Abstract
Domination is a structural complexity of chemical molecular graphs. A dominating set in a (molecular) graph is a subset such that each vertex in is adjacent to at least one vertex in . The domination number of a graph is the minimum size of a dominating set in . In this paper, computeraided approaches for obtaining bounds for domination number on torus graphs are here considered, and many new exact values and bounds are obtained.
1. Introduction
The structure of molecular graphs is a broad area among chemistry, biology, and mathematics. In particular, domination number or dominating complexity has attracted considerable attention in the general case [1–7], like coding theory and combinatorial design. Due to a variety of applications, it has been studied on various types of graphs such as generalized Petersen graphs [8–11], hypercubes [12, 13], Fibonacci cubes [14], Kneser graphs [15–18], torus graphs [19–21], and grid graphs [22, 23]. Others are referred to [24–26].
In this work, we consider certain graphs without loops or multiple edges. A dominating set in a graph is a subset such that each vertex in is adjacent to at least one vertex in . The domination number of a graph is the minimum cardinality of a dominating set in . A dominating set of a graph is a set if it has cardinality . A dominating set is an independent dominating set, if no two vertices in are adjacent. The independent domination number of a graph is the minimum cardinality of an independent dominating set in . For , the cardinality of a minimum dominating set of containing is denoted by , that is,
For two graphs and , the Cartesian product is a graph with vertex set and two vertices (,) and (,) are adjacent if and only if either and or and . The gird is , the cylinder for is , the torus for and is , and the 3D torus for given , , is . The subgraph of induced by is isomorphic to . It is called a Gfiber and is denoted by .
The domination number has attracted considerable attention in the general case [1, 2]. Due to a variety of applications, it has been studied on various types of graphs such as generalized Petersen graphs [8–11], hypercubes [12, 13], Fibonacci cubes [14], Kneser graphs [15, 16, 18], torus graphs [19–21], and grid graphs [22, 23]. Others are referred to [24–26].
Dominating sets in some special graphs have applied in coding theory and combinatorial design. For instance, there is a natural relation between dominating sets of torus graphs and covering codes in the Lee metric [21], and dominating sets in a type of Kneser graphs are also related to Steiner system [18].
The following result is a special case of Theorem 3.3 in [19]:
Theorem 1.
Let be the automorphism group of graph . We say two subsets and are equivalent if there exists a mapping such that . For a subset , the orbit of is the family .
2. The Approaches
2.1. ILPBased Search
Let be a graph and be a dominating set of . We use a Boolean variable to denote if a vertex is in , that is, if .
The following is the wellknown integer linear programming model for finding a minimum dominating set of :
ILPDominatingSet
2.2. Extending Dominating Sets from Inequivalent Seeds
Sometimes, we can succeed to obtain the independent domination number of a graph in a reasonable time but it fails to exhaustively search the domination number . In such cases, based on Observation 1, we can obtain first to assume there are two adjacent vertices in a minimum dominating set.
Observation 1. Let be a graph. If there is a minimum dominating set such that , then contains two adjacent vertices.
Observation 2. Let be a graph and be a family of inequivalent subsets of . If there is a minimum dominating set of containing a subset isomorphic to one element in , then .
Based on the above observations, we first select a family of inequivalent subsets of the vertex set of a considered graph then extend each subset to a minimum dominating set, that is, we compute . We apply the following procedure to compute the domination number of (using a 2.66 GHz Intel Core (TM) i75600U CPU).

2.3. Numerical Results
In this section, we determine the domination number of five torus.
Theorem 2.
Proof. The upper bound follows from the dominating set depicted in Figure 1.
Next, we show that . Consider a family of inequivalent subset where
We run Procedure MinDomination to extend each for a minimum dominating set and determine for each . It can be seen that for each . The results are presented in Table 1.
Moreover, we obtain that in only 2774 seconds by running MinDomination. From the above computational results, we have .
Theorem 3.
Proof. The upper bound follows from the dominating set depicted in Figure 2.
From the computational results above, we have .
Theorem 4.
Proof. The upper bound follows from the dominating set depicted in Figure 3.
Next, we show that . First, it takes 12520 seconds to obtain using Gurobi optimizer. Since , we deduce that any minimum dominating set has two adjacent vertices. Now, we may assume w.l.o.g. that are in a minimum dominating set. We consider the following inequivalent family , where
We run Procedure MinDomination to extend each for a minimum dominating set and determine for each . It can be seen that for each , and the results are presented in Table 2.
Moreover, we obtain that in 16753 seconds by running MinDomination. From the above computational results, we have .
Theorem 5.
Proof. The upper bound follows from the dominating set depicted in Figure 4.
Next, we show that . Suppose, to the contrary, that .
First, it takes 18974 seconds to obtain using Gurobi optimizer. By assumption, . By Observation 1, we conclude that any minimum dominating set has two adjacent vertices. Now, we may assume w.l.o.g. that are in a minimum dominating set. We consider the inequivalent family defined in the proof of Theorem 4. We run Procedure MinDomination to extend each for a minimum dominating set and determine for each . It can be seen that for each , and the results are presented in Table 3.
Moreover, we obtain that in 23764 seconds by running MinDomination. From the above computational results, we have , contradicting with the initial assumption that . Thus, .
Theorem 6.
Proof. The upper bound follows from the dominating set depicted in Figure 5.
Next, we show that . Suppose, to the contrary, that .
First, it takes 938743 seconds to obtain using Gurobi optimizer. By assumption, . By Observation 1, we conclude that any minimum dominating set has two adjacent vertices. Now, we may assume w.l.o.g. that are in a minimum dominating set. We consider the inequivalent family defined in the proof of Theorem 4. We run Procedure MinDomination to extend each for a minimum dominating set with a time limit 200000 seconds. Although we failed to determine the exact values of within 200000 seconds, we are able to confirm that for each . This suffices to confirm the lower bound 144, and the results are presented in Table 4.
Moreover, we obtain that in 465326 seconds by running MinDomination. From the above computational results, we have , contradicting with the initial assumption that . Thus, .
By Theorem 1, we have that if (mod 5), then . Inspired by the result in [21] and the above computational results, the following conjecture is proposed.
Conjecture 1. Let . Then, we have (i)if (mod 5), then ,(ii)if (mod 5), then .
3. Results and Discussions: Domination Number of 3D Torus
In this section, we determine the domination number of some 3D torus , where , , . Denote the vertices in by for and . For any set , let for.
Theorem 7. For , .
Proof. It is easy to verify that for . Let and be a set such that is as small as possible. First, we show that for each . Suppose, to the contrary, that for some , say . To dominate the vertices of , we must have . Clearly, the set
is a dominating set of of size less than which is a contradiction. Thus, for each .
Next, we show that if for some , then and . Let for some , say . Assume without loss of generality that . If , then to dominate the vertices for , we must have and so for .
If (the cases , , and are similar), then to dominate the vertices , we must have and . If (the cases , , and are similar), then to dominate the vertices , we must have for .
Let for each and define on by
Clearly, and for each . It follows that
yielding . Since is integer, we obtain the desired bound.
Theorem 8. Let be an integer. Then(i) if or (mod 4),(ii) if (mod 4) and .
Proof. Let and (i)If (mod 4), then is a dominating set of of size , if (mod 4), then is a dominating set of with cardinality , and if (mod 4), then is a dominating set of G of size . If , then set is a dominating set of with cardinality 9. Therefore, we have when or (mod 4).(ii)If (mod 4), then the set is a dominating set of with cardinality . Therefore, we have if for .
Remark 1. We have succeed to compute all the exact values of for and only if .
Theorem 9. For , .
Proof. Let . First, we show that . Let
If (mod 3), then is a dominating set of with cardinality . If (mod 3), then is a dominating set of with cardinality . If (mod 3), then is a dominating set of with cardinality . Therefore, we have .
Now, we show that . It is not hard to see that for . Assume and let S be a set. Denote the vertices in by for and . First, we show that for each . Suppose, to the contrary, that for some , say . To dominate the vertices of , we have . Clearly, the set
is a dominating set of of size less than which is a contradiction. Thus, for each . Next, we show that there is no such that . Suppose, to the contrary, that for some , say . We may assume without a loss of generality that and and that . To dominate the vertices of , we must have . Now, the set
is a set contradicting the choice of . Hence, for each . Thus,
and this completes the proof.
Applying analogous approaches described in Section 2, we can obtain substantial results on domination of multidimensional torus. The exact values of some 3D torus are presented in Table 5. Let Note that when , the domination of corresponds to the classic ternary covering codes of length . By the results of [27], it is clear that , , and . Using the above approaches, we obtained some new bounds and exact values of . The results are presented in Table 6.
Moreover, some dominating sets corresponding to upper bounds are also listed below. Inspired by the computational results, the following conjecture is proposed.
Conjecture 2. (i)For , ,(ii)For , , .
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that there is no conflict of interest regarding the publication of this paper.
Acknowledgments
This work is supported by the National Key Research and Development Program under Grant 2017YFB0802300, the National Natural Science Foundation of China under the Grant 11361008, and the Applied Basic Research (Key Project) of Sichuan Province under Grant 2017JY0095.