EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 15, No. 1, 2022, 207-223
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Restrained Disjunctive Domination in Graphs under
some Binary Operations

Rolando P. Malalay1,∗, Ferdinand P. Jamil2

1 College of Engineering and Technology, Zamboanga Peninsula Polytechnic State
University, 7000 Zamboanga City, Philippines
2 Department of Mathematics and Statistics, College of Science and Mathematics,
Center of Graph Theory, Algebra and Analysis, Premier Research Institute of Science and
Mathematics, Mindanao State University-Iligan Institute of Technology, 9200 Iligan City,
Philippines

Abstract. A set S ⊆ V (G) is a disjunctive dominating set of a graph G if for every v ∈ V (G)\S, v
is a neighbor of a vertex in S or S has at least two vertices each at distance 2 from v. We say that a
disjunctive dominating set S of G is a restrained disjunctive dominating set if for each v ∈ V (G)\S
there exists u ∈ V (G) \ S such that uv ∈ E(G) or there exist distinct vertices u,w ∈ V (G) \ S
such that dG(u, v) = 2 = dG(w, v). The minimum cardinality γd

r (G) of a restrained disjunctive
dominating set of G is the restrained disjunctive domination number of G. In this paper, we
characterize the restrained disjunctive dominating sets in some binary operations such as the join,
corona and lexicographic product of graphs and, as a result, obtain the values of their corresponding
restrained disjunctive domination numbers.

2020 Mathematics Subject Classifications: 05C69
Key Words and Phrases: Disjunctive dominating set, restrained disjunctive dominating set,
restrained disjunctive domination number, join, corona and lexicographic product

1. Introduction

The study of restrained domination in graphs was first initiated by Domke et al. [6]
in 1997. Accordingly, they established the best possible upper and lower bounds for the
restrained domination number of a connected graph G and characterized those graphs
achieving these bounds. In [16], the restrained dominating set of trees of order n were
characterized, and the exact values of the restrained domination number were determined.
Some studies in restrained domination can be found in [7, 17, 19].

On the other hand, disjunctive domination in graphs, specifically b-disjunctive, first
introduced by Goddard et al. [8] in 2014 was motivated by the concept of domination

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v15i1.4187

Email addresses: orlandomalalay@gmail.com (R. Malalay),
ferdinand.jamil@g.msuiit.edu.ph (F. Jamil)

http://www.ejpam.com 207 © 2022 EJPAM All rights reserved.



R. Malalay, F. Jamil / Eur. J. Pure Appl. Math, 15 (1) (2022), 207-223 208

in graphs (see [4, 10]), the distance domination (see [5]) and the secondary domination
(see [11]). While most of the variations on dominating sets which have been introduced
recently tend to increase the domination number, which in effect raise implementation
costs, disjunctive domination is a relaxation of the domination number [14]. In [8], sharp
bounds for the disjunctive domination number were established for general graphs, and
exact values were determined for specific graphs.

In 2016, Henning and Naicker also extended the concept of total domination by defining
the disjunctive total domination. Accordingly, it allows for greater flexibility by modeling
networks where one trades off redundancy and backup capability with resource optimiza-
tion [14]. The above-mentioned authors established in [14] tight upper bound on the
disjunctive total domination number of a graph in terms of its order and characterized
the extremal graphs, and then proved that this bound can be significantly improved if
claw-freeness of a graph is imposed. The same authors also investigated the variant on
the class of trees in [12, 13].

Motivated by the concepts of b-disjunctive domination and disjunctive total domina-
tion in graphs, Jamil and Malalay [15] in 2019 extended the study of disjunctive domi-
nating, particularly 2-disjunctive domination, and the disjunctive total dominating sets
under some binary operations in graphs. Specifically, they characterized in [15] the dis-
junctive dominating sets and the disjunctive total dominating sets in the join, corona and
lexicographic product of graphs and obtained the values of the respective corresponding
disjunctive domination numbers.

In this paper, we introduce another variant of domination, namely restrained disjunc-
tive domination, and characterize the restrained disjunctive dominating sets in the join,
corona and lexicographic product of graphs.

In here, by a graph we mean a finite, simple and undirected connected graph G =
(V (G), E(G)). All basic terminologies used here are adapted from [2]. The symbols V (G)
and E(G) are the vertex-set and edge-set, respectively, of G. For S ⊆ V (G), |S| is the
cardinality of S. In particular, |V (G)| is called the order of G. The complement of G is
that graph G, where V (G) = V (G) and for its edges, xy ∈ E(G) if and only if xy /∈ E(G).

Given graphs G and H, the join of G and H is the graph G + H with vertex set
V (G) ∪ V (H) and edge set E(G) ∪ E(H) ∪ {uv : u ∈ V (G), v ∈ V (H)}. The corona of G
and H is the graph G ◦H obtained by taking one copy of G and |V (G)| copies of H, and
then joining the ith vertex of G to every vertex in the ith copy of H. In particular, we call
G ◦K1 the corona of G, and write cor(G) = G ◦K1. The lexicographic product of G and
H is the graph G[H] with V (G[H]) = V (G) × V (H) and (u, v)(u′, v′) ∈ E(G[H]) if and
only if either uu′ ∈ E(G) or u = u′ and vv′ ∈ E(H). In any of these graphs, G and H are
referred to as their basic component graphs.

Vertices u and v of a graph G are neighbors if uv ∈ E(G). The open neighborhood of
v ∈ V (G) refers to the set NG(v) consisting of all neighbors of v. The degree of v ∈ V (G)
refers to the cardinality |NG(v)| of the open neighborhood of v, δ(G) is the minimum
degree of a vertex of G, and ∆(G) is the maximum degree of G. The closed neighborhood

of v ∈ V (G) is NG[v] = NG(v) ∪ {v}. Customarily, for S ⊆ V (G), NG(S) =
⋃
v∈S

NG(v)



R. Malalay, F. Jamil / Eur. J. Pure Appl. Math, 15 (1) (2022), 207-223 209

and NG[S] =
⋃
v∈S

NG[v]. A subset S ⊆ V (G) is a dominating set of G if NG[S] =

V (G). In case, S is a dominating set of G and every vertex in V (G) \ S is adjacent to
another vertex in V (G) \ S, then S is a restrained dominating set of G. The minimum
cardinality γ(G) of a dominating set ofG is the domination number ofG, and the minimum
cardinality γr(G) of a restrained dominating set is the restrained domination number of
G. A dominating set of cardinality γ(G) is called a γ-set of G. Similarly, a γr-set is a
restrained dominating set of cardinality γr(G). The reader is referred to [1, 4, 6, 10, 18]
for the history, fundamental concepts and recent developments of domination in graphs as
well as its various applications.

A dominating set S ⊆ V (G) is a 2-dominating set of G if for each v ∈ V (G) \ S,
|NG(v) ∩ S| ≥ 2. The minimum cardinality of a 2-dominating set is the 2-domination
number of G, denoted by γ×2(G). Any 2-dominating set with cardinality γ×2(G) is called
a γ×2-set of G. The 2-domination in graphs is being studied in [3, 7, 9].

For a vertex v of G, NG(v, 2) = {u ∈ V (G) \ {v} : dG(u, v) ≤ 2}. For S ⊆ V (G),
NG(S, 2) = ∪v∈SNG(v, 2). A set S ⊆ V (G) is a distance-two dominating set of G provided
V (G)\S ⊆ NG(S, 2), i.e., if for every v ∈ V (G)\S there exists u ∈ S such that dG(u, v) ≤
2. The minimum cardinality γ2(G) of a distance-two dominating set is the distance-two
domination number of G. A distance-two dominating set of cardinality γ2(G) is called a
γ2-set. Some studies in distance-two domination can be found in [20].

A set S ⊆ V (G) is a disjunctive dominating set of G if for every v ∈ V (G) \ S, v is a
neighbor of a vertex in S or S has at least two vertices each at distance 2 from v. The
minimum cardinality of a disjunctive dominating set is the disjunctive domination number
of G, and is denoted by γd(G). A disjunctive dominating set of cardinality γd(G) is called
a γd-set. For convenience, the symbol Nd

G(S) denotes the set of all x ∈ V (G) such that
xy ∈ E(G) for some y ∈ S or there exist distinct u, v ∈ S with dG(x, u) = 2 = dG(x, v).
Then S is a disjunctive dominating set of G if and only if V (G) \ S ⊆ Nd

G(S). Since
NG(S) ⊆ Nd

G(S), dominating sets are disjunctive dominating sets. In particular, γd(G) =
1 if and only if γ(G) = 1.

A disjunctive dominating set S of G is a disjunctive total dominating set of G provided
V (G) = Nd

G(S). That is, S is a disjunctive total dominating set if for every v ∈ S, v is
adjacent to a vertex of S or S has at least two vertices each at distance 2 from v. The
minimum cardinality of a disjunctive total dominating set of G is the disjunctive total
domination number of G, and is denoted by γdt (G). Any disjunctive total dominating set
of cardinality γdt (G) is called γdt -set.

Provided G has no isolated vertices, a disjunctive dominating set S of G is a restrained
disjunctive dominating set of G if for each v ∈ V (G) \ S there exists u ∈ V (G) \ S such
that uv ∈ E(G) or there exist distinct vertices u,w ∈ V (G) \ S such that dG(u, v) =
2 = dG(w, v). The minimum cardinality of a restrained disjunctive dominating set of
G is the restrained disjunctive domination number of G, and is denoted by γdr (G). Any
restrained disjunctive dominating set of cardinality γdr (G) is called γdr -set. Since restrained
disjunctive dominating sets are disjunctive dominating sets, 1 ≤ γd(G) ≤ γdr (G) for all
graphs G without isolated vertices.



R. Malalay, F. Jamil / Eur. J. Pure Appl. Math, 15 (1) (2022), 207-223 210

2. Preliminaries and Known Results

The following are some known results for paths and cycles.

Theorem 1. (i) [8] For all n, γd(Pn) = ⌈n+1
4 ⌉;

(ii) [8] For cycle Cn, n ≥ 3, γd(Cn) = 2 for n = 4, and γd(Cn) = ⌈n4 ⌉ for n ̸= 4.

Next are our preliminary results which have exact values or bounds of some special
graphs in terms of the restrained disjunctive domination number. Let G be a connected
graph and u, v ∈ V (G) such that uv ∈ E(G). If u is the end vertex of graph G and
degG(v) ≤ 2, then u ∈ S for all restrained disjunctive dominating sets S of G.

Proposition 1. (i) For path Pn, n ≥ 1,

γdr (Pn) =

{
3, if n = 3

⌈n−1
4 ⌉+ 1, if n ̸= 3.

(ii) For cycle Cn, n ≥ 3,

γdr (Cn) =


1, if n = 3

2, if n = 4

⌈n4 ⌉, if n ≥ 5.

Proof. To prove (i), let Pn = [x1, x2, . . . , xn]. The case where 1 ≤ n ≤ 4 is obvious.
Suppose that n ≥ 5. Let k be the largest integer for which 4k + 1 ≤ n. If n− 4k − 1 ̸= 2,
take S = {x1, x5, . . . , x4k+1, xn}. Otherwise, take S = {x1, x5, . . . , x4k, xn}. Then S is a
restrained disjunctive dominating set of Pn. Thus, γdr (Pn) ≤ ⌈n−1

4 ⌉ + 1. Now, suppose
that S∗ ⊆ V (G) is a restrained disjunctive dominating set of G with |S∗| ≤ ⌈n−1

4 ⌉. In
view of Observation 2, x1, xn ∈ S∗. In particular, |S∗| = 1 if n = 5, which is impossible.
Let n ≥ 6. Pick u, v ∈ S such that dPn(u, v) is maximum among all pairs (x, y) ∈ S × S
for which S \ {x, y} does not contain a vertex lying in the x-y path. Then dPn(u, v) ≥ 5.
Thus, there exists w ∈ V (Pn) \ S such that w /∈ Nd

Pn
(S), a contradiction.

The case where 3 ≤ n ≤ 4 is obvious for Cn. Let n ≥ 5. By Theorem 1, ⌈n4 ⌉ =
γd(Cn) ≤ γdr (Cn). Let Cn = [x1, x2, . . . , xn, x1]. Define S = {x1, x5, . . . , x4k+1}, where k
is the largest integer for which 4k+ 1 ≤ n. Then S is a restrained disjunctive dominating
set of Cn. Thus, γ

d
r (Cn) ≤ |S| = ⌈n4 ⌉.

Theorem 2. Let G be a connected graph of order n ≥ 2. Then

(i) γdr (G) ≤ n−∆(G) provided ∆(G) ≥ 3.

(ii) γdr (G) = 1 if and only if n ≥ 3, G ̸= P3 and γ(G) = 1.



R. Malalay, F. Jamil / Eur. J. Pure Appl. Math, 15 (1) (2022), 207-223 211

Proof. To prove (i), suppose that ∆(G) ≥ 3, and let v ∈ V (G) for which degG(v) =
∆(G). Put S = V (G) \ NG(v). Then S is a restrained disjunctive dominating set of G,
and the result follows.

To prove (ii), suppose that γdr (G) = 1. In view of the remark above, 1 ≤ γd(G) ≤
γdr (G) = 1 showing that γd(G) = 1. By the remark above, γ(G) = 1. Since γdr (P2) = 2,
n ≥ 3. In the case where n = 3, G = K3. To prove the converse, it is enough to consider
only the case where n ≥ 4. Let S = {v} be a dominating set of G. Then S is a disjunctive
dominating set of G. Let u ∈ V (G) \ S. Since S is a dominating set, uv ∈ E(G). Since
n ≥ 4, we can pick distinct x, y ∈ V (G) \ {u, v}. Then vx ∈ E(G) and vy ∈ E(G). If
ux ∈ E(G) or uv ∈ E(G), then we are done. Suppose that ux, uy /∈ E(G). Then [u, v, x]
is a u-x geodesic in G. Thus, dG(u, x) = 2. Similarly, dG(u, y) = 2. This shows that S is
a restrained disjunctive dominating set of G. Accordingly, γdr (G) = 1.

Corollary 1. For a connected graph G of order n ≥ 3, γdr (G) = 1 if and only if G = K3

or G = K1 +H for some graph H with |V (H)| ≥ 3.

Note that since γdr (K1,n) = 1 for all n ≥ 3, the bound in Theorem 2(i) is sharp.

3. Exploratory Examples

For any connected graph G, γdr (G) ≤ γr(G) and γd(G) ≤ γdr (G) , and the following
example shows that up to some extent, the values of these parameters can be pre-assigned.

Example 1. For any positive integers a, b with 2 ≤ a ≤ b where b ≤ 3a − 1, there exists
a connected graph G such that γdr (G) = a and γr(G) = b.

To see this, consider the following cases:

Case 1. a = b
In this case, we take G as the graph G1 obtained by connecting a copies of 3-pan

graph using the path [u1, u2, . . . , ua], where ui is a pendant of the ith copy of 3-pan graph
for each i ∈ {1, 2, . . . , a} as shown in Figure 1. Denote by P u that copy of 3-pan graph
corresponding to vertex u.

Figure 1: A graph G with 2 ≤ γd
r (G) = γr(G)

Since the set S = {xi : i ∈ {1, 2, . . . , a}} is both a restrained disjunctive dominating
set and a restrained dominating set of G, γdr (G) = γr(G) ≤ |S| = a. Let D ⊆ V (G)



R. Malalay, F. Jamil / Eur. J. Pure Appl. Math, 15 (1) (2022), 207-223 212

such that |D| < a. Then there exists 1 ≤ k ≤ a such that D ∩ V (3 − P uk) = ∅. In
particular, V (P uk)\{xk, uk} is not dominated by D in the sense of disjunctive domination
and the usual domination. Thus, γdr (G) ≥ a and γr(G) ≥ a. Consequently, γdr (G) = a
and γr(G) = a = b.

Case 2: a < b
Here, we write b = a+ k, where k ≥ 1. Obtain the graph G∗ as graph G2 from graph

G1 in the first case by adding to G1 the pendant edges x1yi for each i ∈ {1, 2, . . . , k} as
shown in Figure 2.

Figure 2: A graph G∗ with 2 ≤ γd
r (G

∗) < γr(G
∗)

Since S1 = {xi : i ∈ {1, 2, . . . , a}} is a γdr -set of G
∗, γdr (G

∗) = |S1| = a. Observe that the
set S2 = S1 ∪ {yj : j ∈ {1, 2, . . . , k}} is a restrained dominating set of G∗ showing that
γr(G

∗) ≤ |S2| = a+ k = b. Let S ⊆ V (G∗) be a γr-set of G
∗. Then S contains S1. Being

a restrained disjunctive dominating set of G∗ and S contains S1, necessarily yj ∈ S, for
each j ∈ {1, 2, . . . , k}. Thus, γr(G∗) ≥ |S| = a+ k = b. Consequently, γr(G

∗) = b.

Example 2. For a pair of positive integers (a, b) with 2 ≤ a ≤ b where either b ≤ 3a
2 if a

is even or b ≤ 3a+1
2 when a is odd, there exists a connected graph G such that γd(G) = a

and γdr (G) = b.

Consider the following cases:

Case 1. a = b
Take G to be the graph G1 obtained from the path P2a−1 = [u1, u2, . . . , u2a−1] as shown

in Figure 3, by adding to P2a−1 the pendant edges xju2j−1 for all j = 1, 2, . . . , a. The

Figure 3: A graph G with γd(G) = γd
r (G)

set {xi : i = 1, 2, . . . , a} is both a disjunctive dominating set and a restrained disjunctive
dominating of G. Thus, γd(G) ≤ a and γdr (G) ≤ a = b. Let S ⊆ V (G) be a γd-set of
G. Then either xj ∈ S or u2j−1 ∈ S for all j = 1, 2, . . . , a. Also, being a disjunctive
dominating set of G, if u2j−2 ∈ S for all j = 2, 3, . . . , a, then necessarily u1u2a−1 ∈ S.



R. Malalay, F. Jamil / Eur. J. Pure Appl. Math, 15 (1) (2022), 207-223 213

This means that γd(G) ≥ a. Thus, γd(G) = a. Next, if S ⊆ V (G) is a γdr -set of G,
then xi ∈ S for all i = 1, 2, . . . , a or x1xa ∈ S and u2j−1 ∈ S for all j = 2, 3, . . . , a − 1.
Being a restrained disjunctive dominating set of G, if u2j−2 ∈ S for all j = 2, 3, . . . , a,
then necessarily u1, x1, u2a−1, xa ∈ S. Similarly, if u2j−1 ∈ S for all j = 1, 2, . . . , a, then
necessarily x1, xa ∈ S. Moreover, if xju2j−1 ∈ S for all j = 1, 2, . . . , a, then u2i−2 ∈ S for
all i = 2, 3, . . . , a. In any case, γdr (G) ≥ a. Thus, γdr (G) = a = b.

Case 2. a < b
If a is even, then by hypothesis, b ≤ 3a

2 = a + a
2 . Write b = a + k, where 1 ≤ k ≤ a

2 .
Take G = G2 obtained from G1 by adding pendant edges xiyi for all i = 1, 2, . . . , 2k − 1
as shown in Figure 4. Since the set {xi : i = 1, 2, . . . , a} is a γd-set of G, γd(G) = a.

Figure 4: A graph G with γd(G) < γd
r (G)

Next, to determine the restrained disjunctive domination number of G, first consider
the case where 4k − 2 = 2a − 2. Let S = {xi, yj : i = 2k, 2k + 1, . . . , a; j = 1, 2, . . . , 2k −
1} ∪ {u4k−4, u2(2j−1) : j = 1, 2, . . . , k − 1}. Then S is a restrained disjunctive dominating

set of G. Hence, γdr (G) ≤ |S| = [a − (2k − 1)] + (2k − 1) + [(k − 1) + 1] = a + k = b.
Let S ⊆ V (G) be a γdr -set of G. Necessarily, xa, yj ∈ S for each j ∈ {1, 2, . . . , 2k − 1}.
Observe that, if xi ∈ S for each i ∈ {1, 2, . . . , a − 1} or u2a−1, u2(2j−1) ∈ S for each
j ∈ {1, 2, . . . , k}, then |S| ≥ a+ k + 1 which is a contradiction. This implies that the set
{xi, yj : i = 2k, 2k+1, . . . , a; j = 1, 2, . . . , 2k−1}∪{u4k−4, u2(2j−1) : j = 1, 2, . . . , k−1} is a

γdr -set of G. Thus, γdr (G) = a+ k = b. Suppose that 4k− 2 < 2a− 2. Let S = {xi, yj : i =
2k, 2k + 1, . . . , a; j = 1, 2, . . . , 2k − 1} ∪ {u2(2j−1) : j = 1, 2, . . . , k}. Then S is a restrained

disjunctive dominating set ofG. Thus, γdr (G) ≤ |S| = [a−(2k−1)]+(2k−1)+k = a+k = b.
Following similar arguments as above show that γdr (G) = b.

Now, suppose that a is odd. Then b ≤ 3a+1
2 = a + a+1

2 . Let b = a + k, where
1 ≤ k ≤ a+1

2 . If 4k−2 = 2a, then γdr (G) = b as determined by the set {xi, yj : i = 2k, 2k+
1, . . . , a; j = 1, 2, . . . , 2k − 1} ∪ {u4k−4, u2(2j−1) : j = 1, 2, . . . , k − 1} and if 4k − 2 < 2a,
then S = {xi, yj : i = 2k, 2k + 1, . . . , a; j = 1, 2, . . . , 2k − 1} ∪ {u2(2j−1) : j = 1, 2, . . . , k} is

a γdr -set of G. Thus, γdr (G) = b.

4. Main Results

4.1. On join of graphs

Theorem 3. [15] Let G and H be any graphs, and S ⊆ V (G+H). Then S is a disjunctive
dominating set of G+H if and only if one of the following holds:



R. Malalay, F. Jamil / Eur. J. Pure Appl. Math, 15 (1) (2022), 207-223 214

(i) S ⊆ V (G) for which either |S| ≥ 2 or S = {x} where NG[x] = V (G).

(ii) S ⊆ V (H) for which either |S| ≥ 2 or S = {x} where NH [x] = V (H).

(iii) S ∩ V (G) ̸= ∅ and S ∩ V (H) ̸= ∅.

For any nontrivial graphs G and H, if S ⊆ V (G+H) intersects both V (G) and V (H),
then S is a (restrained) dominating set, hence is a (restrained) disjunctive dominating set
of G+H.

For any graph G,

γdr (G+K1) =


1, if G = K2 or |V (G)| ≥ 3

2, if G = K1

3, if G = K2.

Theorem 4. Let G and H be nontrivial connected graphs. Then S ⊆ V (G + H) is a
restrained disjunctive dominating set of G+H if and only if one of the following holds;

(i) S ⊆ V (G) and either |S| ≥ 2 or S = {x} where NG[x] = V (G);

(ii) S ⊆ V (H) and either |S| ≥ 2 or S = {x} where NH [x] = V (H); and

(iii) S intersects both V (G) and V (H) satisfying the following:

(a) If V (G) ⊆ S, then V (H) ⊆ S or ⟨V (H) \ S⟩ ∼= K2 or |V (H) \ S| ≥ 3.

(b) If V (H) ⊆ S, then V (H) ⊆ S or ⟨V (H) \ S⟩ ∼= K2 or |V (H) \ S| ≥ 3.

Proof. Let S ⊆ V (G+H) be a restrained disjunctive dominating set of G+H. Then
S is a disjunctive dominating set of G + H. By Theorem 3, if S ⊆ V (G), then either
|S| ≥ 2 or S = {x} where NG[x] = V (G). Similarly, if S ⊆ V (H), then either |S| ≥ 2
or S = {x} where NH [x] = V (H). Now suppose that S intersects both V (G) and V (H).
Suppose further that V (G) ⊆ S. Let v ∈ V (H) \ S. Since S is a restrained disjunctive
dominating set, there exists u ∈ V (H)\S such that uv ∈ E(G+H) or there exist distinct
u,w ∈ V (H) \ S such that dG+H(u, v) = 2 = dG+H(w, v). If the former holds, then
V (H) ⊆ S or ⟨V (H) \ S⟩ ∼= K2. If the latter holds, then |V (H) \ S| ≥ 3. Similarly, if
V (H) ⊆ S, then (b) holds.

Conversely, suppose that (i) holds for S. By Theorem 3, S is a disjunctive dominating
set of G+H. Let v ∈ V (G+H) \S. If v ∈ V (G), then uv ∈ E(G+H) for any u ∈ V (H).
If v ∈ V (H), then pick any u ∈ V (H) for which uv ∈ E(H). Accordingly, S is a restrained
disjunctive dominating set of G+H. Similarly, if (ii) holds for S, then S is a restrained
disjunctive dominating set of G + H. Suppose that (iii) holds for S. By Theorem 3, S
is a disjunctive dominating set of G + H. If V (G) \ S ̸= ∅ and V (H) \ S ̸= ∅, then
the conclusion follows because xy ∈ E(G + H) for all x ∈ V (G) and for all y ∈ V (H).
Suppose that V (G) ⊆ S, and let v ∈ V (G+H) \S. Since V (H) ⊆ S or ⟨V (H) \S⟩ ∼= K2,
v ∈ V (H) \ S. Suppose that, there does not exist u ∈ V (H) \ S for which uv ∈ E(H),



R. Malalay, F. Jamil / Eur. J. Pure Appl. Math, 15 (1) (2022), 207-223 215

then |V (H) \ S| ≥ 3. In particular, |V (H) \ (S ∪ {v}) | ≥ 2, say u,w ∈ V (H) \ (S ∪ {v}),
with u ̸= w. Then dG+K(u, v) = 2 = dG+H(w, v). Similarly, if V (H) ⊆ S, then for each
v ∈ V (G + H) \ S, there exists u ∈ V (G) \ S for which uv ∈ E(G + H) or there exist
distinct u,w ∈ V (G) \ S such that dG+H(u, v) = 2 = dG+H(w, v). Accordingly, S is a
restrained disjunctive dominating set of G+H.

Corollary 2. For nontrivial connected graphs G and H,

γdr (G+H) =

{
1, if γ(G) = 1 or γ(H) = 1

2, otherwise.

Proof. Suppose that γ(G) = 1, and let S = {v} be a dominating set of G. Then S is a
restrained disjunctive dominating set of G+H by Theorem 4. In this case, γdr (G+H) = 1.
Similarly, if γ(H) = 1, then γdr (G + H) = 1. Suppose that γ(G) ≥ 2 and γ(H) ≥ 2. In
view of Theorem 4, γdr (G+H) ≥ 2. Pick u ∈ V (G) and v ∈ V (H). By Theorem 4, {u, v}
is a restrained disjunctive dominating set of G+H. Thus, γdr (G+H) ≤ 2.

4.2. On corona of graphs

It is worth noting that G ◦H is composed of the subgraphs Hv + ⟨v⟩ joined together
by the edges of G. Thus,

V (G ◦H) = V (G) ∪
(
∪v∈V (G)V (Hv)

)
= ∪v∈V (G)V (Hv + v).

Theorem 5. [15] Let G be a nontrivial connected graph and H be any graph, and let
S ⊆ V (G ◦H). Then S is a disjunctive dominating set of G ◦H if and only if each of the
following holds for S:

(i) |S ∩NG(v)| ≥ 2 for all v ∈ V (G) \ S with S ∩ V (Hv) = ∅;

(ii) |S ∩ V (Hv)| ≥ 1 for all v ∈ V (G) \ S with |S ∩NG(v)| = 1; and

(iii) S ∩ V (Hv) is a disjunctive dominating set of Hv + v for all v ∈ V (G) \ S with
S ∩NG(v) = ∅. In particular, if γ(H) > 1, then |S ∩ V (Hv)| ≥ 2.

Corollary 3. [15] Let G be a nontrivial connected graph. Then for any graph H, γd(G ◦
H) = γ×2(G).

Proposition 2. Let G be a nontrivial connected graph and S ⊆ V (G ◦K1). Then S is a
restrained disjunctive dominating set of G ◦K1 if and only if each of the following holds:

(i) uv ∈ S for all v ∈ End(G);

(ii) For each uv /∈ S,

(a) |NG(v) ∩ S| ≥ 2 whenever v /∈ S;



R. Malalay, F. Jamil / Eur. J. Pure Appl. Math, 15 (1) (2022), 207-223 216

(b) |NG(v) \ S| ≥ 2 whenever v ∈ S.

(iii) For each v /∈ S with NG(v) ∪ {uv} ⊆ S,

(a) |NG(v, 2) \ S| ≥ 2 whenever uw ∈ S for all w ∈ NG(v); and

(b) |NG(v, 2) \ S| ≥ 1 whenever uw /∈ S for exactly one w ∈ NG(v). In particular,
v ∈ End(G) if and only if |NG(v)| = 1.

Proof. Let S ⊆ V (G◦K1) be a restrained disjunctive dominating set of G◦K1. Let v ∈
End(G). Suppose that uv /∈ S. If v ∈ S, then since S is a restrained disjunctive dominating
set, there exist distinct w, z ∈ NG(v) \ S for which dG◦K1(u

v, w) = 2 = dG◦K1(u
v, z), and

consequently, dG(w, v) = 1 = dG(z, v). This is impossible since v ∈ End(G). If v /∈ S,
then since S is a disjunctive dominating set, there exist distinct w, z ∈ NG(v) ∩ S for
which dG◦K1(u

v, w) = 2 = dG◦K1(u
v, z), which for the same reason is impossible. Thus,

uv ∈ S and (i) holds. Let uv /∈ S. By (i), v /∈ End(G). If v /∈ S, then since S is a
disjunctive dominating set, there exist distinct w, z ∈ NG(v)∩S for which dG◦K1(u

v, w) =
2 = dG◦K1(u

v, z). Thus, |NG(v) ∩ S| ≥ 2 and Property (ii)(a) holds for S. On the
other hand, if v ∈ S, then since S is a restrained disjunctive dominating set, there exist
distinct w, z ∈ V (G) \ S for which dG◦K1(w, u

v) = 2 = dG◦K1(z, u
v). This proves that

Property (ii)(b) holds for S. Now, let v ∈ V (G) \S with NG(v)∪{uv} ⊆ S. Suppose that
uw ∈ S for all w ∈ NG(v). Since S is a restrained disjunctive dominating set, there exist
distinct w, z ∈ S for which dG◦K1(v, w) = 2 = dG◦K1(v, z). The hypothesis implies that
w, z ∈ V (G), proving (iii)(a). Similarly, (iii)(b) holds for S.

Conversely, suppose that all the properties hold for S. First, using Theorem 5 we prove
that S is a disjunctive dominating set of G ◦K1. Let v ∈ V (G) \ S. By Property (ii)(a),
if uv /∈ S, then |S ∩ NG(v)| ≥ 2. By the same property, if |S ∩ NG(v)| = 1, then uv ∈ S
so that |S ∩ V (Kv

1 )| = 1. Now suppose that S ∩ NG(v) = ∅. Note that either uv ∈ S
or uv /∈ S. If uv /∈ S, then by Property (ii)(a), |S ∩ NG(v)| ≥ 2, a contradiction. Thus,
uv ∈ S and S ∩ V (Kv

1 ) = {uv} is a disjunctive dominating set of Kv
1 + v. Therefore, S is

a disjunctive dominating set of G ◦K1. Now, we show that S is a restrained disjunctive
dominating set. Let x ∈ V (G ◦ K1) \ S. Suppose that x = uv for some v ∈ V (G). If
v /∈ S, then we are done. If v ∈ S, then we pick distinct w, z ∈ NG(v) \ S, as guaranteed
by Property (ii)(b). Then dG◦K1(x,w) = 2 = dG◦K1(x, z). Suppose that x ∈ V (G). If
ux /∈ S or NG(x) \ S ̸= ∅, then we are done. Suppose that ux ∈ S and NG(x) ⊆ S.
Properties (iii)(a) and (iii)(b) imply that there exists distinct w, z ∈ V (G ◦ K1) \ S for
which dG◦K1(x,w) = 2 = dG◦K1(x, z).

Corollary 4. For a nontrivial connected graph G,

|End(G)| ≤ γdr (G ◦K1) ≤ |V (G)|,

and these bounds are sharp.

Proof. The left-hand inequality follows immediately from Property (i) of Proposition
2 for any restrained disjunctive dominating set. Let S = {uv : v ∈ V (G)}. Then S is



R. Malalay, F. Jamil / Eur. J. Pure Appl. Math, 15 (1) (2022), 207-223 217

a restrained disjunctive dominating set of G ◦ K1 by Proposition 2, and the right-hand
inequality follows. For the sharpness of the bounds, verify that γdr (K1,n ◦ K1) = n =
|End(K1,n)| for any n ≥ 2 and γdr (P3 ◦K1) = 3 = |V (P3)|.

Corollary 5. (i) For n ≥ 3, γdr (Pn ◦K1) = ⌊n2 ⌋+ 2,

(ii) For n ≥ 3,

γdr (Cn ◦K1) =

{
⌊n2 ⌋, if n ≡ 0(mod 2)

⌊n2 ⌋+ 2, otherwise;

and

(iii) For m,n ≥ 2, γdr (Km,n ◦K1) = min{m,n, 4}.

Proof. Let Pn = [x1, x2, . . . , xn] and k = ⌊n2 ⌋. Since by Proposition 2, the set
{ux1 , x2, x4, . . . , x2k, u

xn} is a restrained dominating set, γdr (Pn) ≤ k + 2. Suppose that
S ⊆ V (G ◦ K1) is a restrained disjunctive dominating set of G ◦ K1 with |S| ≤ k + 1.
Accordingly, {ux1 , uxn} ⊆ S. Apparently, there exist w, v ∈ V (Pn) for which wv ∈ E(G)
and all vertices w, v, uv, uw /∈ S. In particular, |NG(v) ∩ S| ≥ 2 by Proposition 2, which
is impossible since w /∈ S. Thus, γdr (Pn ◦ K1) ≥ k + 2. This proves Property (i). To
prove (ii), let Cn = [x1, x2, . . . , xn]. Let k = ⌊n2 ⌋. If n = 2k, then the set {x2, x4, . . . , x2k}
is a restrained disjunctive dominating set of Cn ◦ K1 so that γdr (Cn ◦ K1) ≤ k. On the
other hand, if n = 2k+1, then the set {ux1 , x2, x4, . . . , x2k, u

xn} is a restrained disjunctive
dominating set of Cn ◦K1, and so γdr (Cn ◦K1) ≤ k + 2. Following similar arguments as
in the proof for Pn, the desired equality is attained. Let U and V be the partite sets of
Km,n with |U | = m and |V | = n. By Proposition 2, U or V are restrained disjunctive
dominating sets of Km,n ◦K1. Suppose that m ≥ 4 and n ≥ 4. Pick x, y ∈ U and a, b ∈ V .
By Proposition 2, S = {x, y, a, b} is a restrained disjunctive dominating set of Km,n ◦K1.
Thus,

γdr (Km,n ◦K1) ≤ min {m,n, 4}.

Now, let S ⊆ V (Km,n ◦K1) be a restrained disjunctive dominating set of Km,n ◦K1.
Suppose that |S| ≤ 3. Then there exists v ∈ V (Km,n ◦K1) for which uv ∈ S. In view of
Proposition 2, this is only possible if S = U or S = V . Thus,

γdr (Km,n ◦K1) ≥ min {m,n, 4}.

Theorem 6. Let G and H be nontrivial connected graphs, and let
S ⊆ V (G ◦H). Then S is a restrained disjunctive dominating set of G ◦H if and only if
S satisfies all the properties in Theorem 5 as well as each of the following properties:

(i) For each v ∈ S ∩ V (G), at least one of the following holds:



R. Malalay, F. Jamil / Eur. J. Pure Appl. Math, 15 (1) (2022), 207-223 218

(a) S ∩ V (Hv + v) is a restrained disjunctive dominating set of Hv + v whenever
NG(v) ⊆ S; or

(b) |V (Hv) \ S| ≥ 2 whenever |NG(v) \ S| = 1 and V (Hv) \ S ̸= ∅.

(ii) For each v ∈ V (G) \S for which S ∩V (Hv) ̸= ∅, at least one of the following holds:

(a) V (Hv) \ S ̸= ∅;

(b) NG(v) \ S ̸= ∅; or

(c) |NG(v, 2) ∩ [V (G) \ S]| ≥ 2.

Proof. Suppose that S is a restrained disjunctive dominating set of G ◦ H. Since S
is a disjunctive dominating set, all properties in Theorem 5 hold for S. To prove (i), let
v ∈ S ∩ V (G). Suppose that NG(v) ⊆ S. Since V (Hv) ⊆ NHv+v(v), Sv = S ∩ V (Hv + v)
is a disjunctive dominating set of Hv + v. Let u ∈ V (Hv + v) \ Sv. Then u ∈ V (Hv) \ S.
Since S is a restrained disjunctive dominating set, there exists w ∈ V (G◦H)\S such that
uw ∈ E(G ◦H) or there exist distinct w, z ∈ V (G ◦H) \ S for which dG◦H(u,w) = 2 =
dG◦H(u, z). In case the former holds, w ∈ V (Hv) \ Sv and uw ∈ E(Hv + v). If the latter
holds, then w and z are distinct vertices in V (Hv)\Sv with dHv+v(u,w) = 2 = dHv+v(u, z).
Thus, Sv is a restrained disjunctive dominating set of Hv + v. This proves (i)(a). Now,
suppose that V (Hv) \ S ̸= ∅ and |NG(v) \ S| = 1. Let u ∈ V (Hv) \ S, and suppose
that |V (Hv) \ S| = 1. Since S is a restrained disjunctive dominating set and w ∈ S for
all w ∈ V (G ◦ H) for which uw ∈ E(G ◦ H), there exist distinct x, y ∈ V (G ◦ H) \ S
such that dG◦H(u, x) = 2 = dG◦H(u, y). Necessarily, x, y ∈ NG(v), a contradiction. Thus
|V (Hv) \ S| ≥ 2, proving (i)(b).

To prove (ii), let v ∈ V (G) \ S for which S ∩ V (Hv) ̸= ∅. Since S is a restrained
disjunctive dominating set and v ∈ V (G ◦H) \ S, there exists u ∈ V (G ◦H) \ S such that
uv ∈ E(G ◦ H) or there exist distinct x, y ∈ V (G ◦ H) \ S for which dG◦H(v, x) = 2 =
dG◦H(v, y). Note that for such u, either u ∈ V (Hv) or u ∈ NG(v). That is, (a) or (b) holds.
On the other hand, if (a) and (b) do not hold, x, y ∈ V (G) and dG(x, v) = 2 = dG(y, v)
showing that |NG(v, 2) ∩ [V (G) \ S]| ≥ 2.

Conversely, note first that because S satisfies the properties in Theorem 5, S is a
disjunctive dominating set of G ◦H. Let u ∈ V (G ◦H) \ S, and let v ∈ V (G) such that
u ∈ V (Hv + v). Suppose that u = v. Then u = v ∈ V (G) \ S. If S ∩ V (Hv) = ∅, then
ux ∈ E(G ◦H) for any x ∈ V (Hv). Suppose that S ∩ V (Hv) ̸= ∅. Pick x ∈ V (Hv) \ S
or x ∈ NG(v) \ S in case (ii)(a) or (ii)(b) holds, respectively. Then ux ∈ E(G ◦ H). If
(ii)(c) holds, then pick distinct x, y ∈ V (G) for which dG(u, x) = 2 = dG(u, y). Then
dG◦H(u, x) = 2 = dG◦H(u, y). Suppose that u ∈ V (Hv). If v /∈ S, then v is the required
vertex for which uv ∈ E(G ◦H). Suppose that v ∈ S. Suppose further that NG(v) ⊆ S.
By (i)(a), there exists w ∈ V (Hv) \ S such that uw ∈ E(Hv + v) or there exist distinct
w, z ∈ V (Hv) \ S for which dHv+v(u,w) = 2 = dHv+v(u, z). This implies that there exists
w ∈ V (G ◦H) \S such that uw ∈ E(G ◦H) or there exist distinct w, z ∈ V (G ◦H) \S for
which dG◦H(u,w) = 2 = dG◦H(u, z). Finally, suppose that NG(v)\S ̸= ∅. If |NG(v)\S| ≥



R. Malalay, F. Jamil / Eur. J. Pure Appl. Math, 15 (1) (2022), 207-223 219

2, then there exist distinct x, y ∈ NG(v) \ S such that dG◦H(u, x) = 2 = dG◦H(u, y).
Suppose that |NG(v) \ S| = 1, say x ∈ NG(v) \ S. We may also pick y ∈ V (Hv) \ S with
u ̸= y by (i)(b). Then uy ∈ E(G ◦H) or dG◦H(u, y) = 2 = dG◦H(u, x). Accordingly, S is
a restrained disjunctive dominating set of G ◦H.

Corollary 6. For nontrivial connected graphs G and H, γdr (G ◦H) = γ×2(G).

Proof. Let S ⊆ V (G) be a 2-dominating set of G. By Corollary 3, S is a disjunctive
dominating set of G ◦H. Moreover, since all the properties in Theorem 6 hold for S, S
is a restrained disjunctive dominating set of G ◦ H. Thus, γdr (G ◦ H) ≤ |S|. Since S is
arbitrary, γdr (G ◦H) ≤ γ×2(G). Further, using Corollary 3 again,

γ×2(G) ≤ γdr (G ◦H) ≤ γ×2(G).

This proves the equality.

4.3. On lexicographic product of graphs

[15] For any graphs G and H and for any C ⊆ V (G[H]), there exists S ⊆ V (G) for
which C = ∪x∈S ({x} × Tx), where Tx ⊆ V (H) for each x ∈ S.

Theorem 7. [15] Let G and H be nontrivial connected graphs, and let C = ∪x∈S ({x} × Tx).
Then C is a disjunctive dominating set of G[H] if and only if one of the following holds:

(i) S is a disjunctive total dominating set in G; or

(ii) S is a distance-two dominating set of G satisfying the following:

(a) For each x ∈ V (G) \Nd
G[S], where Nd

G[S] = S ∪Nd
G(S), there exists u ∈ S for

which dG(u, x) = 2 and |Tu| ≥ 2.

(b) For each x ∈ S \ NG(S, 2), either Tx = {y} and is a dominating set of H or
|Tx| ≥ 2.

Theorem 8. Let G and H be nontrivial connected graphs, and let C = ∪x∈S ({x} × Tx)
be a disjunctive dominating set of G[H]. Then C is a restrained disjunctive dominating
set of G[H] if and only if (i) or (ii) of Theorem 7 holds and each of the following holds
for S:

For each x ∈ S \ (NG(V (G) \ (S, 2)) for which Tx ̸= V (H),

(a) If Tu = V (H) for all u ∈ NG(x, 2), then either |V (H)\Tx| ≥ 3 or ⟨V (H)\Tx⟩ = K2.

(b) If ⟨V (H) \ Tx⟩ = K2, then there exists u ∈ NG(x, 2) for which Tu ̸= V (H).

(c) If |V (H)\Tx| = 1, then one of the following holds: there exists u ∈ NG(x) for which
Tu ̸= V (H); there exists u ∈ NG(x, 2) for which |V (H) \ Tu| ≥ 2; there exist distinct
u, z ∈ NG(x, 2) for which Tu ̸= V (H) and Tz ̸= V (H).



R. Malalay, F. Jamil / Eur. J. Pure Appl. Math, 15 (1) (2022), 207-223 220

Proof. Suppose that C is a restrained disjunctive dominating set of G[H]. Since C is
a disjunctive dominating set of G[H], (i) or (ii) of Theorem 7 holds. Now, let x ∈ S \
(NG(V (G) \ (S, 2))) for which Tx ̸= V (H). Then u ∈ S for all u ∈ NG(x, 2). First, suppose
that Tu = V (H) for all u ∈ NG(x, 2). Pick y ∈ V (H) \ Tx. Then (x, y) /∈ C so that there
exists (u, v) ∈ V (G[H])\C such that (x, y)(u, v) ∈ E(G[H]) or there exist distinct vertices
(u, v), (z, w) ∈ V (G[H]) \ C for which dG[H]((x, y), (u, v)) = 2 = dG[H]((x, y), (z, w)).
Since x /∈ NG(V (G) \ (S, 2)), the preceding statement implies that x = u, in which case
v ∈ V (H) \ Tx and yw ∈ E(H), or u = x = z and y, v and w are distinct vertices in
V (H) \ Tx. This establishes (a).

Next, suppose that ⟨V (H) \ Tx⟩ = K2. let y, w ∈ ⟨V (H) \ Tx⟩. Suppose that
(u, v) ∈ V (G[H]) \ C such that (x, y)(u, v) ∈ E(G[H]). Since y, w ∈ ⟨V (H) \ Tx⟩, it
is necessary that u ̸= x. Hence u ∈ NG(x) and v ∈ V (H) \ Tu. Consequently, Tu ̸= V (H).
Suppose that (u, v), (z, w) ∈ V (G[H]) \ C are distinct for which dG[H]((x, y), (u, v)) =
2 = dG[H]((x, y), (z, w)). The assumption implies that u ∈ NG(x, 2) or z ∈ NG(x, 2). If
u ∈ NG(x, 2), then Tu ̸= V (H), and if z ∈ NG(x, 2), then Tz ̸= V (H). Hence, (b) holds.

Lastly, suppose that |V (H) \ Tx| = 1, say V (H \ Tx = {y}. If (u, v) ∈ V (G[H]) \ C
such that (u, v)(x, y) ∈ E(G[H]), then since Tx = {y}, u ∈ NG(x) and Tu ̸= V (H). On
the other hand, if (u, v), (z, w) ∈ V (G[H]) \C are distinct for which dG[H]((x, y), (u, v)) =
2 = dG[H]((x, y), (z, w)), then dG(x, u) = 2 = dG(x, z). If u = z, then |V (H) \ Tu| ≥ 2.
Otherwise, Tu ̸= V (H) and Tz ̸= V (H).

Conversely, suppose that Conditions (i) or (ii) of Theorem 7 holds. Then C is a
disjunctive dominating set of G[H]. Suppose further that all properties in Theorem 8 hold
for S. Let (x, y) ∈ V (G[H]) \ C.

Case 1: If x /∈ S, then pick z ∈ V (H) \ {y} for which zy ∈ E(H). In this case, (x, z) ∈
V (G[H]) \ C and (x, y)(x, z) ∈ E(G[H]).

Case 2: Suppose that x ∈ S. Then y /∈ Tx. First, consider having x ∈ Nd
G(V (G) \

S). Suppose u ∈ V (G) \ S such that ux ∈ E(G). Pick v ∈ V (H). Then (u, v) /∈
C and (x, y)(u, v) ∈ E(G[H]). Suppose, on the other hand, that u, z ∈ V (G) \ S
for which dG(x, u) = 2 = dG(x, z). Pick v, w ∈ V (H). Then (u, v), (z, w) /∈ C and
dG[H]((x, y), (u, v)) = 2 = dG[H]((x, y), (z, w)). Next, suppose that x /∈ Nd

G(V (G) \ S).
Suppose further that x ∈ NG(V (G) \ (S, 2)). Pick u ∈ V (G) \ S such that 1 ≤ dG(x, u) ≤
2. If ux ∈ E(G), then (u, v) /∈ C and (u, v)(x, y) ∈ E(G[H]) for any v ∈ V (H).
Otherwise, pick distinct v, w ∈ V (H). Then (u, v) and (u,w) are distinct vertices in
V (G[H]) \ C with dG[H]((x, y), (u, v)) = 2 = dG[H]((x, y), (u,w)). Finally, suppose that
x /∈ NG(V (G) \ (S, 2)). We consider the following subcases:

Subcase 2.1: Suppose that Tu = V (H) for all u ∈ NG(x, 2). By Condition (a), either
|V (H) \ Tx| ≥ 3 or ⟨V (H) \ Tx⟩ = K2. Let y, w ∈ V (H) \ Tx. If |V (H) \ Tx| ≥ 3
and v, w ∈ (V (H) \ Tx) \ {y} are distinct, then (x, v) and (x,w) are distinct vertices in
V (G[H])\C with dG[H]((x, y), (x, v)) ≤ 2 and dG[H]((x, y), (x,w)) ≤ 2. On the other hand,
if y, w ∈ ⟨V (H) \ Tx⟩ = K2, then (x,w) ∈ V (G[H]) \ C with (x, y)(x,w) ∈ E(G[H]).

Subcase 2.2: Suppose that Tu ̸= V (H) for some u ∈ NG(x, 2). If |V (H) \ Tx| ≥ 3, then,



R. Malalay, F. Jamil / Eur. J. Pure Appl. Math, 15 (1) (2022), 207-223 221

as done in Subcase 2.1, there exists (x, v) ∈ V (G[H]) \C such that (x, y)(x, v) ∈ E(G[H])
or there exist distinct (x, v), (x,w) ∈ V (G[H]) \ C for which dG[H]((x, y), (x, v)) = 2 =
dG[H]((x, y), (x,w)). Suppose that yw ∈ ⟨V (H) \ Tx⟩ = K2. Then (x,w) ∈ V (G[H]) \ C
with (x, y)(x,w) ∈ E(G[H]). If yw ∈ ⟨V (H)\Tx⟩ = K2, then there exists u ∈ NG(x, 2) for
which Tu ̸= V (H), by Condition (b). Here, dG[H]((x, y), (x,w) = 2. Pick v ∈ V (H) \ Tu.
Then (u, v) /∈ C and dG[H]((x, y), (u, v) ≤ 2. Finally, suppose that |V (H) \ Tx| = 1. By
Condition (c), one of the following holds: there exists u ∈ NG(x) for which Tu ̸= V (H);
there exists u ∈ NG(x, 2) for which |V (H)\Tx| ≥ 2; there exist distinct u, z ∈ NG(x, 2) for
which Tu ̸= V (H) and Tz ̸= V (H). If u ∈ NG(x) and v ∈ V (H) \ Tu, then (u, v) /∈ C and
(x, y)(u, v) ∈ E(G[H]). If u ∈ NG(x, 2) for which |V (H) \ Tu| ≥ 2 and v, w ∈ V (H) \ Tu

are distinct, then (u, v) and (u,w) are distinct in V (G[H])\C and dG[H]((x, y), (u, v)) ≤ 2
and dG[H]((x, y), (u,w)) ≤ 2. If u, z ∈ NG(x, 2) are distinct for which Tu ̸= V (H) and
Tz ̸= V (H), then (u, v) and (z, w) are distinct vertices in V (G[H])\C for all v ∈ V (H)\Tu

and w ∈ V (H) \ Tz with dG[H]((x, y), (u, v)) = 2 = dG[H]((x, y), (z, w)).
Accordingly, C is a restrained disjunctive dominating set of G[H].

Corollary 7. Let G and H be nontrivial connected graphs. Then

γdr (G[H]) ≤ min {γdt (G), 2γ2(G)}.

Proof. Let S ⊆ V (G) be a disjunctive total dominating set of G, and let u ∈ V (H).
Then S × {u} = ∪x∈S ({x} × {u}) is a disjunctive dominating set of G[H] by Theorem
7. Since S satisfies condition (ii) of Theorem 8, S is a restrained disjunctive dominating
set of G[H]. Consequently, γdr (G[H]) ≤ |S|. Since S is arbitrary, γdr (G[H]) ≤ γdt (G).
Similar arguments as above, if S ⊆ V (G) is a distance-two dominating set of G satisfying
condition (ii) of Theorem 7, then 2S is a restrained disjunctive dominating set of G[H].
Thus, γdr (G[H]) ≤ 2γ2(G). Therefore, γdr (G[H]) ≤ min {γdt (G), 2γ2(G)}.

Since γdr (C7[P8]) = γdt (C7) = 2γ2(C7) = 4, the inequality in Corollary 7 is sharp.

5. Conclusion

Let G andH be nontrivial connected graphs. It is shown that the restrained disjunctive
domination number in the join of two graphs (G+H) is the min{γ(G), γ(H), 2} and the
restrained disjunctive domination number in the corona of two graphs (G ◦ H) is the
(γ×2(G)). Under the same condition, the minimum of the sets between disjuntive total
domination number of graph G and twice of distance-2 domination number of graph G,
denoted by min{γdt (G), 2γ2(G)} proves to be a sharp bound for the restrained disjunctive
domination number of the lexicographic product of graphs G and H.



REFERENCES 222

Acknowledgements

This research was undertaken at the Technische Universitat Kaiserslautern, Germany,
under the Faculty and Student Exchange Program of the project ”Graph Theory and
Optimization (GraThO) with Applications in Industry and Society” and funded by the
Zamboanga Peninsula Polytechnic State University, Philippines. Also, The authors would
like to recognize the efforts of the anonymous reviewers whose suggestions and recommen-
dations contributed to the big improvement of the paper.

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