The Double Roman Domination Number of Sunlet Graphs S<sub>n</sub>

Wensheng Li

doi:doi:10.11648/j.pamj.20261504.12

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The Double Roman Domination Number of Sunlet Graphs S_n

Wensheng Li^*

Published in Pure and Applied Mathematics Journal (Volume 15, Issue 4)

Received: 21 September 2026 Accepted: 4 October 2026 Published: 21 October 2026

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Abstract

A double Roman dominating function (DRDF) f on a given graph G is a mapping from V(G) to {0, 1, 2, 3} in such a way that a vertex u for which f(u) = 0 has at least a neighbor labeled 3 or two neighbors both labeled 2 and a vertex u for which f(u) = 1 has at least a neighbor labeled 2 or 3. The weight of a DRDF f is the value w(f) = ∑_u_∈_V₍_G₎ f(u). The minimum weight of a DRDF on a graph G is called the double Roman domination number γ_dR(G) of G. A Sunlet graph S_n is constructed from an n-cycle by attaching one pendant vertex to each vertex of the cycle. This paper investigates the double Roman domination of Sunlet graphs S_n for integer n≥3. Using three lemmas, we derive the exact formula γ_dR(S_n)=⌈7n/3⌉. We further analyze whether Sunlet graphs belong to the family of double Roman graphs. By comparing the domination number γ(S_n)=n with the known inequality 2γ(G)≤γ_dR(G)≤3γ(G) for double Roman graphs, we prove that S_n fails to satisfy the required inequality, which implies no Sunlet graph is a double Roman graph. The obtained result gives an explicit expression for the double Roman domination number of this class of unicyclic graphs and enriches the research achievements of double Roman domination theory for unicyclic graphs.

Published in	Pure and Applied Mathematics Journal (Volume 15, Issue 4)
DOI	10.11648/j.pamj.20261504.12
Page(s)	73-78
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2026. Published by Science Publishing Group

Keywords

Double Roman Domination, Double Roman Domination Number, Sunlet Graphs

1. Introduction

Let

be a graph without loops and multiple edges, where

and

are the vertex set and edge set of

, respectively. If

, we say that vertices

and

are adjacent, and

is a neighbor of

. The open neighborhood of

, denoted by

, is a set

．And the close neighborhood of

, denoted by

, is a set

．A function

is a double Roman dominating function, shorted by DRDF, if a vertex

for which

has at least a neighbor labeled 3 or two neighbors both labeled 2 and a vertex

for which

has at least a neighbor labeled 2 or 3．The sum of the function values of all vertices in

, which is shortly denoted by

. The weight of

, denoted by

, is the sum of the function values of all vertices in

．The double Roman domination number, denoted by

, equals the minimum weight of a DRDF on

, and a DRDF of

with weight

is called a

-function of

The double Roman domination was introduced in

[1]

and has been studied in

[2-9]

. Double Roman domination for cardinal products of graphs was studied in

[2]

, and double Roman trees were characterized in

[3]

. It is known that the decision problem associated with

is NP-complete for bipartite and chordal graphs, undirected path graphs, chordal bipartite graphs, and circle graphs

[4-6]

. Closely related problems to double Roman domination were studied in

[7-13]

. As observed in

[10],

the double Roman domination problem can be solved in linear time for unicyclic graphs, but explicit formulas for the double Roman domination number of general unicyclic graphs have not been established. Many results concern variants of double Roman domination on unicyclic graphs

[11, 12]

. However, most existing studies mainly focus on algorithmic complexity and bound analysis, while few studies provide closed-form exact values for specific unicyclic graph subclasses. In particular, the precise double Roman domination number of Sunlet graphs has not been formally characterized in previous literature. To fill this research gap, in this paper, we focus on a special family of unicyclic graphs—the Sunlet graphs, and determine the exact value of the double Roman domination number of Sunlet graphs.

The Sunlet graph

is obtained by taking a simple

-gon, that is, a cycle with vertices

,...,

, and attaching to each vertex

a new pendant vertex

, where

. Figure 1 shows a Sunlet graph with

vertices.

Download: Download full-size image

Figure 1. Sunlet graph

with

vertices. Sunlet graph

with

vertices.

[1]

, Beeler et al. established the following result.

Proposition A

[1]

In a double Roman dominating function of weight

, no vertex needs to be assigned the value 1.

In view of Proposition A, we only consider double Roman dominating functions of G with no vertex labeled 1 in the subsequent discussion. The following observation will be frequently utilized throughout the paper.

Proposition B

[13]

Let

be a DRDF on

and

is a leaf of

. Then

2. Double Roman Domination Number of S_n

Let

be a continuous local subgraph of

and

for

, where subscripts are reduced modulo

as shown in Figure 2.

Download: Download full-size image

Figure 2.

, a continuous local component of

Proposition C Let

be a

-function of

with no vertex assigned value 1. Then

where

Proof. By Proposition B, we have

We now prove that the case

is impossible via proof by contradiction. Suppose for contradiction that

. Since no vertex in

is labeled with value 1, the non-zero labels assigned to vertices in

can only take the form {2,2,2} or {3,3}.

Case 1: The non-zero labels are {2,2,2}.

By the definition of the labeling function, if a leaf vertex carries label 0, its supporting vertex must be assigned label 3. It immediately follows that no leaf vertex can take the label 0. Thus, the three leaf vertices

and

must each receive label 2 exclusively. Under this configuration, vertex

bears label 0, and its open neighborhood contains exactly one vertex with a non-zero label (value 2), which directly violates the labeling axiom.

Case 2: The non-zero labels are {3,3}.

Among the three pairs consisting of supporting vertices and their corresponding leaf vertices

and

, at least one pair is entirely assigned label 0. This configuration contradicts Proposition B.

Combining the above two contradictory cases, our initial hypothesis

is invalid. Consequently, we derive

, i.e.,

Lemma 1 Let

, where

.Then

Proof. Consider the mapping

specified in Table 1, where

for indices and all subscripts are taken modulo n. Then, for every vertex labeled 0, there exists a vertex labeled 3 adjacent to it. Therefore,

is a DRDF on

and

, which implies

Table 1. The mapping f :V(S_n)→{0,2,3}with

Vertex	Label	Neighbors	Neighbors with Label 3
v_i, i≡1(mod 3)	0	u_i, v_i_-₁, v_i₊₁	v_i₊₁
v_i, i≡2(mod 3)	3
v_i, i≡0(mod 3)	0	u_i, v_i_-₁, v_i₊₁	v_i_-₁
u_i, i≡1 (mod 3)	2
u_i, i≡2(mod 3)	0	v_i	v_i
u_i, i≡0(mod 3)	2

The mapping is illustrated in Figure 3 with

Download: Download full-size image

Figure 3. The mapping

with

The mapping

with

On the other hand, let

be a

-function of

with no vertex assigned value 1. By Proposition C, we have

From the above arguments, we obtain

for

Lemma 2 Let

, where

.Then

Proof. Consider the mapping

specified in Table 2, where

for indices and all subscripts are taken modulo n. Then, for every vertex labeled 0, there exists a vertex labeled 3 adjacent to it. Therefore,

is a DRDF on

and

, which implies

Table 2. The mapping f :V(S_n)→{0,2,3}with

Vertex	Label	Neighbors	Neighbors with Label 3
v_i, i≡1(mod 3) and i≠n	0	u_i, v_i_-₁, v_i₊₁	v_i₊₁
v_i, i≡2(mod 3)	3
v_i, i≡0(mod 3)	0	u_i, v_i_-₁, v_i₊₁	v_i_-₁
v_n	3
u_i, i≡1(mod 3) and i≠n	2
u_i, i≡2(mod 3)	0	v_i	v_i
u_i, i≡0(mod 3)	2
u_n	0	v_n	v_n

The mapping is illustrated in Figure 4 with

Download: Download full-size image

Figure 4. The mapping

with

. The mapping

with

On the other hand, let

be a

-function of

with no vertex assigned value 1. By Proposition B and Proposition C we have

We next demonstrate that the case

is unattainable via proof by contradiction. Assume for contradiction that

, which implies

Assume

. The sum of weights over the

consecutive local subgraph configurations equals

. By the pigeonhole principle, at least one of these configurations has weight less than 7. This contradicts Property C, which states that the weight of any consecutive local subgraph configuration is at least 7. Accordingly, we obtain

From the above reasoning, we conclude

with

Lemma 3 Let

, where

.Then

Proof. Consider the mapping

specified in Table 3, where

for indices and all subscripts are taken modulo n. Then, for every vertex labeled 0, there exists a vertex labeled 3 adjacent to it. Therefore,

is a DRDF on

and

, which implies

Table 3. The mapping f :V(S_n)→{0,2,3} with

Vertex	Label	Neighbors	Neighbor with Label 3
v_i, i≡1 (mod 3) and i≠n	0	u_i, v_i_-₁, v_i₊₁	v_i₊₁
v_i, i≡2(mod 3)	3
v_i, i≡0(mod 3)	0	u_i, v_i_-₁, v_i₊₁	v_i_-₁
v_n-₁	3
v_n	0	u_n, v_n_-₁, v₁	v_n-₁
u_i, i≡1(mod 3) and i≠n	2
u_i, i≡2(mod 3)	0	v_i	v_i
u_i, i≡0(mod 3)	2
u_n_-1	0	v_n-₁	v_n-₁
u_n	2

The mapping is illustrated in Figure 5 with

Download: Download full-size image

Figure 5. The mapping

with

. The mapping

with

On the other hand, let

be a

-function of

with no vertex assigned value 1. By Proposition B and Proposition C, we have

We now show that the case

cannot hold. We proceed by proof by contradiction. Suppose for contradiction that

, which yields

Assume

. The sum of weights over the

consecutive local subgraph configurations equals

Combining the above arguments, we obtain

with

3. Discussion

By Lemma 1, Lemma 2 and Lemma 3, we obtain the following theorem.

Theorem 1 Let

be a Sunlet graph where

. Then

[1]

, Beeler et al. proposed the concept of the double Roman domination. They showed that 2γ(G) ≤ γdR(G) ≤ 3γ(G). Moreover, they suggested to ﬁnd double Roman graphs.

[14]

, It was proven that

Theorem 2 For a graph

with even order

and no isolated vertices,

if and only if the components of

are the cycle

or the corona

for any connected graph

By Theorem 2,

. Consequently,

is not a double Roman graph for all integers

4. Conclusions

This paper investigates the double Roman domination number and related properties of Sunlet graphs

. We obtain the exact value

, and our results reveal that every Sunlet graph

with

fails to satisfy the required inequality for double Roman graphs, implying no Sunlet graph belongs to the family of double Roman graphs. Several directions deserve further exploration. First, it is worthwhile to extend the discussion to generalized Sunlet graphs with varied pendant structures. We aim to characterize the exact double Roman domination numbers of generalized Sunlet graphs and determine the parameter range under which generalized Sunlet graphs can be double Roman graphs. In deriving the double Roman domination number of Sunlet graphs, we employ local configuration decomposition based on their topological structure. For generalized Sunlet graphs, certain properties of path graphs may be required to handle their structural features. Second, one can establish tighter bounds relating the domination number and double Roman domination number for these graphs. Additionally, further research may develop universal criteria to judge whether unicyclic graphs, including generalized Sunlet graphs, are double Roman graphs.

Abbreviations

DRDF	Double Roman Dominating Function

Author Contributions

Wensheng Li: Conceptualization, Methodology, Resources, Validation, Writing – original draft, Writing – review & editing

Conflicts of Interest

The author declares no conflicts of interest.

References

[1]	R. A. Beeler, T. W. Haynes, and S. T. Hedetniemi. Double Roman domination, Discrete Appl. Math., 2016, vol. 211, 23–29, Oct. 2016. https://doi.org/10.1016/j.dam.2016.03.017
[2]	A. Klobucar; A. Klobucar. Properties of double Roman domination on cardinal products of graphs, Ars Math. Contemp., vol. 19, 337–349, 2020. https://doi.org/10.26493/1855-3974.2022.44a
[3]	H. A. Ahangar, J. Amjadi, M. Atapour, M. Chellali, S. M Sheikholeslami. Double Roman trees, Ars Combin., vol. 145, 173–183, 2019. https://combinatorialpress.com/ars/vol145/
[4]	H. A. Ahangar, M. Chellai, S. M. Sheikholeslam. On the double Roman domination in graphs, Discrete Appl. Math., vol. 232, 1–7, 2017. https://doi.org/10.1016/j.dam.2017.06.014
[5]	S. Banerjee, M. A. Henning, D. Pradhan. Algorithmic results on double Roman domination in graphs, J. Comb. Opt., vol. 39, 90–114, 2020. https://doi.org/10.1007/s10878-019-00457-3
[6]	N. J. Rad, H. Rahbani. Some progress on the double Roman domination in graphs, Discuss. Math. Graph Theory, vol. 39, 41–53, 2019. https://doi.org/10.7151/dmgt.2069
[7]	J. Meena, T. N. M. Malini Mai, M. L. Suresh, L. Rathour, L. N. Mishra. Double Roman domination in some graphs, Discrete Mathematics, Algorithms and Applications, vol. 18(1), 2550016, 2026. https://doi.org/10.1142/s1793830925500168
[8]	S. M. Sheikholeslami, M. Esmaeili, J. J. Hamja., C. L. Armada, I. S. Aniversario. Independent double Roman domination stability in graphs, European Journal of Pure & Applied Mathematics, vol. 18, 1-16, 2025. https://doi.org/10.29020/nybg.ejpam.v18i2.5984
[9]	T. Zec, D. Matić, M. Djukanović. On double Roman domination problem for several graph classes, Aequationes mathematicae, vol. 99, 439-463, 2025. https://doi.org/10.1007/s00010-024-01071-3
[10]	D. P. Rupnik, J. Žerovnik. Double Roman domination: A Survey, Mathematics, vol. 11, 351, 2023. https://doi.org/10.3390/math11020351
[11]	S. Nazari-Moghaddam, M. Chellali, S. M. Sheikholeslami. Outer independent double Roman domination in unicyclic and bicyclic graphs. Ars Combinatoria, vol. 162, 51–70, 2025. https://doi.org/10.61091/ars162-05
[12]	S. M. Sheikholeslami, M. Esmaeili, M. Chellali, Y. Shang. Lower bounds for signed double Roman k-domination in unicyclic graphs, Research in Mathematics, vol. 13(1), 2690319, 2026. https://doi.org/10.1080/27684830.2026.2690319
[13]	W. S. Li, Z. S. Huang, Z. F. Feng. On the double Roman domination in Spider graphs, Open Journal of Discrete Mathematics, vol. 16(2), 13-18, 2026. https://doi.org/10.4236/ojdm.2026.162002
[14]	T. W. Haynes, S. T. Hedetniemi, P. J. Slater. Fundamentals of Domination in Graphs, 1998.

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APA Style

Li, W. (2026). The Double Roman Domination Number of Sunlet Graphs Sn. Pure and Applied Mathematics Journal, 15(4), 73-78. https://doi.org/10.11648/j.pamj.20261504.12

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ACS Style

Li, W. The Double Roman Domination Number of Sunlet Graphs Sn. Pure Appl. Math. J. 2026, 15(4), 73-78. doi: 10.11648/j.pamj.20261504.12

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AMA Style

Li W. The Double Roman Domination Number of Sunlet Graphs Sn. Pure Appl Math J. 2026;15(4):73-78. doi: 10.11648/j.pamj.20261504.12

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@article{10.11648/j.pamj.20261504.12,
  author = {Wensheng Li},
  title = {The Double Roman Domination Number of Sunlet Graphs Sn},
  journal = {Pure and Applied Mathematics Journal},
  volume = {15},
  number = {4},
  pages = {73-78},
  doi = {10.11648/j.pamj.20261504.12},
  url = {https://doi.org/10.11648/j.pamj.20261504.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20261504.12},
  abstract = {A double Roman dominating function (DRDF) f on a given graph G is a mapping from V(G) to {0, 1, 2, 3} in such a way that a vertex u for which f(u) = 0 has at least a neighbor labeled 3 or two neighbors both labeled 2 and a vertex u for which f(u) = 1 has at least a neighbor labeled 2 or 3. The weight of a DRDF f is the value w(f) = ∑u∈V(G) f(u). The minimum weight of a DRDF on a graph G is called the double Roman domination number γdR(G) of G. A Sunlet graph Sn is constructed from an n-cycle by attaching one pendant vertex to each vertex of the cycle. This paper investigates the double Roman domination of Sunlet graphs Sn for integer n≥3. Using three lemmas, we derive the exact formula γdR(Sn)=⌈7n/3⌉. We further analyze whether Sunlet graphs belong to the family of double Roman graphs. By comparing the domination number γ(Sn)=n with the known inequality 2γ(G)≤γdR(G)≤3γ(G) for double Roman graphs, we prove that Sn fails to satisfy the required inequality, which implies no Sunlet graph is a double Roman graph. The obtained result gives an explicit expression for the double Roman domination number of this class of unicyclic graphs and enriches the research achievements of double Roman domination theory for unicyclic graphs.},
 year = {2026}
}

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TY  - JOUR
T1  - The Double Roman Domination Number of Sunlet Graphs Sn
AU  - Wensheng Li
Y1  - 2026/10/21
PY  - 2026
N1  - https://doi.org/10.11648/j.pamj.20261504.12
DO  - 10.11648/j.pamj.20261504.12
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 73
EP  - 78
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20261504.12
AB  - A double Roman dominating function (DRDF) f on a given graph G is a mapping from V(G) to {0, 1, 2, 3} in such a way that a vertex u for which f(u) = 0 has at least a neighbor labeled 3 or two neighbors both labeled 2 and a vertex u for which f(u) = 1 has at least a neighbor labeled 2 or 3. The weight of a DRDF f is the value w(f) = ∑u∈V(G) f(u). The minimum weight of a DRDF on a graph G is called the double Roman domination number γdR(G) of G. A Sunlet graph Sn is constructed from an n-cycle by attaching one pendant vertex to each vertex of the cycle. This paper investigates the double Roman domination of Sunlet graphs Sn for integer n≥3. Using three lemmas, we derive the exact formula γdR(Sn)=⌈7n/3⌉. We further analyze whether Sunlet graphs belong to the family of double Roman graphs. By comparing the domination number γ(Sn)=n with the known inequality 2γ(G)≤γdR(G)≤3γ(G) for double Roman graphs, we prove that Sn fails to satisfy the required inequality, which implies no Sunlet graph is a double Roman graph. The obtained result gives an explicit expression for the double Roman domination number of this class of unicyclic graphs and enriches the research achievements of double Roman domination theory for unicyclic graphs.
VL  - 15
IS  - 4
ER  -

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Author Information

Wensheng Li

School of Science, Langfang Normal University, Langfang, China

Biography: Wensheng Li received the B. S. degree in Information and Computing Science from the Hebei University of Technology, Hebei, China, in 2003. He obtained the M. S. degree in Computational Mathematics from the Dalian University of Technology, Dalian, China, in 2006. He worked as a lecturer in Department of Mathematics and Infor-mation Science, Langfang Teachers College. His research interest is in the area of graph theory.

Research Fields: Optimization, Computational mathematics, Mathematical Modelling, Graph Theory and Applied mathematics

Contact Email

http://orcid.org/0009-0004-0872-2515

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Figure 1

Figure 1. Sunlet graph with vertices.
Figure 2

Figure 2. , a continuous local component of .
Figure 3

Figure 3. The mappingwith
Figure 4

Figure 4. The mappingwith .
Figure 5

Figure 5. The mappingwith .

Plain Text BibTeX RIS

APA Style

Li, W. (2026). The Double Roman Domination Number of Sunlet Graphs Sn. Pure and Applied Mathematics Journal, 15(4), 73-78. https://doi.org/10.11648/j.pamj.20261504.12

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ACS Style

Li, W. The Double Roman Domination Number of Sunlet Graphs Sn. Pure Appl. Math. J. 2026, 15(4), 73-78. doi: 10.11648/j.pamj.20261504.12

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AMA Style

Li W. The Double Roman Domination Number of Sunlet Graphs Sn. Pure Appl Math J. 2026;15(4):73-78. doi: 10.11648/j.pamj.20261504.12

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@article{10.11648/j.pamj.20261504.12,
  author = {Wensheng Li},
  title = {The Double Roman Domination Number of Sunlet Graphs Sn},
  journal = {Pure and Applied Mathematics Journal},
  volume = {15},
  number = {4},
  pages = {73-78},
  doi = {10.11648/j.pamj.20261504.12},
  url = {https://doi.org/10.11648/j.pamj.20261504.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20261504.12},
  abstract = {A double Roman dominating function (DRDF) f on a given graph G is a mapping from V(G) to {0, 1, 2, 3} in such a way that a vertex u for which f(u) = 0 has at least a neighbor labeled 3 or two neighbors both labeled 2 and a vertex u for which f(u) = 1 has at least a neighbor labeled 2 or 3. The weight of a DRDF f is the value w(f) = ∑u∈V(G) f(u). The minimum weight of a DRDF on a graph G is called the double Roman domination number γdR(G) of G. A Sunlet graph Sn is constructed from an n-cycle by attaching one pendant vertex to each vertex of the cycle. This paper investigates the double Roman domination of Sunlet graphs Sn for integer n≥3. Using three lemmas, we derive the exact formula γdR(Sn)=⌈7n/3⌉. We further analyze whether Sunlet graphs belong to the family of double Roman graphs. By comparing the domination number γ(Sn)=n with the known inequality 2γ(G)≤γdR(G)≤3γ(G) for double Roman graphs, we prove that Sn fails to satisfy the required inequality, which implies no Sunlet graph is a double Roman graph. The obtained result gives an explicit expression for the double Roman domination number of this class of unicyclic graphs and enriches the research achievements of double Roman domination theory for unicyclic graphs.},
 year = {2026}
}

Copy | Download

TY  - JOUR
T1  - The Double Roman Domination Number of Sunlet Graphs Sn
AU  - Wensheng Li
Y1  - 2026/10/21
PY  - 2026
N1  - https://doi.org/10.11648/j.pamj.20261504.12
DO  - 10.11648/j.pamj.20261504.12
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 73
EP  - 78
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20261504.12
AB  - A double Roman dominating function (DRDF) f on a given graph G is a mapping from V(G) to {0, 1, 2, 3} in such a way that a vertex u for which f(u) = 0 has at least a neighbor labeled 3 or two neighbors both labeled 2 and a vertex u for which f(u) = 1 has at least a neighbor labeled 2 or 3. The weight of a DRDF f is the value w(f) = ∑u∈V(G) f(u). The minimum weight of a DRDF on a graph G is called the double Roman domination number γdR(G) of G. A Sunlet graph Sn is constructed from an n-cycle by attaching one pendant vertex to each vertex of the cycle. This paper investigates the double Roman domination of Sunlet graphs Sn for integer n≥3. Using three lemmas, we derive the exact formula γdR(Sn)=⌈7n/3⌉. We further analyze whether Sunlet graphs belong to the family of double Roman graphs. By comparing the domination number γ(Sn)=n with the known inequality 2γ(G)≤γdR(G)≤3γ(G) for double Roman graphs, we prove that Sn fails to satisfy the required inequality, which implies no Sunlet graph is a double Roman graph. The obtained result gives an explicit expression for the double Roman domination number of this class of unicyclic graphs and enriches the research achievements of double Roman domination theory for unicyclic graphs.
VL  - 15
IS  - 4
ER  -

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