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Topological Indices, and Applications of Graph Theory

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Volume 2021 |Article ID 5515250 | https://doi.org/10.1155/2021/5515250

M. Kheibari, H. Abdollahzadeh Ahangar, R. Khoeilar, S. M. Sheikholeslami, "Total Roman -Reinforcement of Graphs", Journal of Mathematics, vol. 2021, Article ID 5515250, 7 pages, 2021. https://doi.org/10.1155/2021/5515250

Total Roman -Reinforcement of Graphs

Academic Editor: Ahmet Sinan Cevik
Received28 Feb 2021
Revised22 Mar 2021
Accepted25 Mar 2021
Published12 Apr 2021

Abstract

A total Roman -dominating function (TR2DF) on a graph is a function , satisfying the conditions that (i) for every vertex with , either is adjacent to a vertex labeled 2 under , or is adjacent to at least two vertices labeled 1; (ii) the subgraph induced by the set of vertices with positive weight has no isolated vertex. The weight of a TR2DF is the value . The total Roman -domination number (TR2D-number) of a graph is the minimum weight of a TR2DF on . The total Roman -reinforcement number (TR2R-number) of a graph is the minimum number of edges that have to be added to the graph in order to decrease the TR2D-number. In this manuscript, we study the properties of TR2R-number and we present some sharp upper bounds. In particular, we determine the exact value of TR2R-numbers of some classes of graphs.

1. Introduction

A total dominating set (TDS) in an isolated-free graph is a set of vertices of such that each vertex in is adjacent to a vertex of . The total domination number of is the minimum cardinality of a TDS. Sridharan et al. [1] introduced the total reinforcement number of a graph as the minimum number of edges that have to be added to the graph in order to decrease the total domination number.. Since the domination number of any isolated-free graph is greater than or equal to 2, by convention, Sridharan, Elias, and Subramanian defined if .

A function is a R2DF of if each vertex labeled 0 under satisfies , where . The minimum value of a R2DF on is called the R2D-number of . The R2DF was introduced in [2] and has been studied by several authors [313]. For more details on R2D-number, we refer the reader to the recent book chapter [14] and survey paper [15].

As a new variant of Roman -domination, total Roman -domination was investigated in [16, 17], where it was called total Roman -domination. A TR2DF on a graph is defined as a function , satisfying the conditions: (i) for every vertex with , , that is, either there is a vertex with , or at least two vertices with ; (ii) the subgraph induced by the set of vertices with positive weight under has no isolated vertex. The weight of a TR2DF is the value . The TR2D-number of a graph , denoted by , is the minimum weight of a TR2DF on . A TR2DF on with weight is called a -function. For a sake of simplicity, at TR2DF on will be represented by the ordered partition (or to refer ) of induced by , where , for . Since labeling 2 to the vertices of any minimum total domination set of a graph introduces a TR2DF of , we have

In this paper, we extend the idea of Roman {2}-reinforcement number to TR2R-number as follows: for a graph , a subset of is a total Roman {2}-reinforcement set (TR2RS) of if . The TR2R-number of a graph , denoted by , is the minimum cardinality of a TR2RS of . A TR2Rtotal Roman {2}-reinforcement set (TR2RS of is called a -set if . Observe that if , then addition of edges does not reduce the TR2D-number. We put if . Hence, we always suppose that when we discuss , all graphs involved satisfy . If is a graph such that , then there is always a nonempty set such that , i.e., . For instance, if we take , then we have that .

Our purpose in this manuscript is to initiate the study of TR2R-number in graphs. We derive some sharp upper bounds on and we also determine exact values of TR2R-number of paths and complete multipartite graphs.

We close this section with some useful results.

Observation 1. For any graph with and -function , , for every .

Proposition 1. For any graph with , let be a -set and let be a -function. Then, the following hold:(i)For any edge , .(ii).

Proof. (i)Suppose, to the contrary, that there exists an edge such that for each . Then, is a TR2DF on and so is a TR2DS of , implying that , a contradiction. Hence, (i) holds.(ii)Since is a -set, . Now, we prove the lower bound. If , then we are done. Now, we assume that and suppose, to the contrary, that . Let . By (i), we can suppose that . If , then the function , defined by and for the other vertices, is a TR2DF on with , and so is a TR2-set of , implying that , a contradiction. Assume that . Let be a neighbor of in for . Then, the function , defined by for and for the other vertices, is a TR2DF on with , and so is a TR2-set of , implying that , a contradiction. So (ii) also holds.

Proposition 2. (see [16, 17]). Let be a nontrivial connected graph of order . Then, if and only if or any vertex of is either a leaf or a weak stem.

Theorem 1. (see [18]). If is a connected graph of order and , then .

2. Bounds

The aim of this section is to obtain basic properties of the TR2R-number in graphs. Let be a graph and be a TR2DF. For each , the -private neighborhood of , denoted by , consists of all vertices such that .

Theorem 2. Let be a graph with and let be a -function. Then, the following assertions hold.(i)For any , . This bound is sharp for double star .(ii)If , then for each ,when is the set of all isolate vertices of induced subgraph .

Proof. (i)Let . By definition, there exists . Let . Notice that by Observation 1. Then, the function defined by and , for the other vertices, is a TR2DF of and so .(ii)Let . By definition, there exists . Let if and when . Since , for any vertex , there is a vertex such that . Let when is odd and if is even. If joins at least two vertices in , then the function defined by , , for the other vertices, is a TR2DF of . If joins only one and , then the function , defined before, is a TR2DF of . Hence,

Proposition 3. Let be a graph. Then, .

Proof. If , then . Hence, assume that and let be a -function. If and , then by Theorem 2(i), we have . Assume that . Then, we have . In this case, the result holds by Theorem 2(ii).

Proposition 4. Let be a graph of order with . Then,

Proof. Let be a vertex of degree . Since , we have . Let and . Then, the function defined by and , for the other vertices, is a TR2DF on with . Thus, is a TR2-set of and so , establishing the desired upper bound.

Next result is an immediate consequence of Propositions 3 and 4.

Corollary 1. For any graph of order with ,

This bound is sharp for .

Proof. If , then Proposition 3 yields . If , then by Proposition 4, we obtain .

Let be the graph obtained from two copies of by joining exactly two vertices of degree 2 to obtain a connected graph (for , see Figure 1). Clearly, and , and it is not hard to see that . This shows that the bounds established in Theorem 2(ii), Propositions 3 and 4, and Corollary 1 are sharp.

Theorem 3. Let be a graph of order . If , then

Proof. It is enough to show that . Since (see Proposition 4), we add edges incident with a vertex of maximum degree and call such that a set of edges . Clearly, . Clearly, and by Theorem 1, . Using inequality (1), we obtain

Next, we consider graphs with .

Proposition 5. Let be a graph with . Then,

Proof. Let be a -set. By inequality (1), we have and this implies that .

3. Graphs with Small

In this section, we provide sufficient conditions for the graph to have small . We first give a characterization of graphs with .

Lemma 1. Let be a connected graph of order with . Then, .

Proof. Let . If is the , then is a complete graph and so . Assume that is the corona graph of some connected graphs of order greater than or equal to two. Let and be two leaves in . Then, is not a corona graph and it follows from Proposition 2 that . Thus, .

Theorem 4. Let be a connected graph of order . Then, if and only if or there exists a function of weight less that for which one of the following conditions holds.

(i)For each , , , and the induced subgraph has at most two isolated vertices.(ii) has no isolated vertices and for each vertex except one, say , such that if or if .

Proof. According to Lemma 1, we can suppose that . Let there exist a function of weight less that , satisfying (i) or (ii). Since , we have . First, let satisfy (ii) and let be a vertex for which . Suppose is a vertex for which and is as large as possible. Clearly, is a TR2DF of and this implies that . Next, assume that satisfies (i). If has two isolated vertices , then is a TR2DF of and if has exactly one isolate vertex, say , then is a TR2DF of , where . This implies that .
Conversely, assume that , and let be a -set. If , then we are done. Hence, we assume that . Let be a -function. If , then satisfies item (i) and if or , then satisfies item (ii).

Next, we consider connected graphs with minimum degree 1.

Proposition 6. Let be a connected graph of order . If has a stem , then . This bound is sharp for .

Proof. If , then and the result follows from Proposition 2 and Lemma 1. Assume that . Let be a leaf-neighbor of and be a -function such that is as small as possible. If , then , and by Theorem 2, we have . Assume that . Then, by the choice of , we must have . Since and is a leaf, we have . Let . Then, the function , defined by and for the other vertices, is a TR2DF of of weight less than . Hence, .

Proposition 7. If is a connected graph containing a path in which for , then .

Proof. If or is a leaf, then the desired result follows by Proposition 6. Hence, we suppose and . Let be a -function. If , then to totally dominate , we can suppose without loss of generality that and the function , defined by and for the other vertices, is a TR2DF of of weight less than . Assume that . Likewise, we can suppose that . We consider two cases.
Case 1..
 To totally dominate , we can suppose that . If , then the function , defined by and for the other vertices, is a TR2DF of of weight less than . Suppose that . Then, to Roman -dominate , we must have and the function , defined by and for the other vertices, is a TR2DF of of weight less than .
Case 2..
Since , to Roman -dominate , we must have and to totally dominate , we must have and , respectively. Define by and for the other vertices. Clearly, is a TR2DF of of weight less that of . All in all, we have .

4. Exact Values

In this section, we determine the TR2R-number of paths and complete multipartite graphs. The TR2D-number of paths is determined in [17].

Proposition 8. (see [17]). For any positive integer ,(i)(ii).

Theorem 5. For any integer ,

Proof. Let . If , then by Proposition 8, we have and hence .
Suppose next that . Clearly, the function defined by for , and otherwise, is a TR2DF on with implying that . Now, we show that . In order to this, we show that there is no function of weight less than , satisfying one of the conditions (i) and (ii) of Theorem 4.
Suppose, to the contrary, that there exists a function of weight less than , satisfying one of the conditions (i) and (ii) of Theorem 4. Choose such a function such that is as few as possible. Let for some positive integers . First, let satisfy (ii). We note that ; otherwise, if , then by total, we can suppose that and the function defined by , and otherwise, is a function on of weight less than , satisfying (ii), a contradiction with the choice of . Then, we must have and so . Assume without loss of generality that such that and for each . Then, we must have . Since has no isolated vertex, we have for each . It follows that which is a contradiction.
Now, let satisfy (i). Using the above argument, we can see that each vertex in is isolated in . Thus, . First, let . Since , we have and this implies that . Clearly, four vertices in are dominated by vertices with weight 2. Assume without loss of generality that in which and that has no neighbor in . As mentioned above, we can see that which leads to the contradiction .
Next, assume that . Let . We claim that . If , then the function defined by and is a TR2DF of of weight less than which is a contradiction. If , then we must have and (note that ) and the function , defined before, is a TR2DF of of weight less than , a contradiction again. Thus, . Similarly, we can see that . Since , we must have . Then, the function , defined by and otherwise, is a function on of weight less than , a contradiction.
Finally, let . We distinguish two situations.(1) has exactly one isolated vertex, say . If (the case is similar), then and the function , defined by and otherwise, is a function of weight less than which satisfies (ii) and the result follows as above. Thus, . Then, . To Roman -dominate and , we must have . On the other hand, since is the only isolated vertex of , we have . Let be the path obtained from by deleting the vertices and adding the edge . Clearly, the restriction of on is a TR2DF of , and by Proposition 8, we have which is a contradiction.(2) has exactly two isolated vertices. Let be the isolated vertices of such that . Note that if , then , and to Roman -dominate , we must have . Now, if , then we must have and then the function , defined by and otherwise, is a TR2DF of implying that which is a contradiction. Assume that . Then, the function , defined by and otherwise, is a function of weight , satisfying the condition (i) of Theorem 4. Thus, we can suppose that . Similarly, we can suppose that . Then, we must have . To Roman -dominate and , we must have , and to totally dominate and , we have . If , then the restriction of on , obtained from by deleting the vertices and adding the edge , is a TR2DF of , and by Proposition 8, we have which is a contradiction. If , then , and the restriction of on , obtained from by deleting the vertices and adding the edge , is a TR2DF of , and by Proposition 8, we have , a contradiction again. Let . Then, we must have and the restriction of on , obtained from by deleting the vertices and adding the edges , is a TR2DF of , and by Proposition 8, we have which is a contradiction.Thus, there is no function of weight less than , satisfying one of the conditions (i) and (ii) of Theorem 4. So, .

The proof of the following result is straightforward and then omitted.

Proposition 9. For any positive integers with ,