Abstract
Domination is a well-known graph theoretic concept due to its significant real-world applications in several domains, such as design and communication network analysis, coding theory, and optimization. For a connected graph , a subset of is called a dominating set if every member present in is adjacent to at least one member in . The domatic partition is the partition of the vertices into the disjoint dominating set. The domatic number of the graph is the maximum cardinality of the disjoint dominating sets. In this paper, we improved the results for the middle and central graphs of a cycle, respectively. Furthermore, we discuss the domatic number for some other cycle-related graphs and graphs of convex polytopes.
1. Introduction
The paper considers connected simple, nontrivial, and undirected graphs. For the notations and terminologies in graph theory, we follow the book by Diestel [1]. Let be a graph with the vertex set and the edge set . The neighborhood of a vertex , denoted by , is defined as the set . The degree (valency) of a vertex is denoted by . If the text is clear and there is no ambiguity, we omit the subscript . The minimum and maximum degrees of a vertex in are represented by and , respectively, where and .
The scarcity of energy, the most critical resource, is one of the key aspects of wireless ad hoc and sensor networks. Sensor nodes are typically battery-powered and can only be active for a short time before the battery runs out. Improved network lifetime (i.e., the time interval during which the network is capable of executing its intended activity) is an essential aspect of the system design in sensor networks, and the techniques for energy-saving are highly desirable.
One of the most well-known strategies is to plan vertex (node) activity so that redundant vertices (nodes) can enter some sleep mode as frequently and for as long as possible. Vertices cannot send or receive messages while in this sleeping mode, although they consume hardly any energy. The amount of energy used in active mode when the CPU is working with total capacity is often orders of magnitude more than in sleep mode.
Clustering the network, likewise, data collection has shown to be one of the most effective strategies when working with the complexity of sensor and wireless ad hoc networks. Clustering realizes spatial multiplexing in nonoverlapping clusters and enhances the use of scarce resources such as energy and bandwidth. Different types of clustering have been proposed in recent years, depending on the unique network organization problem under investigation. One popular technique is to select cluster heads so that each vertex is either a cluster head or has at least one cluster head in its general neighborhood. This type of clustering relates to the well-known dominating set problem when the given network is represented as a graph . A set is the dominating set of the graph if occupies some vertex in the set . The minimum cardinality of the dominating set of is called the domination (dominating) number. It is denoted as .
Graph theory has attracted the attention of researchers from almost all domains, and the notion of domination possesses an essential role in graph theory in terms of the applicability of domination and its related parameters in various fields, see Haynes et al. [2, 3] for comprehensive treatment and some detailed surveys on (earlier) domination results in graphs. Node failure is a nonnegligible probability event in sensor and ad hoc wireless networks, one of their essential characteristics. A basic dominating set may not be the ideal type of clustering for applications where fault tolerance is crucial. The concept of a -dominating set accounts for this increased fault tolerance by requiring that every vertex in its proximity has at least cluster heads. A -dominating set is a subset of in which each vertex has at least dominators in its immediate proximity (including itself) in .
The concept adopted in this paper is that, by having many disjoint dominating sets, we may activate them sequentially. For instance, the only vertices in the active set take over information-collecting tasks. In contrast, the rest of the vertices go into low-energy sleep mode for several plane networks. The maximum domatic partition problem is a classical graph-theoretic problem in which the main objective is to determine the maximum number of disjoint dominating sets. The domatic partition of the graph is the partition of into a disjoint dominating set of . The maximum cardinality of the disjoint dominating set of is called the domatic number of the graph . The domatic number can be denoted as . To elaborate this point, a domatic partition of the graph is the partition into a pairwise disjoint dominating set of the graph . The domatic number can be considered in the following way as well. The domatic number of the graph is the maximum number of classes of a partition of such that each class is a dominating set of the graph . In particular, the domatic problem is analogous to allocating resources in the computer network and also locating facilities in a communication network.
Cockayne and Hedetniemi [4] were the first to establish the concept of domatic partition. This concept later proved to be quite beneficial in numerous situations, for instance, locating facilities in a network by Fujita et al. [14], cluster head rotation in sensor networks by Misra and Mandal [5], prolonging the life and conserving the energy of networks by Islam et al. [6] and Pemmaraju and Pirwani [7], respectively.
The following result is due to Cockayne and Hedetniemi [4].
Theorem 1. For any graph of order , we have
The graphs, which satisfy equation (1), are known as domatically full graphs.
The paper is organized as follows. In Section 2, the improved results are depicted for the middle and central graphs of cycles, respectively. In Section 3, we consider some more families of cycle-related graphs and obtain their domatic numbers. In Section 4, graphs of convex polytopes are considered for the domatic number. The improved result for the middle graph of a cycle is then utilized to study domatic numbers in some classes of convex polytopes.
2. Improved Results
The result for the middle graph of cycle (where represents the cycle graph on vertices) was proved in [8], and is.
Theorem 2. [8] Let and . Then,(i) for (ii) for and The improved result is shown for the middle graph of cycle in the following section.
2.1. Middle Graph of Cycle
Let be the cycle graph on vertices. The vertex set . When the definition of the middle graph of cycle is employed, then the set with the newly added vertices is as follows:
The corresponding edge set for the middle graph of cycle is given as follows:
The order and size of the graph are and , respectively. For the sake of simplicity, the the vertex set are called the inner vertices, and are called the outer vertices. The graph of is shown in Figure 1. Moreover, where necessary .
Theorem 3. The domatic number for the middle graph of cycle , when , is
Proof. In order to show this, we consider the following three cases:
Case 1. When and , the partition of set of into disjoint dominating sets is shown as follows.
Let . . .
Case 2. When and , the partition of set of into disjoint dominating set is shown as follows.
Let . .
Case 3. When and , the partition of set in into disjoint dominating set is shown as follows.
Let . . . It is not hard to observe that when , . Moreover, when , we have . Furthermore, to show that the classes of the domatic partition set are also the dominating sets, Table 1 is exhibited.
It is quite clear from Table 1, and the classes of the domatic partition set are also dominating.
Observation 1. Let be the graph of middle graph of cycle , then is domatically full for and .
Example 1. For better understanding, the case for is discussed graphically. The graph of is exhibited, which shows that the . Let us consider a partition dominating set . Now, let us consider the classes as , , and vertices, respectively. Now, it is obvious from Figure 2 that . Furthermore, the classes of disjoint dominating set are dominating.
The improved result is shown for the central graph of cycle in the following section.
2.2. Central Cycle Graph
This graph is constructed by considering the graph of a cycle on vertices. Then, be the vertex set. The corresponding edge set is . From the definition of central graph [8], the vertex set of central graph of cycle . In the central graph of cycle, it is imperative to note that the vertex is not adjacent to the vertices and for .
The result for the central graph of cycle (where represents the cycle graph on vertices) was proved in [8], and is.
Theorem 4 (see [8]). For any ,(i) and(ii)
Proof. We first examine the statement of Theorem 4, before improved result is presented.(i), holds for even , but not for odd (ii) , holds for odd , but not for even As an illustration, we show graphically in Figure 3, when is even , but . Let us consider a domating partition set . Now, let us consider the classes as , , and vertices, respectively. Thus, , and this is the maximum disjoint dominating sets, and these classes of domatic partition show domination.
An improved result is shown for the central cycle graph .
Theorem 5. The domatic number for the central cycle graph when is
Proof. The proof is divided into six cases as illustrated.
Case 4. When and , the partition of the set of into disjoint dominating set is shown as follows.
Let . . .
Case 5. When and , the partition of the set of into disjoint dominating set is shown as follows.
Let .
Case 6. When and , the partition of the set of into disjoint dominating set is shown as follows.
Let .
Case 7. When and , the partition of the set of into disjoint dominating set is shown as follows.
Let .
Case 8. When and , the partition of the set of into disjoint dominating set is shown as follows.
Let ,
Case 9. When and , the partition of the set of into disjoint dominating set is shown as follows.
Let . .
It is not difficult to observe that when , , and when , , respectively.
Observation 2. Let be the graph of central graph of cycle , then is domatically full for .
3. Cycle-Related Graphs
3.1. Wheel Graphs
The domatic number for cycle graphs is recently studied by Mary and Amutha [8] and is obtained as follows.
Theorem 6. For ,
Next, it is quite natural to study the notion of domatic number for wheel graphs. The wheel graph, denoted by , consists of a cycle , which is called the , and a vertex is called the hub (or universal vertex), which is joined with all the vertices of . The wheel graph has order , and the universal vertex is denoted as , and the rim as .
Theorem 7. The domatic number for the wheel graph , when , is
Proof. In order to show this, we consider the following three cases.
Case 10. When and , the partition of the set of into disjoint dominating set is shown as follows.
Let . . . .
Case 11. When and , the partition of the set of into disjoint dominating set is shown as follows.
Let . . .
Case 12. When and , the partition of the set of into disjoint dominating set is shown as follows.
Let . . .
It is not difficult to observe that when , and when , , respectively.
Observation 3. Let be the wheel graph , then is domatically full for .
3.2. Squared Cycle Graph
The squared cycle graphs are the extension of the cycle graphs. The squared cycle graph, denoted by , is of order . The vertex set for simplicity can be written as . The graph of is shown in Figure 4. For each , we join to and to . Furthermore, if these vertices are arranged cyclically , then each vertex is adjacent to the vertices which follows that and vertices that immediately precede . Thus, is a four regular graph.
The graph of is the complete graph so , , , , .
Theorem 8. The domatic number for the squared cycle graph , for , is
Proof. The proof of the theorem is split into the following five cases.
Case 13. Let and . The partition of the set of into disjoint dominating set is shown as follows.
Let . . . . .
Case . 14. Let and . The partition of the set of into disjoint dominating set is shown as follows.
Let . . .
Case 15. Let and . The partition of the set of into disjoint dominating set is shown as follows.
Let . . .
Case 16. Let and . The partition of the set of into disjoint dominating set is shown as follows.
Let . . .
Case 17. Let and . The partition of the set of into disjoint dominating set is shown as follows.
Let . . . .
In all the cases considered for the squared cycle graph, there exist disjoint dominating of different cardinalities. Furthermore, it is not hard to analyze that all these mentioned partitions are dominating set as well.
Observation 4. The minimum degree for the squared cycle graph is . So for the case of , , it is domatically full graph.
4. Convex Polytopes
A convex polytope is a polytope that is also a convex set of points in the -dimensional Euclidean space . The graph of a convex polytope is constructed by the vertices and edges with the same incidence relation. The graphs of convex polytopes were put forward by Bača [9, 10]. The author studied graceful and antigraceful labeling problems for these geometrically essential graphs. Recently, the graphs of convex polytopes are considered for the domination-related parameters. The binary locating domination number (resp. locating domination number) is studied by Simić et al. [11]. The open locating domination number in graphs of convex polytopes was studied by Savić et al. and Raza in [12, 13].
4.1. The Graph of Convex Polytopes
The graph of convex polytope defined in [10] consists of -sided faces as shown in Figure 5. From the mathematical point of view, the vertex and edge set for are as follows.
Vertex set of isand the edge set of is
In order to validate the vertex and edge set of graph , we fix and draw the graph as shown in Figure 6.
Theorem 9. The domatic number for the graph of convex polytope , when , is
Proof. The proof is divided into three case0073 as illustrated.
Case 18. When and . The partition of the set of into disjoint dominating set is shown as follows.
Let , , and .
Case 19. When and , the partition of the set of into disjoint dominating set is shown as follows.
Let and .
Case 20. When and , the partition of the set of into disjoint dominating set is shown as follows.
Let and .
It is not hard to see that the classes of the domatic partition for , and for , respectively. To show that these are dominating sets as well, Table 2 is exhibited.
The family of convex polytope can be attained by expending the above-studied family of convex polytope . The graph of a convex polytope consists of 2 5-sided faces, 6-sided faces, and -sided faces, as shown in Figure 7. The domatic number for the graph of convex polytope is the same as the graph of convex polytope .
Theorem 10. The domatic number of graph of convex polytope , for , is
Proof. The proof is similar as shown for the graph of convex polytope .
To validate the result in Theorem 3 for the middle graph of a cycle, the graph of convex polytope is considered, which is generated from the middle graph of a cycle. There are infinite classes of convex polytopes that show the same structural similarity.
4.2. The Graph of Convex Polytope
The graph of convex polytope , as shown in Figure 8, consists of 4 3-sided faces, 4-sided faces, and a pair of -sided faces. The vertex and edge set for can be written as follows:
Theorem 11. The domatic number of graph of convex polytope , when , is
Proof. In order to show this, we consider the following three cases.
Case 21. When and , the partition of the set of into disjoint dominating set is shown as follows.
Let , , and .
Case 22. When and , the partition of the set of into disjoint dominating set is shown as follows.
Let and .
Case 23. When and , the partition of the set of into disjoint dominating set is shown as follows.
LetIt is not hard to observe that when , . Moreover, when , we have . Furthermore, to show that classes of the domatic partition set are also dominating set, Table 3 is presented which clearly indicates that the chosen classes are dominating.
5. Conclusion
This article first improves the results for the domatic number of a middle graph and central graphs of a cycle graph. Next, the domatic number of squared cycle graphs and wheel graphs is determined. The graphs of convex polytopes are also considered, and for these, the concept of domatic number is studied. We also found that the two families of convex polytopes, namely, and are with the same domatic number, i.e., . Future research can be directed towards studying domatic numbers for more complex planar and chemical graphs.
Data Availability
No data were used to support the findings of the study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
All the authors contributed equally to this manuscript.