ORIGINAL RESEARCH article
Volume 6 - 2020 | https://doi.org/10.3389/fams.2020.559132
On the Outer-Independent Double Roman Domination of Graphs
- 1Institute of Computing Science and Technology, Guangzhou University, Guangzhou, China
- 2Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, Iran
- 3LAMDA-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
We consider only simple connected graphs G with vertex set
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
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
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) γoidR (G) is the minimum weight of an OIDRD-function on G. Clearly, γdR (G) ≤ γoidR (G) holds for every graph G. Recently, Mojdeh et al., Ref. 20, proved that the decision problem associated with γoidR (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,
Proposition 1.2. For every tree T on n ≥ 4 vertices, γoidR (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, γoidR (G) ≤ 4n/3.
2 Trees T of Order n With γoidR (T) = 5n/4
With the aim of characterizing the trees T of order n ≥ 3 with γoidR (T) = 5n/4, let
Lemma 2.1. Let
Proof. Let T be a tree of
Theorem 2.2. Let T be a tree on n ≥ 4 vertices. Then, γoidR (T) = 5n/4 if and only if
Proof. We prove only the necessity. Let T be a tree of n ≥ 4 such that γoidR (T) = 5n/4. Clearly,
which leads to a contradiction. Hence,
Therefore, by the induction hypothesis on
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
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
Proposition 3.3. Let T be a tree of order
Proposition 3.4. Let T be a tree of order n ≥ 3, where
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
Proposition 4.3. For integers
Proof. Assume first that
Let g be an OIDRD-function of the path
Proposition 4.4. For any graph
If k = 2, then the function g defined on
From now on, we can assume that
Next, assume that
Considering the above situations, we may assume that
Assume first that
Henceforth, we can assume that
Subcase 2.1.u is adjacent to at least two A-ear paths of
Suppose now that
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
Assume that, for each i, the edge
This completes the proof. □
Theorem 4.5. If G is a connected n-vertex graph with
This bound is sharp for
Proof. We use an induction on the order n. Clearly,
Assume first that there exists an A-ear path P such that
Next, we can assume that
In this article, we continued the study of outer-independent double Roman domination number and we characterized the trees T of order
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.
All authors listed have made a substantial, direct, and intellectual contribution to the work and approved it for publication.
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.
3. Abdollahzadeh Ahangar, H, Amjadi, J, Chellali, M, Nazari-Moghaddam, S, and Sheikholeslami, SM. Trees with double Roman domination number twice the domination number plus two. Iran J Sci Technol Trans A Sci (2019). 43:1081–1088. doi:10.1007/s40995-018-0535-7
8. Khoeilar, R, Chellali, M, Karami, H, and Sheikholeslami, SM. An improved upper bound on the double Roman domination number of graphs with minimum degree at least two. Discrete Appl Math (2019). 270:159–167. doi:10.1016/j.dam.2019.06.018
10. Maimani, HR, Momeni, M, Nazari-Moghaddam, S, Rahimi Mahid, F, and Sheikholeslami, SM. Independent double Roman domination in graphs. Bull Iranian Math Soc (2020). 46:543–555. doi:10.1007/s41980-019-00274-8
15. Abdollahzadeh Ahangar, H, Álvarez, MP, Chellali, M, Sheikholeslami, SM, and Valenzuela-Tripodoro, JC. Triple Roman domination in graphs. Appl Math Comput (2021). 391:125444. doi:10.1016/j.amc.2020.125444
16. Chellali, M, Jafari Rad, N, Sheikholeslami, SM, and Volkmann, L. Roman domination in graphs In: TW Haynes, ST Hedetniemi, and MA Henning, editors Topics in domination in graphs. Springer International Publishing (2020). doi:10.1007/978-3-030-51117-3
17. Chellali, M, Jafari Rad, N, Sheikholeslami, SM, and Volkmann, L. Varieties of Roman domination. In: TW Haynes, ST Hedetniemi, and MA Henning, editors Structures of domination in graphs. Springer International Publishing (2021). doi:10.1007/978-3-030-58892-2
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, China
Reviewed by:Zepeng Li, Lanzhou University, China
Sarfraz 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, firstname.lastname@example.org