Abstract
A subset of is called a total dominating set of a graph if every vertex in is adjacent to a vertex in . The total domination number of a graph denoted by is the minimum cardinality of a total dominating set in . The maximum order of a partition of into total dominating sets of is called the total domatic number of and is denoted by . Domination in graphs has applications to several fields. Domination arises in facility location problems, where the number of facilities (e.g., hospitals and fire stations) is fixed, and one attempts to minimize the distance that a person needs to travel to get to the closest facility. In this paper, the numerical invariants concerning the total domination are studied for generalized Petersen graphs.
1. Introduction
Graph theory has been used to study various concepts of partition of the vertex set for the graph [1–3]. The domination (total domination) problem for the graph was studied from 1950s onward, and domination (total domination) in graphs is said to be NP-complete problem [4]. Domination (total domination) sets are of practical interest in several areas of mathematics and other branches of science [5, 6]. In wireless networks, dominating (total dominating) sets are used to find efficient routes within ad hoc mobile networks. In documents’ summarization, domination (total domination) sets are used. Dominating (total dominating) sets are closely related to independent sets. Total dominating set is independent if and only if it is a maximal input set [7].
Moreover, in computer science, weighted graphs are widely used in the developments of data mining, software testing, image processing, communication networks, and information security [8–11].
In this paper, the numerical invariants concerning the total domination are studied for generalized Petersen graphs [12–16].
2. Preliminaries
Let be a simple, undirected, and finite graph with the vertex set and the edge set .
The (open) neighborhood of a vertex , denoted by , is the set of all vertices adjacent to , i.e., . The closed neighborhood of is defined as [17–19]. For a set , its open neighborhood is the set , and its closed neighborhood is the set . A dominating set of a graph , abbreviated as , is a set of vertices in such that every vertex in is adjacent to a vertex in . The domination number of a graph , denoted by , is the minimum cardinality of a dominating set. A total dominating set of a graph , abbreviated as , is a set of vertices in such that every vertex in is adjacent to a vertex in . If no proper subset of is a of , then is a minimal of . Every graph without isolated vertices has a . The total domination number of a graph , denoted by , is the minimum cardinality of a . A of of minimum cardinality is called a -set.
The concept of total domination was introduced by Cockayne et al. [20]. For more information, see [21, 22].
Let be the total number of -sets. For each vertex , we define the total domination value of , denoted by , to be the number of -sets to which belongs. Evidently, for any and any .
Generalized Petersen graphs are important classes of commonly used interconnection networks and have been intensively studied. The generalized Petersen graph , and , consists of an outer -cycle , a set of spokes , , and edges , , with indices taken as modulo , see Figure 1. Thus, its vertex set is the union of and . We say that the first set consists of vertices and the second one of vertices. By a -path in , we mean a path whose vertices consist of just -vertices. A -path is defined similarly. The generalized Petersen graph is shown in Figure 1.
Investigation of the total number of total domatic sets and the total number of total dominating sets of the generalized Petersen graph is the sole aim and interest of writing this paper. Cao et al. in [23] studied the total domination number of generalized Petersen graphs and obtained the exact value of . For , they proved
In this paper, we determine the exact values for when and and when . We also investigate total number of total dominating sets for these cases.
3. Main Results
The following result provides a trivial lower bound on the total domination number of a graph in terms of the maximum degree of the graph.
Theorem 1. (see [24]). If is a graph with no isolated vertex and maximum degree , then
Theorem 2. Let be a generalized Petersen graph with , . If , then
Proof. Let . According to Theorem 1, we getThus, to prove the equality, we just need to find a total dominating set of cardinality . Let us consider a setThe vertex , , dominates vertices , , and , and the vertex , , dominates vertices , , and with indices taken as modulo . It is easy to see that, as , all vertices in are dominated. Thus, is a total dominating set, and . This concludes the proof.
Now, we proceed for .
Theorem 3. Let be a generalized Petersen graph; then,
Proof. Let us consider three cases: Case 1: when , , the result follows from Theorem 2. Case 2: when , , without loss of generality, consider a set . The vertex , , dominates vertices , , and , and the vertex , , dominates vertices , , and with indices taken as modulo . Thus, the vertices from this set dominate all but two vertices in . The two vertices which are not dominated are and . To totally dominate and , we add their neighbors and to because these two vertices are already adjacent vertices in . This means that the set totally dominates all vertices in . Thus, . For necessity, we need to prove the reverse inequality, i.e., . The proof will be by mathematical induction. For , the result is obvious. Since the result is true for all of the number of vertices less than when , for , let be any total dominating set of . Then, , where is the total dominating set of if only , and we get . Thus, we proved our required assertion. Case 3: when , , without loss of generality, consider a set . The vertex , , dominates vertices , , and , and the vertex , , dominates vertices , , and with indices taken as modulo . Thus, the vertices from this set dominate all but two vertices in . The two vertices which are not dominated are and . To totally dominate and , we add these two vertices to because these two vertices are adjacent. This means that the set totally dominates all vertices in . Thus, .For necessity, we need to prove the reverse inequality, i.e., . The proof will be by mathematical induction.
Assume the result is true for all of the number of vertices less than . For , let be any total dominating set of . Then, , where is the total dominating set of if only , and then , i.e., .
Theorem 4. Let be a generalized Petersen graph. Then, if (mod ) and .
Proof. Let be the total domatic partitions since no vertex of any partition is contiguous to any other partitions and all the three domatic partitions totally dominate . Also, we know that . We proved in Theorem 6 in the following that if only (mod 3). It is observed that there is no recurrence in the total number of total dominating sets. Hence, we can write if only (mod ).
Henning and Yeo [24] proved the following.
Theorem 5. (see [24]). For any connected graph ,
Theorem 6. Let and be positive integers. If , thenwhere for , must be odd.
Proof. Let be a positive integer, . Case 1: when , , then , and a -set comprises ’s, and is fixed by the choice of the first . There is exactly one -set comprising vertices and . Thus, . Similarly, . Using Theorem 5, we get Hence, in this case. Case 2: when or , then a -set is composed of in exactly one of the following ways:(1) comprises ’s and one (2) comprises ’s and two ’s(3) comprises ’s and one (4) comprises ’s and two ’s(5) comprises ’sNow, we discuss these subcases.(i)Subcase 1: when , note that is fixed by the choice of single . Selecting is the same as selecting its initial vertex in the counterclockwise order; thus, .(ii)Subcase 2: when , the set is fixed by the choice of two ’s, and there are options for choosing the first two ’s; thus, .(iii)Subcase 3: when , there is only one chance for choosing single and chances for choosing . Each vertex has equal number of chances. Each vertex is equally repeated in the total number of total dominating sets three times; thus, the total number of total dominating sets will be .(iv)Subcase 4: when , then is fixed by the choice of double . Each vertex is repeated three times in the total number of dominating sets, and for each vertex, there are equal number of chances for total domination; hence, each will be repeated times.(v)Subcase 5: when , there are exactly two vertices in each . Each vertex of dominates three vertices including the vertex itself because this is a 3-regular graph. Thus, there are number of choices for the total number of dominating sets; thus, in all cases, .
Corollary 1. Let be a generalized Petersen graph; then,
Proof. The proof is similar to the proof of Theorem 3.
Corollary 2. Let be a generalized Petersen graph; then,
Proof. The proof is similar to the proof of Theorem 3.
4. Total Domatic Number of a Graph
The total domatic number of a graph is denoted by and is defined as , where and are the total dominating sets and are the maximum disjoint partition of the vertex set of the graph.
Proposition 1. For any simple graph with minimum degree , the total domatic number .
In the following result, our proof is simple from the proof of [25].
Theorem 7. (see [25]). For a graph of order and minimum degree , the lower bound for the total domatic number , where .
Proof. Let be the total dominating set of the graph and be the complement of . If is the minimum degree of , then is the maximum degree of , and due to this, any vertex of the graph is adjacent to at most vertices of in . If the vertex , then a vertex of the graph exists which is not adjacent to in but ; this confirms that , and hence, .
Corollary 3. For generalized Petersen graph of order and , we have .
5. Open Problems
We conclude the paper with the following open problems for further investigation.
Open Problem 1. Characterize for all feasible values of and .
Open Problem 2. Characterize for each .
5.1. Discussion
In this paper, the generalized Petersen graph is investigated for the total domination number , total number of dominating sets , and total domatic number . Here, we found the total dominating number for some restricted values of . For detailed applications of our results, refer [26–28].
Data Availability
All data required for this paper are included within this paper.
Conflicts of Interest
The authors do not have any conflicts of interest.
Authors’ Contributions
Taiyin Zhao gave applications of results and improved the presentation of the paper, Gohar Ali supervised this work, Nabila Hameed proved the results, Syed Inayat Ali Shah wrote the first version of the paper, and Yu-Ming Chu wrote the final version of the paper and arranged funding for this paper.
Acknowledgments
This research was supported by the National Natural Science Foundation of China (Grant nos. 11971142, 11871202, 61673169, 11701176, 11626101, and 11601485).