#### Abstract

In this paper, we derive sharp upper and lower bounds on the sum and product , where is the complement of graph . We also show that for each tree of order , and , where and are the number of support vertices and leaves of .

#### 1. Terminology and Introduction

For notation and graph theory terminology, we, in general, follow Haynes et al. [1]. Throughout this paper, denotes a simple graph, with vertex set and edge set . The order of is denoted by . The open neighborhood of is the set , and the closed neighborhood of is the set . The degree of a vertex is . The minimum and maximum degree of a graph are denoted by and , respectively. The complement of a graph is denoted by . The distance between two vertices and of a (connected) graph is the length of a shortest -path in . The maximum distance among all pairs of vertices in is the diameter of , which is denoted by diam . For a subset of vertices of , we denote by the subgraph induced by . A vertex of degree one is called a leaf and its neighbor a support vertex. For a vertex in a rooted tree , the maximal subtree at is the subtree of induced by and its descendants and is denoted by .

In the field of chemistry, graph theory has provided many useful tools, such as topological indices. Cheminformatics is one of the latest concepts which is a join of chemistry, mathematics, and information science.

Topological indices and domination in graphs are the essential topics in the theory of graphs. Topological indices are numerical parameters of the graph, such that these parameters are the same for the graph in which they are isomorphism. In [2], the study introduced the concept of domination degree of the vertex . Some of the major classes of topological indices are distance-based topological indices and degree-based topological indices. Degree-based topological indices are of great significance. Relationships between various topological indices and domination number of graphs have been the focus of interest of the researchers for quite many years, and this direction is continuously vital (see [35]). Specifically, Ahmad Jamri et al. [6, 7] investigated extremal Zagreb indices of graphs with a given Roman domination number.

Let be an integer and be a finite and simple graph with vertex set . Let be a function that assigns label from the set to the vertices of a graph . For a vertex , the active neighborhood of , denoted by , is the set of vertices such that . A -RDF is a function satisfying the condition that for any vertex with . The weight of a -RDF is . The -Roman domination number of is the minimum weight of an -RDF on . The case is the usual Roman domination which has been introduced in [8]. For more details on Roman domination and its variants, we refer the reader to [9]. The case is called double Roman domination and has been studied in [1012], while the case is called the triple Roman domination and has been studied in [13, 14].

The Roman domination in graphs is well studied in graph theory. The topic is related to a defensive strategy problem in which the Roman legions are settled in some secure cities of the Roman Empire. The deployment of the legions around the Empire is designed in such a way that a sudden attack to any undefended city could be quelled by a legion from a strong neighbor. There is an additional condition: no legion can move and doing so leaves its base city defenceless. In this paper, we continue the study of a variant of Roman domination in graphs: the triple Roman domination. We consider that any city of the Roman Empire must be able to be defended by at least three legions. These legions should be either in the attacked city or in one of its neighbors. We determine various bounds on the triple Roman domination number for general graphs, and we give exact values for some graph families. Moreover, complexity results are also obtained.

For the graph parameter , bounds for the sum or product are called results of “Nordhaus–Gaddum” type, honoring the paper of Nordhaus–Gaddum [15] obtaining such bounds when is the chromatic number.

In this paper, we first present Nordhaus–Gaddum-type bounds for the triple Roman domination number. For each tree of order , we also show that and .

We will use the next results in this paper. Let be the family of all trees that can be built from k ≥ 1 paths by adding k − 1 edges incident with the ’s, so that they induce a connected subgraph. (see Figure 1).

Theorem 1. For any connected graph of order , with equality if and only if .

Theorem 2. For each connected graph , there exists a -function that does not value the label one to each vertex in .

Theorem 3. Let be a connected graph on vertices. Then, if and only if .

Theorem 4. If is a connected graph on vertices, then if and only if .

Theorem 5. There is no connected graph of order for which .

Theorem 6. If is a connected graph on vertices, then if and only if there are two nonjoin vertices in with degree .

#### 2. Nordhaus–Gaddum Inequality for the 3RD Numbers

In this section, we present Nordhaus–Gaddum-type inequalities for the triple Roman domination numbers of graphs. We begin with a simple result.

Lemma 1. If is a graph of order and maximum degree , then

Moreover, if and , then .

Proof. Suppose is a vertex of maximum degree . Clearly is a 3RD-function of weight and so .
Let and . Let be a vertex with maximum degree , and let . Suppose that is a vertex with maximum degree in . If has a neighbor in , then the function is a 3RD-function of of weight smaller than , a contradiction. Therefore, . If , then is a 3RD-function of of weight which is a contradiction. Hence, there is a vertex for which . Then, is a 3RDF of of weight as desired.

Theorem 7. If is a graph of order and , then

Equality holds for the lower bound if and only if or is , and equality holds for the upper bound if and only if or is a complete graph.

Proof. We can suppose that . First, we show the left inequality. If , then as desired. Let . By Theorems 3 and 5, we find , and Theorem 4 follows that . Hence, is disconnected and so . Thus, . If the equality holds, then we should have . As noted, has a vertex of degree , say . Since is an isolated vertex in , we find that and, consequently, .
Next, we show the upper bound. By Lemma 1, we find and . If and , then we findSuppose that . If , then Lemma 1 follows that and soSuppose . Then, we have is equal to four. If , then we haveSuppose that . Then, is complete graph, and clearly, we have . This completes the proof.
To prove an upper bound on the product , we use the following result in [13].

Theorem 8. Let be a graph of order and minimum degree . Then,

Theorem 9. Let be a graph of order for which and are equal to two. Then,

Proof. Let be a graph of order with , and suppose is a vertex of minimum degree in . If , then the diameter constraint follows that is a 3RDF for and is a 3RDF of , and so when is greater than or equal to six. Hence, we can suppose that , and similarly, is greater than or equal to three.
Let . We choose a family of disjoint subsets of dominating as follows. Initialize ; note that dominates ; then, let be a minimal subset of dominating , and let . If does not dominate , then stop setting and . Otherwise, repeat the process. Note that is a partition of , with each being a minimal set that dominates .
Since is a minimal dominating set for , there is a vertex having just one neighbor in . Let be this neighbor. Since does not dominate , there exists for which . Let and . Now, is a 3RDF for , since dominates , dominates , and dominates . Thus, , which reduces to if .
Let , where . Note that is a dominating set for . If is equal to two, then . Since is connected and , Theorem 1 yields and then clearly . Hence, we can suppose that , which requires . If , then and and so . Hence, we can suppose that , but now, Theorem 8 yields . Easily, it can be seen that , when , and thus, .
Hence, we can suppose that . Using the 3RDF , we haveSince when , we can suppose that and similarly . By Theorem 8, . Hence, , and this leads to the result since .

#### 3. New Bounds for the Triple Roman Domination Number of Trees

We establish some bounds on the triple Roman domination number of trees. For each tree , let and denote the number of support vertices and leaves of , respectively. We begin with some simple lemmas.

Lemma 2. Let be a nonnegative integer, and let be a rooted tree and be a vertex different from its root. Then, we have the following:(1)If , , and , then (2)If , , and , then

Observation 1. Let be a tree and . If is the tree obtained from by adding a star and joining to one leaf of , then .

Next, we improve the upper bound of Theorem 1 for trees. Let be the family of all trees that can be built from copies by adding edges between the vertices to connect the graph.

Theorem 10. If is a tree of order , then

Furthermore, this bound is sharp for all trees in .

Proof. We proceed by induction on . If , then clearly . Suppose that is greater than or equal to five, and let the result hold for all nontrivial trees of order less than . Let be a tree of order . If is equal to two, then is a star and we have . If is greater than or equal to three, then is a double star and we have . Henceforth, we suppose is greater than or equal to four. Let be a diametral path in for which is as large as possible and root at . If , then ; if , then ; and if , then , and using the induction hypothesis and setting in Lemma 2-(1), we obtain in all of the above cases. Suppose that . By the choice of diametral path, we can suppose that all children of with depth one have degree two.
If , then by Observation 1, we have , and by the induction hypothesis on and setting in Lemma 2-(1), we have . Suppose that . If is a strong support vertex or is a support vertex and , then clearly , and using the induction hypothesis on and setting in Lemma 2-(1), we have . Next, we consider the following cases:Case 1. is a support vertex and .If , then the result is immediate. Suppose that . If , then each -function can be extended to a 3RDF of by assigning a 4 to , a 3 to , and a 0 to the other vertices of following that , and applying the induction hypothesis on and setting in Lemma 2-(1), we obtain . Suppose that . Then, clearly and as above, we have , and using the induction hypothesis on and setting in Lemma 2-(1), we have .Case 2. is not a support vertex and .Then, is a healthy spider with at least three feet. We first suppose that , then , and applying the induction hypothesis on and setting in Lemma 2-(1), we have . Now, we suppose that . If , then each -function can be extended to a 3RDF of by assigning a 4 to , a 3 to leaves of except , and a 0 to the other vertices of following that , and applying the induction hypothesis on and setting in Lemma 2-(1), we obtain .If , then each -function can be extended to a 3RDF of by assigning a 3 to , a 3 to leaves of , and a 0 to the other vertices of following that , and applying the induction hypothesis on we have .Case 3. is not a support vertex and .If , then the result is immediate. Let and and since each -function can be extended to a 3RDF of by assigning a 0 to and the leaves of and a 4 to the children of . Using the induction hypothesis on , we have . This completes the proof. Now, we provide a lower bound on the triple Roman domination number of a tree in terms of its order and the number of its leaves and support vertices.

Theorem 11. Let be a tree of order with leaves. Then,

Furthermore, this bound is sharp for stars.

Proof. We proceed by induction on . If , then clearly , and if , then the only tree is and clearly . Suppose , and let the result hold for all nontrivial trees of order at most . Suppose is a tree of order . If diameter is equal to 2, then is a star and . If diameter is equal to 3, then is a double star and we have . Henceforth, we suppose . Let be a diametral path in for which is as large as possible and root at . Suppose is a -function of for which is as large as possible. If , then clearly and so is a triple Roman dominating function of . Since , applying the induction hypothesis on and setting in Lemma 2-(2), we obtain .
Suppose that . By the choice of diametral path, we can suppose that each child of with depth 1 has degree 2. We consider the following cases:Case 1. has a child with depth one different from or is a support vertex.Let . If has a child with depth one different from , then we can suppose that . Clearly, and the function , restricted to , is a triple Roman dominating function of , and this follows that . Since , using the induction hypothesis on and setting in Lemma 2-(2), we have .Now, let be a support vertex. It is obvious that . We can suppose that and . Now, the result follows as above.Case 2..Let . By Observation 5, we have . Since , using the induction hypothesis on and setting in Lemma 2-(2), we obtain .

#### 4. Conclusion

The main objective of this paper was to study the triple Roman domination number in graphs. We first presented Nordhaus–Gaddum inequalities for the triple Roman domination number, and then, we focused on trees and proved that, for each tree of order , and , where and are the number of support vertices and leaves of . Characterizing trees attaining the bounds of Theorems 10 and 11 may be considered for future works.

#### Data Availability

No data were used to support this study.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### Acknowledgments

H. Abdollahzadeh Ahangar was supported by the Babol Noshirvani University of Technology under research grant number BNUT/385 001/00.