Abstract
Let be a graph. A function is said to be a signed clique dominating function (SCDF) of if holds for every nontrivial clique in . The signed clique domination number of is defined as . In this paper, we investigate the signed clique domination numbers of join of graphs. We correct two wrong results reported by Ao et al. (2014) and Ao et al. (2015) and determine the exact values of the signed clique domination numbers of and .
1. Introduction
We use Bondy and Murty [1] for terminology and notation not defined here and consider only simple and undirected graphs.
The theory of domination is an important content in graphs, and its applications are more and more widely. In 1995, Dunbar et al. [2] first introduced the signed domination of graphs, and now, there are a lot of variations, such as the total domination [3]. However, most of them belong to the vertex domination of graphs, and a few of the edge dominations were studied. In 2001, B. Xu puts forward the signed edge domination of graphs [4]. Since then, many variations based on edge domination have become more and more abundant, such as signed total domination [5], Roman domination [6], signed cycle domination [7], and signed clique domination [8]. The emergence of these concepts enriched and completed the domination theory of graphs. In this paper, we consider the signed clique domination numbers of joint graphs.
Let be a graph with vertex set and edge set . Every maximal complete subgraph of graph is called a clique of ; that is, there are no other complete subgraphs contain . A clique is called nontrivial if .
Given a graph , for any vertex , let be the set of edges in incident to . If , , and , then we write .
For any two disjoint graphs and , then denotes the joint graph of and , where
For ease of description, let be a graph, and be a real-valued function defined on , then we put .
Definition 1 (see [8]). Let be a graph, a function is said to be a signed clique dominating function (SCDF) of if holds for every nontrivial clique in . The signed clique domination number of is defined asIf is an SCDF such that , then the function is said to be a minimum SCDF of .
Lemma 1 (see [8]). For any graph , .
Lemma 2 (see [8]). For any connected graph of order , if and . Then,
This lower bound is the best possible.
For the joint graphs of the two graphs, we have the following results.
Lemma 3 (see [9]). (1) For any two positive integers and , then(2) For any two positive integers and , then
Lemma 4 (see [10]). For any positive integer , then
However, we find the following two lemmas are wrong.
Lemma 5 (see [10]). For any positive integer , then
For example, in fact, when and , then . The labeling of is shown in Figure 1, and when and , then . The labeling of is shown in Figure 2 where unlabeled edges are assigned as -1.
Lemma 6 (see [11]). For any positive integer and , then
In fact, the above conclusion is not true for , and is odd. For example, we may see and . The labeling of and is shown in Figures 3 and 4, respectively, where unlabeled edges are assigned as –1.
In this note, we mainly correct two wrong conclusions in [10, 11] and determine exactly the signed clique domination numbers of and .
2. Main Results
Theorem 1. For any two positive integers and , then
Proof. Let be a joint graph, , we define , , .
Case 1. When .
Let be a minimum SCDF of , that is, . Obviously, there are cliques in , among is a clique with three vertices . The cliques contain a common edge . According to Definition 1, we have held for every clique . Then, we obtain . In this inequality, note that the function value of each edge in is counted exactly times, and the function value of each edge in is counted one time. Then,For every vertex , is the set of edges in incident to . It is obvious that (otherwise, if there exists a vertex such that ). It implies the contradiction that ). Thus, we derive . Combining with (10), we haveMeanwhile, we define such a function of as follows:It is routine to check that is an SCDF of . Hence,Combining with , we finally derive .
Case 2. When .
We know that is connected graph of order , , and . By Lemma 2, we have . Meanwhile, we define such a function of as follows:It is routine to check that is an SCDF of . Then,Combining with , we finally have .
Case 3. When and is odd.
Let be a minimum SCDF of , that is, . Clearly, there are cliques in , among is a clique with three vertices . According to Definition 1, we obtain holds for every clique . Therefore, we have .
In the above inequality, we know that the function value of each edge in is counted exactly times. Then, let be the edge set in where the function value of each edge is counted one time, and is the edge set in where the function value of each edge is counted exactly 2 times, whereThen,For every vertex , it is obvious that (otherwise, there must exist a clique such that , a contradiction). Then, we have . We assume that there are vertices such that , thenwhere .
According to (17), we obtainMeanwhile, we define such a function of as follows:It is clear that is an SCDF of . Therefore,Together with , we have .
Case 4. When and is even.
Let be a minimum SCDF of , i.e., . The same as Case 3, we have , then . We assume that there are vertices such that , thenwhere .
According to (17), we haveIn addition, we define such a function of as follows:It is routine to check that is an SCDF of . Therefore,Combining with , we have . This completes the proof of Theorem 1.
Theorem 2. For any two positive integers and , then
Proof. Let , , and , among ; , .
Let be a minimum SCDF of , that is, . Obviously, there are cliques in , among is a clique with vertices and the vertex set .
Case 5. When . Note that . According to Definition 1, we have held for every clique . Since , by Lemma 1, we have . Then, we obtain . In this inequality, we know that the function value of each edge in is counted exactly times, the function value of each edge in is counted exactly one time, and the function value of each edge in is counted 2 times. Then,Since , we haveAccording to (27), we obtainTogether with (28), we have(i)When ,(ii)When ,Then, we have . Note that, when is odd, is also odd. As per Lemma 1, we derive . Thus, . In addition, we define such a function of as follows:It is not difficult to check that is an SCDF of , then . In summary, when , .
Case 6. When . We know . By the Definition 1, we have held for every clique . Then, we obtain . The same as Case 5, we haveSince , we deriveAccording to (33), we haveCombining with (34), we have(i)When ,(ii)When ,Then, we have . Meanwhile, we define such a function of as follows:Clearly, is an SCDF of , and then . In summary, when , whether is odd or is even, we finally have .
Case 7. When . We know . As per Definition 1, we have held for every clique . Then, we obtain . The same as Case 5, we haveSince , we deriveAccording to (39), we haveCombining with (40), we have(i)When ,(ii)When ,Then, we have . Notice that when is odd, is even. By Lemma 1, we obtain . Thus, . In addition, we define such a function of as follows:It is not difficult to check that is an SCDF of , then . In summary, when , we have .
Case 8. When . Let be a minimum SCDF of , that is, . Write , .
According to Definition 1, we have held for every clique in . Thus, we have . It implies , and then,In addition, since , we know . Now define a function , that is, let the number of +1 edges in is , the other edges are assigned as -1. It is obvious thatIn summary, when , we have . We complete the proof of Theorem 2.
Data Availability
All the results and data in this paper are obtained through theoretical analysis and logical reasoning.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This research was supported by the National Natural Science Foundation of China (Grant nos. 11961026, 11861032). The authors would like to thank the referees for their helpful comments.