Abstract
A perfect Roman dominating function on a graph is a function for which every vertex with is adjacent to exactly one neighbor with . The weight of is the sum of the weights of the vertices. The perfect Roman domination number of a graph , denoted by , is the minimum weight of a perfect Roman dominating function on . In this paper, we prove that if is the Cartesian product of a path and a path , a path and a cycle , or a cycle and a cycle , where , then .
1. Introduction and Preliminaries
All graphs considered in this work are simple, finite, and undirected. Let be a graph. We denote the cardinality of by . Two vertices are adjacent when . The open neighborhood of a vertex is the set while the closed neighborhood of a vertex is the set . The number of vertices of a path is its length. We denote a path of length by . We denote the cycle graph with vertices by .
A dominating set of a graph is a subset of where each vertex in is adjacent to at least one vertex in . The domination number is the minimum cardinality of a dominating set of , and it is usually denoted by . There is a large literature that covered the domination number. For basic definitions and concepts relating to this subject, we refer the reader to[1].
A perfect dominating set of a graph is a subset of where each vertex satisfies . The perfect domination number is the minimum cardinality of a perfect domination set of , and it is usually denoted by . The study of perfect domination has received much attention in the literature, see for example [2, 3].
A Roman dominating function on a graph , denoted by RD-function, is a function such that every vertex with is adjacent to at least one vertex with . For any vertex , the weight of is its value under the function while the weight of , denoted by , is the sum . The Roman domination number of a graph , denoted by , is the minimum weight of a RD-function, i.e.,
Roman domination has been studied well and there are many research papers on this subject, such as [4, 5]. There are some variations on domination number and Roman domination number have been appeared in the literature such as total, week, and perfect [6–10]. In this paper, we continue the investigation of perfect Roman domination.
A perfect Roman dominating function on a graph is a function for which every vertex with is adjacent to exactly one neighbor with . We denote a perfect Roman dominating function by PRD-function. The weight of , denoted by , is the sum . The perfect Roman domination number of a graph , denoted by , is the minimum weight of a PRD-function, i.e.,
The investigation of perfect Roman domination was initiated by Henning et al. in [7] on trees and then Henning and Klostermeyer considered this subject on regular graphs [11]. More recent work on perfect Roman domination can be found in [12–15].
Roman domination in product graphs has become an attractive topic in the study of domination and much work has been done in this area such as Cartesian product [16], lexicographic product [6, 17], rooted product [8, 18], and direct product [18]. Perfect Roman domination in product graphs has been considered for corona product in [13]. In this work, we studied perfect Roman domination in the Cartesian product of paths and paths, paths and cycles, and cycles and cycles.
Let and be two graphs. The Cartesian product graph of and , denoted by , is the graph with as its set of vertices, and two vertices are adjacent if either(1) and or(2) and
The graph is a grid graph with columns and rows, see Figure 1.
We denote the vertex in row and column by . The graph is a cylinder grid graph which is a grid graph, with columns and rows, and some extra edges between the vertices of the first and the last rows, see Figure 2. The graph is a torus grid graph which is a cylinder grid graph with some extra edges between the vertices of the first and last columns, see Figure 3. So and .
2. Discussion
In this section, we present an upper bound for the perfect Roman domination number of a grid graph, a cylinder grid graph, and a torus grid graph.
Theorem 1. Let . If {, , }, .
Proof. The statement will be a result from the following three cases.
Case 1. or for some integer . If , label each vertex in column number with 2, and label the remainder vertices with 0. See Figure 4. It is not hard to see that this labeling produces a perfect Roman domination function of weight equals to where . In a similar way, if , label each vertex in row number with 2, and label the remainder vertices with 0.
We need the following function for the remaining cases. Define a function as follows:The function has a pattern recurring every six columns, and it also has a pattern recurring every three rows, as shown in Figure 5. It is not hard to see that every vertex with has exactly one neighbor labelled 2, except some vertices in the first and last rows and possibly in the first and last columns. So we need to modify slightly depending on the values of and .
Case 2. . If is a multiple of three, we are in a situation symmetric to Case 1, and we are done. So we may assume that is not a multiple of three. We divide this case to two subcases. Case 2.1. for some integer . If , define a function such that Then, is a PRD-function on where {, , }, see Figure 6. Note that if , . If , . If , . If , . If , . If , . Therefore, Inequality (5) follows from the fact that . If , define a function such that Then, is a PRD-function on where {, , }, see Figure 7. If , . If , . If , . If and , . If , . If , . If , . Thus, Inequality (7) follows from the fact that . Case 2.2. for some integer . Assume that . Define a function such that Then, is a PRD-function on where {,, }, see Figure 8. If , . If , . If , . If , . If , . If , . Therefore, Inequality (9) follows from the fact that and . Assume that . Define a function such that Then, is a PRD-function on where {, , }, see Figure 9. If , . If , . If and , . If , . If and , . If , . If , . If , . Therefore, Inequality (11) follows from the fact that .
Remark 1. In both cases, i.e., and , the function is also a PRD-function on the Cartesian product graph.
Case 3. . If then we are in a situation symmetric to Case 2 (see Remark 1). So we may assume that .
Assume that . Define a function such thatThen, is a PRD-function on where {}, see Figure 10. If , . If and then , and if then . If , . If , . If , . If , . Therefore,Inequality (13) follows from the fact that . Assume that . Define a function such thatThen, is a PRD-function on where {, , }, see Figure 11. If , . If , . If , . If , . If , . If , . Therefore,Inequality (15) follows from the fact that .
3. Conclusions
In this paper, we discussed the perfect Roman domination number of the Cartesian product of a path and a path, a path and a cycle, and cycle and a cycle. We found that in these three graphs the perfect Roman domination number is not greater than 2/3 of the cardinality of its vertices. Concerning the further plans for our work, we will investigate the perfect Roman domination number of the Cartesian product of trees.
Data Availability
All data required for this paper are included within this paper.
Conflicts of Interest
The authors declare there are no conflicts of interest.
Acknowledgments
This work was supported through the Raed track by the Deanship of Scientific Research, Vice Presidency for Graduate Studies and Scientific Research, King Faisal University, Saudi Arabia (Project no. RA00033).