# On the Outer-Independent Double Roman Domination of Graphs

^{1}Institute of Computing Science and Technology, Guangzhou University, Guangzhou, China^{2}Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, Iran^{3}LAMDA-RO Laboratory, Department of Mathematics, University of Blida, Blida, Algeria

An outer-independent double Roman dominating function (OIDRDF) of a graph *G* is a function *v* with *v* with *γ*_{oidR} (*G*) is the minimum weight of an OIDRDF on *G*. It has been shown that for any tree *T* of order *n* ≥ 3, *γ*_{oidR} (*T*) ≤ 5n/4 and the problem of characterizing those trees attaining equality was raised. In this article, we solve this problem and we give additional bounds on the outer-independent double Roman domination number. In particular, we show that, for any connected graph *G* of order *n* with minimum degree at least two in which the set of vertices with degree at least three is independent, *γ*_{oidR} (*T*) ≤ 4n/3.

## 1 Introduction

We consider only simple connected graphs *G* with vertex set *order* of *G*. The *open neighborhood* of a vertex *degree* of *v* is *leaf* is a vertex with degree one and its neighbor is called a *stem*. A *strong stem* is a stem adjacent to at least two leaves. The *diameter* of *G*, denoted by diam (*G*), is the maximum value among distances between all pairs of vertices of *G*.

A set *independent* if no two vertices in *S* are adjacent. The *independence number α* (*G*) of a graph *G* is the maximum cardinality among the independent sets of vertices of *G*. A *vertex cover* of a graph *G* is a set of vertices such that each edge of the graph is incident to at least one vertex of the set. A minimum vertex cover is a vertex cover of smallest possible size. The vertex cover number *α*_{0} (*G*) is the minimum cardinality of a vertex cover of *G*.

In 2016, Beeler et al., Ref. 1, introduced the concept of double Roman domination and defined a *double Roman dominating function* (DRDF) on a graph *G* to be a function *weight* of a DRDF *f* is the value *double Roman domination number γ*_{dR} (*G*) equals the minimum weight of a DRDF on *G*. Double Roman domination has been studied by several authors; see, for example, Refs. 2–14. For more details on Roman domination and its variations, we refer the reader to Refs. 15–18.

For double Roman domination, one can think of any vertex representing a location in the Roman Empire and any edge being a road between two locations. A location is said to be protected if at least one army is stationed in it or by sending to it two armies from neighboring location(s) having already more than two armies (according to the decree of Emperor Constantine the Great). A locality without an army is certainly vulnerable, and it will be even more vulnerable if one of its neighbors is without army too. Hence, the best situation for a location with no army is to be surrounded by locations where each with at least one army. This leads us to seek an DRDF *h* is an OIDRDF.

Regarding this, Abdollahzadeh Ahangar et al., Ref. 19, introduced a new variation of double Roman domination called outer-independent double Roman domination. An *outer-independent double Roman dominating function* (OIDRD-function) of a graph *G* is a DRDF *h* such that the set of vertices assigned a 0 under *h* is independent. The *outer-independent double Roman domination number* (*OIDRD-number* for short) *γ*_{oidR} (*G*) is the minimum weight of an OIDRD-function on *G*. Clearly, *γ*_{dR} (*G*) ≤ *γ*_{oidR} (*G*) holds for every graph *G*. Recently, Mojdeh et al., Ref. 20, proved that the decision problem associated with *γ*_{oidR} (*G*) is NP-complete even when restricted to planar graphs with maximum degree at most four. They also characterized the families of all connected graphs with small outer-independent double Roman domination numbers.

In the following, we denote the set

The authors of Ref. 19 provided an upper bound for the OIDRD-number of trees in terms of the order and number of stems.

Theorem 1.1. For each tree T on *n* ≥ 3 vertices,

where *T*.Since the number of stems of any tree does not exceed half the order of the tree, the next result is immediate from Theorem 1.1.

Proposition 1.2. *For every tree T on n* ≥ 4 *vertices*, *γ*_{oidR} (*T*) ≤ 5n/4.Moreover, it should be noted that the problem of characterizing the trees *T* attaining equality in the upper bound of Proposition 1.2 was raised in Ref. 19. This problem will be solved in this article, and additional bounds on the OIDRD-number will be given. In particular, we prove that, for any connected graph *G* of order *n* with minimum degree at least two in which the set of vertices with degree at least three is independent, *γ*_{oidR} (*G*) ≤ 4*n*/3.

## 2 Trees *T* of Order *n* With *γ*_{oidR} (*T*) = 5*n*/4

With the aim of characterizing the trees *T* of order *n* ≥ 3 with *γ*_{oidR} (*T*) = 5*n*/4, let *P*_{4} whose vertices are labeled in order *T* obtained from

Lemma 2.1. *Let**for some integer t* ≥ 1. *Then, there is a γ*_{oidR} (*T*) *function f such that, for every leaf v of T*, *.*

Proof. Let *T* be a tree of *t* ≥ 1. Then, *T* is a bipartite graph, let *X* and *Y* be the partite sets of *T*. Let *T* belonging to *X*, and likewise let *T* is either in *f* on *Y*, a 1 to all vertices in *X*. Then, *f* is an OIDRD-function of *T* of weight *f* is a *γ*_{oidR} (*T*) function with desired property. □

Theorem 2.2. *Let T be a tree on n* ≥ 4 *vertices. Then, γ*_{oidR} (*T*) = 5*n*/4 *if and only if**for some integer**.*

Proof. We prove only the necessity. Let *T* be a tree of *n* ≥ 4 such that *γ*_{oidR} (*T*) = 5*n*/4. Clearly, *t* ≥ 1. To prove that *t*. If *t* = 1, then *t* ≥ 2 and assume that the result is true for any tree *T* with *γ*_{oidR} (*T*) = 5*n*/4, where *T* be a tree with *γ*_{oidR} (*T*) = 5*n*/4 and *T* is the corona of some tree and so *T* has no strong stem. Moreover, *t* ≥ 2. Let *T* and root *T* at *w* adjacent to *x* and its descendants in the rooted tree *T*. We claim that *i*, let *T* by removing every *f* be a *γ*_{oidR}(*f* can be extended to an OIDRD-function of *T* by assigning a 0 to

which leads to a contradiction. Hence, *γ*_{oidR}(*f* can be extended to an OIDRD-function of *T* by assigning a 3 to

Therefore, by the induction hypothesis on *f* such that *h* defined on *T* of weight *T* of weight

## 3 Slightly Improved Bounds for Trees

In this section, we present some sharp bounds on the OIDRD-number. We start with some classes of trees where the upper bound in Proposition 1.2 will be slightly improved.

Proposition 3.1. *Let T be a tree of order n* ≥ 3, *where**. If T contains a strong stem*, *then γ*_{oidR}(*T*) ≤ 5n/4−1.

Proof. Let *T* and let *γ*_{oidR}(*T*_{i})-function for each *T* by assigning a 3 to *s* and to all center vertices of the components of order three, a 2 to each leaf at distance two from *s* belonging to a component of order two, and a 0 to the remaining vertices in the components of order at most three. Observe that if *p*, and thus

as desired. □

A closer look at the proof of Proposition 3.1 shows that it can be used to obtain the next two results too.

Proposition 3.2. *Let T be a tree of order n* ≥ 3, *where**. If T contains one strong stem s such that**contains a component isomorphic to**or**, then γ*_{oidR} (*T*) ≤ 5*n*/4−1*.*

Proposition 3.3. *Let T be a tree of order**where**. If T contains a strong stem having at least three leaves*, *then γ*_{oidR} (*T*) ≤ 5*n*/4−1*.*

Proposition 3.4. *Let T be a tree of order n* ≥ 3, *where**. If T contains more than one strong stem, then γ*_{oidR} (*T*) ≤ 5*n*/4−1*.*

Proof. Let *T* and let *T* of weight at most *γ*_{oidR} (*T*_{i}) function for each *i*. In addition, let *f*_{i}’s together can be extended to an OIDRD-function of *T* by assigning a 3 to *s* that belongs to a component of order two in

as desired. □

## 4 Graphs With Minimum Degree Two

We begin by recalling the question, posed in Ref. 19, on whether the

Proposition 4.1. For *n* ≥ 3,

Proposition 4.2. *For n* ≥ 3, *the path**has an OIDRD-function f that assigns positive weight to the end-vertices of**and**.*

Proof. Let *f* on *f* is an OIDRD-function of

For integers *s* ≥ 1, let

Proposition 4.3. *For integers**and s* ≥ 1, *the graph**has an OIDRD-function f that assigns a positive weight to v*_{s}*and ω*(*f*) ≤ 4(*r*+*s*)/3.

Proof. Assume first that *f* be a *γ*_{oidR} (*C*_{r})function such that *g* defined on *g* defined on

Let *g* be an OIDRD-function of the path *h* on *i* and *h* is an OIDRD-function of

Let *H* without loops with *H* at least once and at most five times. Clearly, any graph in *G* in *n* satisfies

Proposition 4.4. *For any graph**of order n, there exists an OIDRD-function f of G such that**and**for each vertex x of degree at least three*.

Proof. Let *n*. We use an induction on *n*. If *n* = 5, then *f* that assigns a 2 to the vertices of degree 3 and a 0 to the remaining vertices satisfies the conditions as desired. Let *P* of *G* an *A-ear path* if *P* is connected to *A* by either its unique vertex (when *A*-ear paths *P* of *G* of order *i* and let *A*-ear path *P*, let

First, let *G* by first removing all vertices of the path *P* except *k* = 1, then the function *g* defined on *G* such that

If *k* = 2, then the function *g* defined on *G* such that

From now on, we can assume that

Next, assume that *G* by deleting *f* of *g* defined on *G* such that

Considering the above situations, we may assume that

Assume first that *j*. Moreover, if *g* on *j* and *g* is an OIDRD-function of *G* such that

Henceforth, we can assume that

**Case 1.**

Then, *G* is obtained from a loopless multigraph *f* on *f* is an OIDR-function of *G* such that

**Case 2.**

Let *A*-ear paths of

**Subcase 2.1.***u* is adjacent to at least two *A*-ear paths of

Let *A*-ear paths such that *G* by removing first *x* in *a* or *b* provided *a* or *b* is not adjacent to the end-vertex of the *A*-ear path containing *x*. Clearly, *g* on *g* is an OIDR-function of *G* such that

Suppose now that *G* by removing first *x* in *a* or *w* provided *a* or *w* is not adjacent to the end-vertex of the *A*-ear path containing *x*. Clearly, *g* defined above satisfies the desired conditions.

**Subcase 2.2.** All neighbors of *u* but one belong to *A*-ear paths of

By the choice of *u*, we may assume that each vertex in *A* is adjacent to at most one *A*-ear path in *G* is obtained from a multigraph *H* without loops with *H* at least once and at most twice so that the set of edges of *H* subdivided twice is independent (in *H*). Hence, let *H* subdivided twice and let *H* for which all edges that are incident are subdivided once. Therefore, we have *k* edges of *H* are subdivided twice and the remaining edges are subdivided once). Hence, the order of *G* is

Assume that, for each *i*, the edge *H* once subdivided twice produces the path *G*. One can easily see that the function *g* defined on *G* such that

This completes the proof. □

Theorem 4.5. *If G is a connected n-vertex graph with**such that the set of vertices with degree at least three is independent, then*

This bound is sharp for

Proof. We use an induction on the order *n*. Clearly, *n* with minimum degree at least two such that the set of vertices with degree at least three is independent. Let *G* be a graph of order *n* such that *A*-ear paths and keep the same notations as defined in the proof of Proposition 4.4. Note that

Assume first that there exists an *A*-ear path *P* such that *G* is simple, this means that *G* of degree three is adjacent to the end-vertices of *A* is independent), and thus there is a unique *A*-ear path *b* is an end-vertex of *c* be the other end-vertex of *a* and all vertices of *P* and *g* such that *h* defined on

Next, we can assume that *A*-ear path *A*-ear path *P* has an OIDRD-function *g* such that *g* assigns positive weight to the end-vertices of the path *P*. Now, let *G* by removing all vertices of *P*. By the induction hypothesis on *γ*_{oidR}(*G*′)-function *f*, the function *h* defined on *G*, and thus

## 5 Conclusion

In this article, we continued the study of outer-independent double Roman domination number and we characterized the trees *T* of order *G* of order *n* with minimum degree at least two in which the set of vertices with degree at least three is independent, *G* of order *n* with minimum degree remains open.

## Data Availability Statement

The original contributions presented in the study are included in the article/Supplementary Material. Further inquiries can be directed to the corresponding author.

## Author Contributions

All authors listed have made a substantial, direct, and intellectual contribution to the work and approved it for publication.

## Funding

This work was supported by the National Key R&D Program of China (No. 2018YFB1005100).

## Conflict of Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

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Keywords: outer independence double Roman domination, outer-independent double Roman dominating function, independent set, double Roman domination, Roman domination, tree

Citation: Rao Y, Kosari S, Sheikholeslami SM, Chellali M and Kheibari M (2021) On the Outer-Independent Double Roman Domination of Graphs. *Front. Appl. Math. Stat.* 6:559132. doi: 10.3389/fams.2020.559132

Received: 09 November 2020; Accepted: 04 December 2020;

Published: 05 February 2021.

Edited by:

Yong Chen, Hangzhou Dianzi University, ChinaReviewed by:

Zepeng Li, Lanzhou University, ChinaSarfraz Ahmad, COMSATS University Islamabad, Pakistan

Copyright © 2021 Rao, Kosari, Sheikholeslami, Chellali and Kheibari. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Saeed Kosari, saeedkosari38@yahoo.com