, |𝐴_𝑖|=𝑘. Let 𝑆=⋃^𝑛−1_𝑖=0𝐴_𝑖; then |𝑆|=⌈(2𝑘+1)𝑛/2⌉ and there are at most 𝑘 vertices, (1,0),(4,0),(7,0),…,(3𝑘−2,0) of 𝐶_3𝑘+1□𝐶_𝑛 of which each is not dominated by 𝑆. Denoted by 𝐵={(1,𝑛−1),(4,𝑛−1),…,(3𝑘−2,𝑛−1)}, we consider the following cases.
(1)  𝑛≡6ℓ(mod6𝑘+2).  In this case, it is easy to see that |𝐴_𝑛−1∩𝐵|=𝑘−ℓ. By adding at most ℓ vertices to 𝑆, we get a dominating set of 𝛾(𝐶_3𝑘+1□𝐶_𝑛) as required.
(2)  𝑛≡2+6ℓ(mod6𝑘+2).  Since |𝐴_𝑛−1∩𝐵|=ℓ, we will obtain a desired dominating set of 𝛾(𝐶_3𝑘+1□𝐶_𝑛) by adding at most 𝑘−ℓ vertices to 𝑆.
(3)  𝑛≡4+6ℓ(mod6𝑘+2). Since |𝐴_𝑛−1∩𝐵|=∅, we have to add at most 𝑘 vertices so that a dominating set of 𝛾(𝐶_3𝑘+1□𝐶_𝑛) is constructed based on 𝑆.
(4)  𝑛≡3+6ℓ(mod6𝑘+2).  Since 𝑛−1 is an even and |𝐴_𝑛−1∩𝐵|=𝑘−ℓ, we have to add at most ℓ vertices to 𝑆 so that a dominating set of 𝛾(𝐶_3𝑘+1□𝐶_𝑛) will be formed.
(5)  𝑛≡5+6ℓ(mod6𝑘+2). Since |𝐴_𝑛−1∩𝐵|=ℓ, we will obtain a dominating set of 𝛾(𝐶_3𝑘+1□𝐶_𝑛) by adding at most 𝑘−ℓ vertices to 𝑆.
(6)  𝑛≡7+6ℓ(mod6𝑘+2).  Let 𝐴^′_𝑛−3={(1+3ℓ+3𝑘,𝑛−3),(2+3ℓ,𝑛−3),(5+3ℓ,𝑛−3),…,(2+3ℓ+3(𝑘−1),𝑛−3)} when 2+3ℓ>2+3ℓ+3𝑘(mod3𝑘+1); otherwise, let 𝐴^′_𝑛−3={(1+3ℓ,𝑛−3), (5+3ℓ,𝑛−3),(8+3ℓ,𝑛−3),…,(2+3ℓ+3𝑘,𝑛−3)} when 2+3ℓ<2+3ℓ+3𝑘(mod3𝑘+1); let 𝐴^′_𝑛−2=𝐶^𝑛−2_3𝑘+1⧵{(1,𝑛−2),(4,𝑛−2),(7,𝑛−2),…,(3𝑘−2,𝑛−2),(1+3ℓ+3𝑘,𝑛−2)} when 2+3ℓ>2+3ℓ+3𝑘(mod3𝑘+1); otherwise, let 𝐴^′_𝑛−2=𝐶^𝑛−2_3𝑘+1⧵{(1,𝑛−2),(4,𝑛−2),(7,𝑛−2),…,(3𝑘−2,𝑛−2),(1+3ℓ,𝑛−2)} when 2+3ℓ<2+3ℓ+3𝑘(mod3𝑘+1); let 𝐴^′_𝑛−1={(1,𝑛−1),(4,𝑛−1),…,(3𝑘−2,𝑛−1)}, 𝐴^′_𝑖=𝐴_𝑖 for 𝑖∈[𝑛−4]. In this way, if we let 𝐷=⋃^𝑛−1_𝑖=0𝐴^′_𝑖, then 𝐷 is a dominating set of 𝐶_3𝑘+1□𝐶_𝑛. In addition, |𝐴^′_𝑛−3|=𝑘+1, |𝐴^′_𝑛−2|=2𝑘, and |𝐴^′_𝑛−1|=𝑘. So, 𝐷=(((2𝑘+1)𝑛)/2)+((2𝑘−1)/2)=⌊((2𝑘+1)𝑛)/2⌋+𝑘.
(7)  𝑛≡1(mod6𝑘+2).  Let 𝐷=∑^𝑛−2_𝑖=0𝐴_𝑖∪{(1+3𝑞,𝑛−1),(3+3𝑞,𝑛−1)∶0≤𝑞≤𝑘−1}. Then 𝐷 is a dominating set of 𝐶_3𝑘+1□𝐶_𝑛 with cardinality ⌊(2𝑘+1)𝑛/2⌋+𝑘.

The result of Lemmas 5 and 6 show the following.

Corollary 7. Let 𝑘≥1 and 𝑛≥3 be two integers; then (2𝑘+1)𝑛2≤𝛾𝐶_3𝑘+1□𝐶_𝑛≤(2𝑘+1)𝑛2+𝑘. (5)

3. On the Domination of 𝐶₇□𝐶_𝑛

Lemma 8. Let 𝑘≥1 and 𝑛≥3 be two integers. Suppose 𝐷 is a minimum dominating set of 𝐶_3𝑘+1□𝐶_𝑛. Let 𝑆_𝑖=𝐷∩𝐶^𝑖_3𝑘+1 and 𝑎_𝑖=|𝑆_𝑖| for 𝑖=0,1,…,𝑛−1. (a)If there is an integer 𝑖 such that 𝑎_𝑖+𝑎_𝑖+1=2𝑘+1 (considered modulo 𝑛), then one of {𝑎_𝑖,𝑎_𝑖+1} is 𝑘 and the other is 𝑘+1.(b)If there is an integer 𝑖 such that 𝑎_𝑖=𝑘+1, 𝑎_𝑖+1=𝑘, 𝑎_𝑖+2=𝑘+1, and 𝑎_𝑖+3=𝑘, then for any distinct vertices (𝑗₁,𝑖+1), (𝑗₂,𝑖+1) in 𝑆_𝑖+1 and (𝑗₃,𝑖+3), (𝑗₄,𝑖+3) in 𝑆_𝑖+3, 𝑗₁−𝑗₂≡0(mod3), 𝑗₃−𝑗₄≡0(mod3); for any distinct vertex (𝑗^′₁,𝑖), (𝑗^′₂,𝑖) in 𝑆_𝑖 and (𝑗^′₃,𝑖+2), (𝑗^′₄,𝑖+2) in 𝑆_𝑖+2, |𝑗^′₁−𝑗^′₂|≡0(mod3) or |𝑗^′₁−𝑗^′₂|=1 and |𝑗^′₃−𝑗^′₄|≡0(mod3) or |𝑗^′₃−𝑗^′₄|=1, where 𝑗₁<𝑗₂, 𝑗₃<𝑗₄, 𝑗^′₁<𝑗^′₂, 𝑗^′₃<𝑗^′₄, and they are taken modulo 3𝑘+1.

Proof. It follows immediately that (a) holds by Lemmas 3 and 4. In addition, since 𝑎_𝑖=𝑘+1,  𝑎_𝑖+1=𝑘,  𝑎_𝑖+2=𝑘+1,  and 𝑎_𝑖+3=𝑘, each of vertices of 𝐶^𝑖+1_3𝑘+1 and 𝐶^𝑖+3_3𝑘+1 is dominated by only one vertex; also, only one vertex of 𝐶^𝑖+2_3𝑘+1 is dominated by two vertices. Now we will first show that it is impossible that there are three consecutive vertices in 𝑆_𝑖. Otherwise, suppose that (𝑠,𝑖), (𝑠+1,𝑖), and (𝑠+2,𝑖) (modulo 3𝑘+1) are three consecutive vertices. We can deduce that vertices (𝑠+1,𝑖+2) and (𝑠+2,𝑖+3) are in 𝑆_𝑖+2 and 𝑆_𝑖+3, respectively, so (𝑠+1,𝑖+3) is dominated by both (𝑠+1,𝑖+2) and (𝑠+2,𝑖+3), a contradiction.
Suppose that there are (𝑗₁,𝑖+1), (𝑗₂,𝑖+1) (or (𝑗₃,𝑖+3), (𝑗₄,𝑖+3)) in 𝑆_𝑖+1 (or 𝑆_𝑖+3) such that 𝑗₁−𝑗₂≢0(mod3) (or 𝑗₃−𝑗₄≢0(mod3)). Then it is easy to show that there exist three consecutive vertices ∈𝑆_𝑖 or three consecutive vertices (𝑠₁,𝑖+2), (𝑠₁+1,𝑖+2), and (𝑠₁+2,𝑖+2)∈𝑆_𝑖+2 (modulo 3𝑘+1), thus (𝑠₁,𝑖+1), (𝑠₁+1,𝑖+1), and (𝑠₁+2,𝑖+1) (modulo 3𝑘+1) can only be dominated by 𝑆_𝑖, and (𝑠₁+4,𝑖+1)∈𝑆_𝑖+1 since it is impossible that there are four consecutive vertices in 𝑆_𝑖+3 by 𝑎_𝑖+3=𝑘. So, it is an easy task to prove that either (𝑠₁+3,𝑖+2) is not dominated by 𝐷 or (𝑠₁+4,𝑖+2) is dominated by two vertices, a contradiction. By using the same approach as above, we also can prove the cases of vertices in 𝑆_𝑖 and 𝑆_𝑖+2.

Lemma 9. Let 𝑘≥1 and 𝑛≥3. If 𝐷 is a minimum dominating set of 𝐶_3𝑘+1□𝐶_𝑛 such that (𝑎_𝑖,𝑎_𝑖+1,𝑎_𝑖+2,𝑎_𝑖+3,…,𝑎_𝑖+2ℓ)=(𝑘+1,𝑘,𝑘+1,𝑘,…,𝑘+1), where 𝑆_𝑖=𝐷∩𝐶^𝑖_3𝑘+1 and 𝑎_𝑖=|𝑆_𝑖| for some 𝑖, ℓ, then (𝑗,𝑖+1)∈𝑆_𝑖+1 if and only if (𝑗+𝑡−1,𝑖+2𝑡−1)∈𝑆_{𝑖+2𝑡−1} for any 𝑗∈{0,1,…,3𝑘} and 𝑡∈{1,2,…,ℓ}.

Proof. By Lemma 8 and the symmetrical structure of 𝐶_𝑚□𝐶_𝑛, we assume without loss of generality 𝑆_{𝑖+2ℓ−1}={(3𝑞,𝑖+2ℓ−1)∶0≤𝑞≤𝑘−1}. Since each vertex in {(3𝑞−1,𝑖+2ℓ−1)∶0≤𝑞≤𝑘−1}∪(−2,𝑖+2ℓ−1) is not dominated by 𝐶^{𝑖+2ℓ−1}_3𝑘+1, we have 𝑆_{𝑖+2ℓ−2}={(3𝑞−1,𝑖+2ℓ−2)∶0≤𝑞≤𝑘−1}∪(−2,𝑖+2ℓ−2). By observing that each vertex in 𝑆_{𝑖+2ℓ−2} needs to be dominated, we have the desired set 𝑆_{𝑖+2ℓ−3}. Repeating this process until 𝑆_𝑖+1 is determined, we have the desired result.

Theorem 10. Let 𝑛≥7. Then 𝛾𝐶₇□𝐶_𝑛=5𝑛2,𝑛≡0(mod14),5𝑛2+1,𝑛≡3,4,6,8,10,11(mod14),5𝑛2+2,𝑛≡1,2,5,7,9,12,13(mod14). (6)

Proof. Consider the following.
Case 1.  𝑛≡0(mod14).
The lower bound follows from Lemma 5, and the upper bound follows from Lemma 6 with ℓ=0 in Case 1.
Case 2.  𝑛≡3,4,6,8,10,11(mod14).
By using Lemmas 3 and 4, we obtain that 𝛾(𝐶₇□𝐶_𝑛)=⌊5𝑛/2⌋ if and only if 𝑛≡0(mod14). Therefore, the desired lower bound is established. Upper bound can be obtained by applying Lemma 6 with 𝑘=2 in the following cases:(i)  𝑛≡3(mod14), ℓ=0 in case 4;(ii) 𝑛≡6(mod14), ℓ=1 in case 1;(iii) 𝑛≡8(mod14), ℓ=1 in case 2;(iv) 𝑛≡11(mod14), ℓ=1 in case 5.
We give the upper bound for the case 𝑛≡4, 10(mod14) by constructions.
Let 𝑛=14𝑘+4. The pattern depicted in Figure 2 is a dominating set of 𝐶₇□𝐶₁₈ with 46 vertices. Moreover, the leftmost 14 columns induce a dominating set of 𝐶₇□𝐶₁₄ with 35 vertices. By repeating the leftmost 14 columns for 𝑘 times, we get a dominating set of 𝐶₇□𝐶_14𝑘+4 with 35𝑘+11 vertices. Therefore, 𝛾(𝐶₇□𝐶_14𝑘+4)≤35𝑘+11, and thus, 𝛾(𝐶₇□𝐶_𝑛)≤⌊5𝑛/2⌋+1.
Let 𝑛=14𝑘+10. The pattern depicted in Figure 3 is a dominating set of 𝐶₇□𝐶₂₄ with 61 vertices. Moreover, the leftmost 14 columns induce a dominating set of 𝐶₇□𝐶₁₄ with 35 vertices. By repeating the leftmost 14 columns for 𝑘 times, we get a dominating set of 𝐶₇□𝐶_14𝑘+10 with 35𝑘+26 vertices. Therefore, 𝛾(𝐶₇□𝐶_14𝑘+10)≤35𝑘+26, and thus, 𝛾(𝐶₇□𝐶_𝑛)≤⌊5𝑛/2⌋+1.
Case 3.  𝑛≡1,2,5,7,9,12,13(mod14).
The upper bound follows from Lemma 6. We then show the lower bound. Let 𝑛≡1(mod14). If there exists a minimum dominating set of 𝐶_3𝑘+1□𝐶_𝑛 with cardinality ⌊5𝑛/2⌋+1, then by Lemmas 3 and 4, we have such a dominating set 𝐷 with (𝑎₀,𝑎₁,𝑎₂,𝑎₃,…,𝑎_𝑛−2,𝑎_𝑛−1)=(3,2,3,2,…,2,3), where 𝑆_𝑖=𝐷∩𝐶^𝑖₇ and 𝑎_𝑖=|𝑆_𝑖|. We without loss of generality assume 𝑆_𝑛−2={(0,𝑛−2),(3,𝑛−2)}. Then 𝑆₁={(1,1),(4,1)} and so 𝑆₀={(0,0),(3,0),(6,0)}. We now consider 𝑆_𝑛−2, 𝑆_𝑛−1, and 𝑆₀ with 𝑆_𝑛−2 and 𝑆₀ known. Then there are four vertices in 𝑆_𝑛−1 not dominated by 𝑆_𝑛−2∪𝑆₀, a contradiction. Therefore, 𝛾(𝐶₇□𝐶_𝑛)≥⌊5𝑛/2⌋+2 if 𝑛≡1(mod14). By using the same approach, we can obtain the desired lower bound if 𝑛≡2,5,7,9,12,13(mod14) and the proof is omitted.

Figure 3 

A dominating set of 𝐶7□𝐶24.

4. On the Domination of  𝐶₁₀□𝐶_𝑛

We use the dynamic algorithm to provide the result on 𝛾(𝐶_𝑚□𝐶_𝑛) for 𝑚=10 and 13. We first describe the concept needed to describe our computer checking. The idea is introduced in [16] in a more general framework and extended in [17], but for our purpose the following description will be sufficient.

For a graph 𝐺 and 𝐷⊆𝑉(𝐺), we define a function 𝑓∶𝑉(𝐺)→{0,1} such that 𝑓(𝑣)=0 if 𝑣∉𝐷 and 𝑓(𝑣)=1 if 𝑣∈𝐷. We call such a function the 2-coloring of 𝐺. If for 𝑓(𝑣)=0 there exists a vertex 𝑢∈𝑁⁻_𝐺(𝑣) such that 𝑓(𝑣)=1, then we say 𝑣 is dominated. We can see that this notation is equivalent to the dominating set and we will use it to describe the dynamic algorithm for this problem.

Let 𝐶={𝑢_𝑖𝑢_𝑖+1∶𝑖∈[𝑛]} be a directed cycle in a weighted digraph 𝐺. We say that the sum of the weights of all the edges in 𝐶 to be the weight of 𝐶.

Let 𝑢 be a 2-coloring of 𝐶_𝑚□𝑃₂; 𝑢 is said to be an (𝑎,𝑏)-vertex if the number of ones in the first copy of𝐶_𝑚equals 𝑎 and the second equals 𝑏 (see, e.g., the left side of the vertex in Figure 4 is a (3,2)-vertex). An (𝑎,𝑏)-vertex is said to be redundant if it is dominated and any removal of one vertex in the first column destroys the domination.

Figure 4 

Two vertices of 𝐷7.

Let 𝑇={(𝑎_𝑖,𝑏_𝑖)∶1≤𝑖≤𝑡}. We define a weighted digraph 𝐷^𝑇_𝑚 (when no confusion can arise, the set 𝑇 is omitted and the digraph is denoted by 𝐷_𝑚) as follows.

The vertices of 𝐷^𝑇_𝑚 consist of all the nonredundant 2-colorings of 𝐶_𝑚□𝑃₂. Let 𝑢=𝑢₁𝑢₂ be a vertex of 𝐷^𝑇_𝑚. Then 𝑢₁ and 𝑢₂ represent the 2-colorings of 𝐶_𝑚□𝑃₂ restricted to the first and second copies of 𝐶_𝑚. Let 𝑢 and 𝑣 be two vertices of 𝐷^𝑇_𝑚. Then 𝑢𝑣 denotes the 2-coloring of 𝐶_𝑚□𝑃₃ obtained by applying 𝑢₂ and 𝑣₁ to the consecutive copies of 𝐶_𝑚. For each vertex 𝑢 of 𝐷^𝑇_𝑚, we make an arc from 𝑢 to 𝑣 in 𝐷^𝑇_𝑚 if and only if the following conditions are fulfilled:(i)𝑢₂ equals 𝑣₁,(ii)each of the vertices of 𝐶¹_𝑚∪𝐶²_𝑚 is dominated (by 𝐶⁰_𝑚∪𝐶¹_𝑚),(iii)for each (𝑎,𝑏)-vertex of 𝐷^𝑇_𝑚, (𝑎,𝑏)∈𝑇.

Finally, the weight of 𝑢𝑣 is the number of ones in 𝐶¹_𝑚 of 𝑣.

Let 𝑇={(2,3),(3,2)}. Figure 4 shows two vertices of 𝐷^𝑇₇. We can see that the 2-coloring (0010010) of the second copy of 𝐶₇ of the first vertex equals the first copy of 𝐶₇ of the second vertex. Moreover, the weight of the second copy of 𝐶₇ of the first vertex is 2. It follows that 𝐴(𝐷^𝑇₇) has an arc from the first (left) vertex to the second (right) with weight 2.

Lemma 11. Let 𝑚,𝑛≥3. Then 𝐷_𝑚 possesses a directed cycle of length 𝑛 with weight 𝑘 if and only if 𝐶_𝑚□𝐶_𝑛 contains a minimal dominating set 𝑆 with |𝑆|=𝑘.

Proof. If 𝑆 is a minimal dominating set of 𝐶_𝑚□𝐶_𝑛, then 𝑆 restricted to 𝐶^𝑖_𝑚∪𝐶_𝑚^𝑖+1 is a 2-coloring 𝒞 of 𝐶_𝑚□𝑃₂. Since 𝑆 is minimal, we have that 𝒞 is nonredundant.
If 𝐷_𝑚 possesses a directed cycle of length 𝑛, we concatenate the second cycle 𝐶_𝑚 of each vertex of the cycle and then obtain a dominating set 𝑆 of 𝐶_𝑚□𝐶_𝑛. Since each vertex in the cycle is nonredundant, we have that 𝑆 is minimal.

We define a weighted adjacency matrix 𝐴(𝐷_𝑚) of 𝐷_𝑚 as follows. Let 𝑉(𝐷_𝑚)={𝑣₁,𝑣₂,…,𝑣_𝐾}. Then 𝐴𝐷_{𝑚𝑖,𝑗}=𝑤𝑣_𝑖𝑣_𝑗,𝑣_𝑖𝑣_𝑗∈𝐷_𝑚,+∞,𝑣_𝑖𝑣_𝑗∉𝐷_𝑚, (7) where 𝑤(𝑣_𝑖𝑣_𝑗) is the weight of the edge 𝑣_𝑖𝑣_𝑗.

Let 𝐴=(𝑎)_𝑝×𝑞 (𝑝 rows, 𝑞 columns) and 𝐵=(𝑏)_𝑞×𝑟 (𝑞 rows, 𝑟 columns) be two weighted adjacency matrices. We define 𝐴 multiplied by 𝐵 as follows 𝑎_1,1𝑎_1,2⋯𝑎_1,𝑞𝑎_2,1𝑎_2,2⋯𝑎_2,𝑝𝑎⋮⋮⋮⋮_𝑝,1𝑎_𝑝,2⋯𝑎_𝑝,𝑞×𝑏_1,1𝑏_1,2⋯𝑏_1,𝑟𝑏_2,1𝑏_2,2⋯𝑏_2,𝑟𝑏⋮⋮⋮⋮_𝑞,1𝑏_𝑞,2⋯𝑏_𝑞,𝑟=𝑐_1,1𝑐_1,2⋯𝑐_1,𝑟𝑐_2,1𝑐_2,2⋯𝑐_2,𝑟𝑐⋮⋮⋮⋮_𝑝,1𝑐_𝑝,2⋯𝑐_𝑝,𝑟, (8) where 𝑐_𝑠,𝑡=min_{1≤𝑖≤𝑞}{𝑎_𝑠,𝑖+𝑏_𝑖,𝑡} for 𝑠∈{1,2,…,𝑝} and 𝑡∈{1,2,…,𝑟}.

We denote by 𝐴^𝑛 the 𝑛th power of 𝐴; we have the following result.

Theorem 12. Let 𝑚,𝑛≥3 and 𝐴(𝐷_𝑚) be the weighted adjacency matrix of 𝐷_𝑚 with 𝐾 vertices. Then one has 𝛾𝐶_𝑚□𝐶_𝑛=min𝐴𝐷_𝑚^𝑛_𝑖,𝑖∶1≤𝑖≤𝐾. (9)

Lemma 13. Let 𝑛≥10, 𝛾(𝐶₁₀□𝐶_𝑛)≤⌊7𝑛/2⌋+2, and 𝐷 a dominating set with cardinality ⌊7𝑛/2⌋+2. (a)If 𝑛 is odd, then |𝐷∩(𝐶^𝑖₁₀∪𝐶^𝑖+1₁₀)|≤8 for any 𝑖∈𝑉(𝐶_𝑛).(b)If 𝑛 is even, then |𝐷∩(𝐶^𝑖₁₀∪𝐶^𝑖+1₁₀)|≤9 for any 𝑖∈𝑉(𝐶_𝑛) but at most one integer 𝑖₀ such that |𝐷∩(𝐶^𝑖0₁₀∪𝐶^𝑖0+1₁₀)|=9.

Proof. (a) Suppose that there exists 𝑖₀∈[𝑛−1] such that |𝐷∩(𝐶^𝑖0₁₀∪𝐶^𝑖0+1₁₀)|≥9. We can deduce that 𝐷∩_{[],𝑖𝑗∈𝑛−1𝑗∉0,𝑖0+1}𝑉𝐶^𝑗₁₀≤72()−1𝑛−22. (10) Thus, by the pigeonhole principle it can be seen that there exist some 𝑖∈[𝑛−1] such that |𝐷∩(𝐶^𝑖₁₀∪𝐶^𝑖+1₁₀)|<7, which contradicts with Lemma 4.
(b) The proof is similar to (a).

Theorem 14. Let 𝑛≥10. Then 𝛾𝐶₁₀□𝐶_𝑛=7𝑛2,𝑛≡0(mod20),7𝑛2+1,𝑛≡3,6,14,17(mod20),7𝑛2+2,𝑛≡4,7,8,9,10,11,12,13,16(mod20),7𝑛2+3,𝑛≡1,2,5,15,18,19(mod20). (11)

Proof. When 𝑛≡0,3,6,14,17  (mod20), the result follows from Lemmas 6, 8, and 9. Otherwise, let 𝑇={(3,4),(4,3),(3,5),(5,3),(4,4)}. We implemented the algorithm as described above, and the digraph 𝐷^𝑇₁₀ was created and it has 4905 vertices. By Lemma 13, it can be seen that there may exist at most one (𝑥,𝑦)-vertex such that 𝑥+𝑦=9 in the auxiliary graph 𝐷₁₀. We consider each possible auxiliary graph 𝐷₁₀, whose vertex set is the union of that of 𝐷^𝑇₁₀ and one (𝑥,𝑦)-vertex with 𝑥+𝑦=9. Note that 𝐷₁₀ has only 4906 vertices; the matrix multiplication can be done easily. By using matrix multiplication, we found a formula as stated in Theorem 14. (Such parameters 𝑎 and 𝑏 exist; see [17].) If 𝛾(𝐶₁₀□𝐶_𝑛) (for some 𝑛) is greater than ⌊7𝑛/2⌋+2, then 𝛾(𝐶₁₀□𝐶_𝑛)=⌊7𝑛/2⌋+3.

Remark 15. Note that when Lemmas 4 and 13 are applied, we can operate matrix multiplication on the graph 𝐷^𝑇1₁₀ whose row number is 4906. It can be clearly seen that |𝐷∩(𝐶^𝑖₁₀∪𝐶^𝑖+1₁₀)|≤9 for any 𝑖∈𝑉(𝐶_𝑛) with the condition of Lemma 13, and it is natural to consider the graph 𝐷^𝑇2₁₀, where 𝑇₂=𝑇∪{(3,6),(4,5),(5,4),(6,3)}. However, 𝐷^𝑇2₁₀ has 24845 vertices, whose vertex number is much more than that of 𝐷₁₀. Therefore, Lemmas 4 and 13 play the key role in computing the exact values of 𝛾(𝐶₁₀□𝐶_𝑛), and this helps to make many hard instances become easy to solve.

5. Bounds on the Domination Number of  𝐶₁₃□𝐶_𝑛

We will conclude this work with upper bounds for the domination number of 𝐶₁₃□𝐶_𝑛.

Theorem 16. Let 𝑛≥13. Then 𝛾𝐶₁₃□𝐶_𝑛≤9𝑛2,𝑛≡0(mod26),9𝑛2+1,𝑛≡3,6,20,23(mod26),9𝑛2+2,𝑛≡9,10,12,14,16,17(mod26),9𝑛2+3,𝑛≡4,7,8,11,13,15,18,19,22(mod26),9𝑛2+4,𝑛≡1,2,5,21,24,25(mod26). (12)

Proof. We show the desired weight for all cases in Table 1, where 𝑟≡𝑛(mod26) and (𝑐,ℓ) stands for case (𝑐) with ℓ in Lemma 6. For example, the result of the case 𝑛≡0(mod26) follows from case 1 with ℓ=0 in Lemma 6.
Also, patterns 𝐴--𝐺with 13 rows are presented (see Figures 5 and 6). By repeating the pattern of 𝐴 for 𝑘 times and concatenating 𝐻∈{𝐵,𝐶,𝐷,𝐸,𝐹,𝐺}, we will obtain a pattern 𝐴^𝑘𝐻 of 𝐶₁₃□𝐶_𝑛, where 𝑛=26𝑘+𝑐(𝐻). From the above, it can be seen that all the desired upper bounds are established.

Table 1 

Constructions for upper bounds of 𝐶₁₃□𝐶_𝑛.

Figure 5 

𝐴𝐵: dominating set of 𝐶13□𝐶30; 𝐴𝐶: dominating set of 𝐶13□𝐶33; 𝐴𝐷: dominating set of 𝐶13□𝐶36.

Figure 6 

𝐴𝐸: dominating set of 𝐶13□𝐶42; 𝐴𝐹: dominating set of 𝐶13□𝐶45; 𝐴𝐺: dominating set of 𝐶13□𝐶48.

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Acknowledgments

The authors would like to acknowledge the referees for their valuable comments and suggestions. This work was supported by the National Natural Science Foundation of China under Grants nos. 61309015 and 61173121.