EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 16, No. 4, 2023, 2082-2095
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

J-Domination in Graphs

Javier A. Hassan1,∗, Jeffrey Imer Salim1

1Mathematics and Sciences Department, College of Arts and Sciences, MSU-Tawi-Tawi
College of Technology and Oceanography, Bongao, Tawi-Tawi, Philippines

Abstract. Let G be a graph. A subset D = {d1, d2, · · · , dm} of vertices of G is called a J-set if
NG[di] \NG[dj ] ̸= ∅ for every i ̸= j, where i, j ∈ {1, 2, . . . ,m}. A J-set is called a J-dominating
set of G if D = {d1, d2, . . . , dm} is a dominating set of G. The J-domination number of G, denoted
by γJ(G), is the maximum cardinality of a J-dominating set of G. In this paper, we introduce
this new concept and we establish formulas and properties on some classes of graphs and in join
of two graphs. Upper and lower bounds of J-domination parameter with respect to the order of a
graph and other parameters in graph theory are obtained. In addition, we present realization result
involving this parameter and the standard domination. Moreover, we characterize J-dominating
sets in some classes of graphs and join of two graphs and finally determine the exact value of the
parameter of each of these graphs.

2020 Mathematics Subject Classifications: 05C69

Key Words and Phrases: J-set, J-dominating set, J-domination number

1. Introduction

Domination in a graph has been one of the most interesting and widely studied topics
in Graph Theory. It has many applications in various fields and in networks.

The study of domination in graphs came about partially as a result of the study of
games and recreational mathematics. In particular, mathematicians studied how chess
pieces of a particular type could be placed on a chessboard in such a way that they would
attack, or dominate, every square on the board.

In 1962, Oystein Ore introduced the concepts of dominating set and domination num-
ber in a graph in his book on Graph Theory [12]. A decade later, Cockayne and Hedetniemi
published a survey paper, in which the notation γ(G) was first used for the domination
number of a graph G [3]. Since then, several mathematicians had studied and introduced
new domination parameters in graphs. Some variants of domination were defined and
further studied by researchers can be found in [1, 2, 4–11, 13, 14].

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v16i4.4883

Email addresses: javierhassan@msutawi-tawi.edu.ph (J. A. Hassan)
jeffreyimersalim@msutawi-tawi.edu.ph (J. I. Salim)

https://www.ejpam.com 2082 © 2023 EJPAM All rights reserved.



J. Hassan, J. Salim / Eur. J. Pure Appl. Math, 16 (4) (2023), 2082-2095 2083

In this paper, new variant of domination called J-domination in a graph will be in-
troduced and investigated. Its relationships with other variants of domination and other
concepts in graph theory will be determined. Moreover, characterizations of J-dominating
sets in some classes of graphs and join of two of graphs will be presented. These results
will be used to determine exact values or bounds of the parameter for these graphs. We
believe that this study and its results will help other researchers in the field for more
research directions in the future.

2. Terminology and Notation

Let G = (V (G), E(G)) be a simple and undirected graph. Two vertices x, y of G
are adjacent, or neighbors, if xy is an edge of G. The open neighborhood of x in G is
the set NG(x) = {y ∈ V (G) : xy ∈ E(G)}. The closed neighborhood of x in G is the
set NG[x] = NG(x) ∪ {x}. If X ⊆ V (G), the open neighborhood of X in G is the set

NG(X) =
⋃
x∈X

NG(x). The closed neighborhood of X in G is the set NG[X] = NG(X)∪X.

A subset D of V (G) is called a dominating of G if for every x ∈ V (G) \ D, there
exists y ∈ D such that xy ∈ E(G). The domination number of G, denoted by γ(G), is the
minimum cardinality of a dominating set in G.

A graph G is connected if every pair of its vertices can be Joined by a path. Otherwise,
G is disconnected. A maximal connected subgraph (not a subgraph of any connected
subgraph) of G is called a component of G.

A dominating set D of G is called a connected dominating if the induced subgraph
⟨D⟩ of D is connected. The connected domination number of G, denoted by γc(G), is the
minimum cardinality of a connected dominating set in G. Any connected dominating set
D with cardinality equal to γc(G) is called a γc-set of G.

The distance dG(u, v) in G of two vertices u, v is the length of a shortest u-v path in
G. The greatest distance between any two vertices in G, denoted by diam(G), is called
the diameter of G.

A subset I of V (G) is called an independent if for every pair of distinct vertices x, y ∈ I,
dG(x, y) ̸= 1. The maximum cardinality of an independent set in G, denoted by α(G), is
called the independence number of G. Any independent set I with cardinality equal to
α(G) is called an α-set of G.

A graph is complete if every pair of distinct vertices are adjacent. A complete graph
of order n is denoted by Kn.

The complement of a graph G, denoted by G, is the graph with V (G) = V (G) and
E(G) = {uv : u, v ∈ V (G) and uv /∈ E(G)}.

Let G and H be any two graphs. The join of G and H, denoted by G+H is the graph
with vertex set V (G+H) = V (G) ∪ V (H) and edge set

E(G+H) = E(G) ∪ E(H) ∪ {uv : u ∈ V (G), v ∈ V (H)}.



J. Hassan, J. Salim / Eur. J. Pure Appl. Math, 16 (4) (2023), 2082-2095 2084

3. Results

We begin this section by introducing the concept of J-domination in a graph.

Definition 1. Let G be a simple and undirected graph and m ∈ N . A subset
D = {d1, d2, · · · , dm} of vertices of G is called a J-set if NG[di] \ NG[dj ] ̸= ∅ for ev-
ery i ̸= j, where i, j ∈ {1, 2, . . . ,m}. A J-set is called a J-dominating set of G if
D = {d1, d2, . . . , dm} is a dominating set of G. The J-domination number of G, denoted by
γJ(G), is the maximum cardinality of a J-dominating set of G. Any J-dominating set D
with |D| = γJ(G) (resp. |D| = γ(G)), is called a γJ -set or the maximum (resp. minimum)
J-dominating set of G. Moreover, if x ∈ NG[di] \NG[dj ], then we say di J-footprinted a
vertex x in G.

Example 1. Consider the graph G in Figure 1. Let D = {a, b, c, d}. Observe that
a ∈ NG[a] \ NG[u], ∀ u ∈ {b, c, d}, b ∈ NG[b] \ NG[v], ∀ v ∈ {a, c, d}, c ∈ NG[c] \ NG[w],
∀ w ∈ {a, b, d} and d ∈ NG[d] \ NG[x], ∀ x ∈ {a, b, c}. It follows that NG[s] \ NG[t] ̸= ∅
for every s, t ∈ D, s ̸= t. Thus, D is a J-set of G. Since NG[D] = V (G), it follows that D
is a dominating set of G. Consequently, D is a J-dominating set of G. Moreover, it can
be verified that γJ(G) = 4.

G :

b

c

f

e

d

a g

Figure 1: A graph G with γJ (G) = 4

Theorem 1. Let G be any graph and m ∈ N. Then T = {t1, t2, . . . , tm} ⊆ V (G) is a
minimum dominating set in G if and only if T is a minimum J-dominating set in G.

Proof. Suppose that T = {t1, t2, . . . , tm} ⊆ V (G) is a minimum dominating set in
G. Then it remains to show that T is a J-set in G. Suppose on the contrary that
T = {t1, t2, · · · , tm} is not a J-set in G. Then there exist a, b ∈ T such that either
NG[a] \ NG[b] = ∅ or NG[b] \ NG[a] = ∅. This means that either NG[a] ⊆ NG[b] or
NG[b] ⊆ NG[a]. If NG[a] ⊆ NG[b], then D = T \ {a} is a dominating set in G, con-
tradicting to the minimality of T . Similarly, when NG[b] ⊆ NG[a] = ∅. Hence, T is a



J. Hassan, J. Salim / Eur. J. Pure Appl. Math, 16 (4) (2023), 2082-2095 2085

J-dominating set in G. Since γ(G) = |T |, the assertion follows.

Conversely, suppose that T = {t1, t2, . . . , tm} is a minimum J-dominating set of G.
Then |T | = γ(G) (by definition). It follows that T is a minimum dominating set in G.

Proposition 1. Let G be any graph and r ∈ N. Then each of the following holds:

(i) A graph G admits a J-domination.

(ii) Given any J-dominating set D = {d1, d2, · · · , dr} of G, we have |D| = r ≤ γJ(G).

(iii) γ(G) ≤ γJ(G), and this bound is sharp.

Proof. (i) Since any graph G admits domination, the result follows from Theorem 1.

(ii) Let D be any J-dominating set of G. If D is the maximum, then γJ(G) = |D| and
we are done. If D is not the maximum, then γJ(G) > |D|. Consequently, |D| ≤ γJ(G) for
any J-dominating set D of G.

(iii) Let D′ be a minimum dominating set of G. Then D′ is a minimum J-dominating
set in G by Theorem 1. Thus, by (ii), γ(G) = |D′| ≤ γJ(G). To see the bound is sharp,
consider P4. Then γJ(P4) = 2 = γ(P4).

Theorem 2. Let G be any graph. Then each of the following is true.

(i) A J-set may not be a J-dominating set. In particular, if the cardinality of a J-set D
is strictly less than the domination number of G, then D cannot be a J-dominating
set of G.

(ii) A vertex set V (G) of G may not be a J-set in G.

(iii) Every J-dominating set in G is a dominating set but the converse is not always true.

Proof. (i) Consider the Example 1 and let D = (b, e). Then NG[b] \NG[e] = {b} ̸= ∅
and NG[e] \NG[b] = {a, d, e, g} ≠ ∅. It follows that D is a J-set in G. However, D is not
a J-dominating set in G since c /∈ NG[D], that is, D not a dominating set in G. For the
particular case, suppose on the contrary that a J-set D can be a J-dominating set of G.
Then D is a dominating set (by definition). Thus, γ(G) ≤ |D|, a contradiction. Therefore,
D cannot be a J-dominating set of G.

(ii) Consider again the graph in Example 1. Observe that

NG[c] \NG[g] = {c, g} \ {c, e, g} = ∅.

This shows that V (G) is not a J-set of G.
(iii) Let G be a graph and let D be a J-dominating set. Then D is a dominating set in

G (by definition). To see that the converse is not true, consider the graph in Example 1
and let à = {d, e, f, g}. Then NG[Ã] = V (G), and so à is a dominating set of G. However,



J. Hassan, J. Salim / Eur. J. Pure Appl. Math, 16 (4) (2023), 2082-2095 2086

NG[d] \ NG[e] = {d, e} \ {a, d, e, f, g} = ∅, showing that à is not a J-set in G. Thus, Ã
cannot be a J-dominating set in G.

Theorem 3. Let G be any graph and k ∈ N. Then T = {v1, v2, · · · , vk} is a maximum
J-set of G if and only if T is a maximum J-dominating set of G. In particular, γJ(G) = k.

Proof. Let T = {v1, · · · , vk} be a maximum J-set of G. Suppose T is not a dominating
set of G. Then there exists a ∈ V (G) \ T such that a /∈ NG[T ]. This implies that
a /∈ NG[b] for every b ∈ T . Let T0 = {v0, v1, · · · , vk}, where v0 = a. Since T is a J-set and
v0 ∈ NG[v0], it follows that NG[vi] \NG[vj ] ̸= ∅ for every i ̸= j, where i, j ∈ {0, 1, . . . , k}.
Hence, T0 is a J-set in G, contradicting the maximality of T . Therefore, T is a dominating
set of G. Since T is a maximum J-set in G, it follows that T is a maximum J-dominating
set in G. Consequently, γJ(G) = k.

The converse is clear.

The following result follows from Theorem 3.

Corollary 1. Let G be a graph and let D = {x1, x2, · · · , xs} be a J-set of G. Then
|D| = s ≤ γJ(G).

Theorem 4. Let G be any graph and D be any J-dominating set of G. Then

(i) u ∈ D if and only if NG[u] ⊈ NG[v] and NG[v] ⊈ NG[u] ∀ v ∈ D \ {u}; and

(ii) a pendant vertex x is in D if and only if its neighbor y is not in D.

Proof. (i) Let D be a J-dominating set of G and u ∈ D. Since D is a J-set, we have
NG[u] \NG[v] ̸= ∅ and NG[v] \NG[u] ̸= ∅ ∀ v ∈ D \ {u}. It follows that NG[u] ⊈ NG[v]
and NG[v] ⊈ NG[u] ∀ v ∈ D \ {u}.

Conversely, suppose that NG[u] ⊈ NG[v] and NG[v] ⊈ NG[u] ∀ v ∈ D \ {u}. Then
NG[u] \NG[v] ̸= ∅ and NG[v] \NG[u] ̸= ∅ ∀ v ∈ D \ {u}. This means that u ∈ D.

(ii) Let x ∈ D be a pendant vertex of G and let y be a neighbor of x. Then
NG[x] ⊆ NG[y]. Suppose on the contrary that y ∈ D. Then by (i), NG[x] ⊈ NG[y]
and NG[y] ⊈ NG[x] ∀ y ∈ D \ {x}, a contradiction. Thus, y /∈ D.

Conversely, suppose on the contrary that x /∈ D. Then by (i), NG[x] ⊆ NG[a] or
NG[a] ⊆ NG[x] for some a ∈ D. Since its neighbor y is not in D, we have x /∈ NG[w]
for every w ∈ D. Thus, NG[D] ̸= V (G), showing that D is not a dominating set in G.
However, this is a contradiction to the fact that D is a J-dominating set in G. Therefore,
x ∈ D.



J. Hassan, J. Salim / Eur. J. Pure Appl. Math, 16 (4) (2023), 2082-2095 2087

Theorem 5. Let G be any graph and let S ⊆ V (G). Then

(i) every independent set S is a J-set; and

(ii) every maximum independent set S is a J-dominating set. Moreover, α(G) ≤ γJ(G).

Proof. (i) Let S = {s1, s2, . . . , sk} be an independent set in G. Then dG(si, sj) ≥ 2 for
every i ̸= j, where i, j ∈ {1, 2, . . . , k}. It follows that si ∈ NG[si] \NG[sj ] for every i ̸= j.
Thus, NG[si] \NG[sj ] ̸= ∅ for every i ̸= j. This shows that S is a J-set of G.

(ii) Next, let S′ = {a1, a2, . . . , am} be a maximum independent set of G. Then S′ is
a J-set in G by (i). Now, suppose on the contrary that S′ is not a dominating set of
G. Then there exists x ∈ V (G) \ S′ such that x /∈ NG[y] ∀ y ∈ S′. This means that
dG(x, y) ≥ 2 for all y ∈ S′. Thus, S∗ = {x} ∪ S′ is an independent set in G, contradicting
the maximality of S′. Therefore, S′ is a dominating set of G, and so S′ is a J-dominating
in G. Consequently, α(G) ≤ γJ(G) by Proposition 1.

Remark 1. The converse of the Theorem 5 is not true. Moreover, the sharpness of the
bound is attainable.

To see this, consider the graph G in Figure 2. Let C = {a, e, h}. Then NG[a] = {a, b},
NG[e] = {b, c, d, e, f, g, h} and NG[h] = {e, g, h, i}. Thus, NG[a] \ NG[e] = {a} ≠ ∅,
NG[a] \ NG[h] = NG[a] ̸= ∅, NG[e] \ NG[a] = {c, d, e, f, g, h} ≠ ∅, NG[e] \ NG[h] =
{b, c, d, f} ≠ ∅, NG[h] \NG[a] = NG[h] ̸= ∅ and NG[h] \NG[e] = {i} ≠ ∅. Therefore, C
is a J-set in G. However, C is not an independent set in G since dG(e, h) = 1. For the
sharpness, consider again the graph G in Figure 2 and let T = {a, c, d, f, g, i}. Then T
is an α-set of G, and so α(G) = 6. Now, since x ∈ NG[x] \ NG[y] ∀ x, y ∈ T , it follows
that T is a J-set in G. Moreover, observe that NG[T ] = V (G). Hence, T is J-dominating
set in G. Since a, d, f and i are pendant vertices of G, it follows that T is the maximum
J-dominating of G by Theorem 4. Consequently, α(G) = 6 = γJ(G).

G :

b c

fed

a

g h i

Figure 2: A graph G with a J-set C which is not an independent set in G



J. Hassan, J. Salim / Eur. J. Pure Appl. Math, 16 (4) (2023), 2082-2095 2088

Theorem 6. Let G be any graph on n ≥ 1 vertices. Then each of the following statements
holds.

(i) 1 ≤ γJ(G) ≤ n.

(ii) γJ(G) = 1 if and only if G is complete.

(iii) If γJ(G) ≤ n − 1, then |NG[v]| ≥ 2 for some v ∈ V (G). However, the converse is
not true.

(iv) If G = Kn, then γJ(G) = |V (G)| = n. However, the converse is not true.

Proof. (i) Since any singleton set {a} is a J-set for any a ∈ V (G), the lower bound
follows by Corollary 1. Moreover, since any γJ -set D is always a subset of V (G), the upper
bound follows. Consequently, 1 ≤ γJ(G) ≤ n.

(ii) Assume that γJ(G) = 1. Suppose G is non-complete graph. Then there exist
a, b ∈ V (G) such that dG(a, b) ≥ 2. Let C = {a, b}. Since dG(a, b) ≥ 2, it follows that
a ∈ NG[a] \NG[b] and b ∈ NG[b] \NG[a]. Thus, NG[a] \NG[b] ̸= ∅ and NG[b] \NG[a] ̸= ∅
C, showing that C is a J-set in G. By Corollary 1, γJ(G) ≥ 2 which is a contradiction.
Hence, G is complete.

Conversely, suppose that G is complete. Then NG[a] = V (G) for any a ∈ V (G). It
follows that for every two distinct vertices x, y ∈ V (G), NG[x]\NG[y] = V (G)\V (G) = ∅.
Thus, γJ(G) ≥ 2 is impossible. Hence, γJ(G) = 1 by (i).

(iii) Assume that γJ(G) ≤ n−1. Suppose on the contrary that |NG[v]| < 2 for every v ∈
V (G). This means that |NG[v]| = 1 for every v ∈ V (G). It follows that G = Kn. Observe
that γ(Kn) = n. Thus, γ(Kn) = n ≤ γJ(G) by Proposition 1, a contradiction. To see that
the converse is not true, consider the graphG in Figure 3. LetK = V (G) = {a, b, c, d, e, f}.
Then NG[a] \ NG[b] = {d}, NG[a] \ NG[c] = {a}, NG[a] \ NG[d] = {b},
NG[a] \ NG[e] = {a, b}, NG[a] \ NG[f ] = {a, d}, NG[b] \ NG[a] = {c, f},
NG[b] \ NG[c] = {a, f}, NG[b] \ NG[d] = {b, f}, NG[b] \ NG[e] = {a, b, c},
NG[b] \ NG[f ] = {a, c}, NG[c] \ NG[a] = {c}, NG[c] \ NG[b] = {d},
NG[c] \ NG[d] = {b}, NG[c] \ NG[e] = {b, c}, NG[c] \ NG[f ] = {d, c},
NG[d] \ NG[a] = {c, e}, NG[d] \ NG[b] = {d, e}, NG[d] \ NG[c] = {a, e},
NG[d] \ NG[e] = {a, c}, NG[d] \ NG[f ] = {a, c, d}, NG[e] \ NG[a] = {e, f},
NG[e] \ NG[b] = {e, d}, NG[e] \ NG[c] = {e, f}, NG[e] \ NG[d] = {f},
NG[e] \ NG[f ] = {d}, NG[f ] \ NG[a] = {e, f}, NG[f ] \ NG[b] = {e},
NG[f ] \ NG[c] = {e, f}, NG[f ] \ NG[d] = {b, f}, NG[f ] \ NG[e] = {b}. Thus, K is a J-
dominating set of G, and so γG(G) = |V (G| = 6.



J. Hassan, J. Salim / Eur. J. Pure Appl. Math, 16 (4) (2023), 2082-2095 2089

G :

b

c

fe

d

a

Figure 3: A graph G with |NG[v]| ≥ 2 for some v ∈ V (G), however, γJ (G) = |V (G)|

(iv). Let G = Kn andD = {v1, v2, . . . , vn} be a vertex set of G. Then NG[vi] = {vi} for
each i ∈ {1, 2, . . . , n}. Thus, vi ∈ NG[vi] \NG[vj ] for each i ̸= j, where i, j ∈ {1, 2, . . . , n}.
This means thatNG[vi]\NG[vj ] ̸= ∅ for each i ̸= j. Thus, V (G) is a J-set ofG. Since V (G)
is a dominating set of G, V (G) is a J-dominating set in G. Hence, γJ(G) = |V (G)| = n
by (i). The converse follows by considering the graph in Figure 3.

Theorem 7. Let G be any non-complete graph. Then every dominating vertex v of G is
not in γJ -set D of G.

Proof. Suppose G is non-complete graph. Then γJ(G) ≥ 2 by Theorem 6(ii). Now, let
v ∈ V (G) be a dominating vertex of G. Suppose on the contrary that v ∈ D. Since v is a
dominating vertex of G, we have NG[v] = V (G). Thus,

NG[u] \NG[v] = NG[u] \ V (G) = ∅ ∀ u ∈ D \ {v}.

It follows that {v} is a maximum J-set of G. By Theorem 3, γJ(G) = 1, a contradiction.
Therefore, v /∈ D.

The next result is a realization result involving J-domination number and domination
number of a graph.

Theorem 8. Let m and n be positive integers such that 2 ≤ a ≤ b. Then there exists a
connected graph G such that γ(G) = a and γJ(G) = b. That is, γJ(G)−γ(G) can be made
arbitrarily large.

Proof. For equality, consider the graph Ka. Since γ(Ks) = s, it follows that
γJ(Ka) = a = γ(Ka) by Theorem 6(iv).

For the inequality(a < b), consider the following cases:
Case 1: a=2
Let m = b − 2 and consider consider the graph G in Figure 4. Let D1 = {y1, y2} and



J. Hassan, J. Salim / Eur. J. Pure Appl. Math, 16 (4) (2023), 2082-2095 2090

D2 = {x1, x2, z1, z2, . . . , zm}. Since for any x ∈ V (G) is not a dominating vertex, it follows
that D1 is a γ-set of G, that is, γ(G) = 2. Now, observe that x1, y1 ∈ NG[x1]\NG[u] ∀ u ∈
D2 \ {x1}, x2 ∈ NG[x2] \NG[v] ∀ v ∈ D2 \ {x2}, and zi ∈ NG[zi] \NG[w] ∀ w ∈ D2 \ {zi},
i ∈ {1, 2, . . . ,m}. Thus, D2 is a J-set of G. Since D2 is a dominating set, D2 is a
J-dominating of G. Since xi and zj are pendant vertices for each i ∈ {1, 2} and j ∈
{1, 2, . . . ,m}, it follows that D2 is a maximum J-dominating set of G by Theorem 4.
Hence, γJ(G) = m+ 2 = b by Theorem 3. Consequently, γ(G) = a < b = γJ(G).

G :

z1x2x1

.
.
.

zm

z2
y1 y2

Figure 4: A graph G with γ(G) < γJ (G)

Case 2: a ≥ 3
Let m = b − a and consider the graph G′ in Figure 5. Let D′ = {y1, y2, . . . , ya} and
D∗ = (x1, x2, · · · , xa, z1, z2, · · · , zm). Then D′ is a γ-set of G. Thus, γ(G) = a. Ob-
serve that xi ∈ NG[xi] \ NG[u] ∀ u ∈ D∗ \ {xi}, zk−1 ∈ NG[zk] \ NG[zl] ∀ l > k,
zs+1 ∈ NG[zs] \ NG[zt] ∀ s > t, and zq ∈ NG[zq] \ NG[xr], where i, r ∈ {1, 2, . . . , a}, and
i, k, l, s, t, q ∈ {1, 2, . . . ,m}. Thus, D∗ is a J-set of G. Since D∗ is a dominating set, D∗

is a J-dominating of G. Since xi is a pendant vertex for each i ∈ {1, 2, . . . , a}, it follows
that D∗ is a maximum J-dominating set of G by Theorem 4. Hence, γJ(G) = m+ a = b
by Theorem 3. Therefore, γ(G) = a < b = γJ(G).



J. Hassan, J. Salim / Eur. J. Pure Appl. Math, 16 (4) (2023), 2082-2095 2091

G′ :

x2
z1

. . .

xaxa−1x3

x1

.
.
.

x4

zm

z2
y1 y2 y3 ya−1 yaya−2

Figure 5: A graph G′ with γ(G′) < γJ (G
′)

Proposition 2. Given any positive integer n ≥ 1, we have

(i) γJ(Pn) =


1 if n = 1, 2

2 if n = 3, 4

n− 2 if n ≥ 5,

(ii) γJ(Cn) =

{
1 if n = 3

n if n ≥ 4.

Proof. (i) Clearly, γJ(Pn) = 1 for n = 1, 2 and γJ(Pn) = 2 for n = 3, 4. Suppose n ≥ 5.
Let Pn = G = [v1, v2, . . . , vn], D = {v2, v3 · · · , vn−2, vn−1} and let i, j ∈ {2, 3, . . . , n − 1}.
Then vi−1 ∈ NG[vi]\NG[vj ] and vj+1 ∈ NG[vj ]\NG[vi] for all j > i, i, j ∈ {2, 3, . . . , n−1}.
Thus, NG[vi]\NG[vj ] ̸= ∅ for all i ̸= j, i, j ∈ {2, 3, . . . , n−1}, showing that D is a J-set in
G. Therefore, γJ(Pn) ≥ n− 2 by Corollary 1. If γJ(Pn) = n, then V (Pn) is the maximum
J-dominating set in Pn. However, NG[vn] ⊆ NG[vn−1]. Thus, NG[vn] \ NG[vn−1] = ∅,
a contradiction. Next, suppose that γJ(G) = n − 1, say A = {a1, a2, · · · , an−1} is a
J-dominating set of G. Then either ⟨A⟩ is connected or disconnected. Assume that
⟨A⟩ is connected. Then either v1 or vn is not in A. If v1 /∈ A, then vn−1, vn ∈ A.
However, NG[vn] \ NG[vn−1] = {vn, vn−1} \ {vn, vn−1, vn−2} = ∅, a contradiction. If
vn /∈ A, then v1, v2 ∈ A. However, NG[v1] \ NG[v2] = {v1, v2} \ {v1, v2, v3} = ∅, a
contradiction. Next, suppose that ⟨A⟩ is disconnected. Then there exists vj ∈ V (G)
such that vj /∈ A for some j ∈ {2, 3, . . . , n − 1}. If v2 /∈ A, then vn−1, vn ∈ A and
NG[vn] \ NG[vn−1] = {vn, vn−1} \ {vn, vn−1, vn−2} = ∅, a contradiction. If vn−1 /∈ A,
then v1, v2 ∈ A and NG[v1] \NG[v2] = {v1, v2} \ {v1, v2, v3} = ∅, a contradiction. Lastly,
suppose that vj /∈ A, where 2 < j < n− 1. Then vn−1, vn ∈ A and v1, v2 ∈ A, a contradic-
tion by the preceding arguments. Therefore, γJ(G) = n − 1 is impossible. Consequently,



J. Hassan, J. Salim / Eur. J. Pure Appl. Math, 16 (4) (2023), 2082-2095 2092

γJ(Pn) = n− 2 for all n ≥ 5.

(ii) Clearly, γJ(C3) = 1. Suppose n ≥ 4. Let Cn = G = [v1, v2, . . . , vn, v1] and
let D∗ = V (G) = {v1, v2, . . . , vn}. Then vn ∈ NG[v1] \NG[vj ] for all j ∈ {2, 3, . . . , n− 2},
v1 ∈ NG[v1] \ NG[vn−1], v2 ∈ NG[v1] \ NG[vn], v1 ∈ NG[vn] \ NG[vs] for all
s ∈ {3, 4, . . . , n − 1}, vn ∈ NG[vn] \ NG[v2], vn−1 ∈ NG[vn] \ NG[v1], vi+1 ∈ NG[vi] \
NG[vj ] ∀ i, j ∈ {1, 2, . . . , n − 1}, i > j, and vk−1 ∈ NG[vk] \ NG[vl] ∀ k, l ∈ {2, 3, . . . , n},
k < l. It follows that NG[vq] \NG[vr] ̸= ∅ for every q ̸= r, q, r ∈ {1, 2, . . . , n}. Therefore,
D∗ = V (G) is a J-dominating set in G, showing that γJ(Cn) = n for all n ≥ 4.

Theorem 9. Let G and H be two non-complete graphs. A subset D of distinct vertices of
G+H is a J-dominating set in G+H if and only if one of the following condition holds:

(i) D is a J-dominating set of G.

(ii) D is a J-dominating set of H.

(iii) D = DG ∪DH , where DG and DH are J-sets in G and H, respectively.

Proof. Suppose that D is J-dominating set in G + H. If DH = ∅, then D = DG is
a J-dominating set in G, and so (i) holds. If DG = ∅, then D = DH is a J-dominating
set in H, showing that (ii) holds. Now, assume that DG and DH are both non-empty.
Suppose on the contrary that DG is not a J-set in G. Then there exist a, b ∈ DG ⊆ D
such that either NG[a] \NG[b] = ∅ or NG[b] \NG[a] = ∅, a contradiction. Therefore, DG

is a J-set in G. Similarly, DH is a J-set in H. Consequently, (iii) holds.
Conversely, suppose that (i) holds. Then D is a J-dominating set in G+H. Similarly,

if (ii) holds, then D is a J-dominating set in G+H. Now, assume that (iii) holds. Since
DG and DH are both non-empty, it follows that D is a dominating set in G+H. It remains
to show that D is a J-set in G+H. Let a, b ∈ D. If a, b ∈ DG ⊆ D, then NG[a]\NG[b] ̸= ∅
and NG[b] \ NG[a] ̸= ∅ by assumption. This implies that NG+H [a] \ NG+H [b] ̸= ∅ and
NG+H [b] \NG+H [a] ̸= ∅, and we are done. Similarly, if a, b ∈ DH ⊆ D, then D is a J-set
in G+H. Now, assume that a ∈ DG and b ∈ DH . If a is a dominating vertex of G, then
DH = ∅, a contradiction. Thus, a is not a dominating vertex of G. Similarly, b is not a
dominating vertex of H. Let x ∈ V (G) and y ∈ V (H), where x /∈ NG[a] and y /∈ NH [b].
Then y ∈ NG+H [a] \ NG+H [b] ̸= ∅ and x ∈ NG+H [b] \ NG+H [a]. Thus, D is a J-set in
G+H. Consequently, D is a J-dominating set in G+H.

Corollary 2. Let G and H be two non-complete graphs. Then

γJ(G+H) = γJ(G) + γJ(H).

In particular, each of the following holds:

(i) γJ(Km,n) = γJ(Km) + γJ(Kn) = m+ n for all m,n ≥ 2.



J. Hassan, J. Salim / Eur. J. Pure Appl. Math, 16 (4) (2023), 2082-2095 2093

(ii) γJ(Pn + Pm) =


4 if n = 3,m = 3

m if n = 3,m ≥ 4

n if n ≥ 4 ≥ m = 3

n+m− 4 if n ≥ 4,m ≥ 4.

(iii) γJ(Pn + Cm) =

{
m+ 2 if n = 3 ≥ m ≥ 4

n+m− 2 if n ≥ 4,m ≥ 4.

(iv) γJ(Cn + Cm) = n+m for all n,m ≥ 4.

Proof. Let D = DG ∪DH be a γJ -set of G+H. Then by Theorem 9, DG and DH are
J-sets in G and H, respectively. It follows that

γJ(G+H) = |D| = |DG|+ |DH | ≤ γJ(DG) + γJ(DH) by Corollary 1.

On the other hand, suppose that D = DG ∪ DH , where DG and DH are maximum
J-sets in G and H, respectively. Then by Theorem 9, D is a J-dominating set in G+H.
Thus, γJ(G) + γJ(H) = |DG|+ |DH | = |D| ≤ γJ(G+H) by Proposition 1. Consequently,
γJ(G + H) = γJ(G) + γJ(H). Moreover, particular cases follow from Theorem 6 and
Proposition 2.

Theorem 10. Let G and H be any complete and any non-complete graphs, respectively.
A subset D = DG ∪DH of distinct vertices of G +H is a J-dominating set in G +H if
and only if one of the following conditions holds:

(i) DH = ∅ and DG is a J-dominating set of G.

(ii) DG = ∅ and DH is a J-dominating set of H.

Proof. Suppose that D is J-dominating set in G + H. Since G is complete, both
DG ̸= ∅ and DH ̸= ∅ are impossible. It follows that either DG ̸= ∅ and DH = ∅ or
DH ̸= ∅ and DG = ∅. If DH = ∅, then D = DG is a J-dominating set in G, and so (i)
holds. If DG = ∅, then D = DH is a J-dominating set in H, and so (ii) holds.

Conversely, suppose that DH = ∅ and DG is a J-dominating set of G. Then D = DG

is a J-set in G +H. Since DG is dominating set in G, NG+H [D] = V (G +H). Thus, D
is a dominating set in G+H, showing that D is a J-dominating set in G+H. Similarly,
if (ii) holds, then D is a J-dominating set in G+H.

Corollary 3. Let G and H be complete and non-complete graphs, respectively. Then

γJ(G+H) = γJ(H).

In particular, each of the following holds:

(i) γJ(K1,n) = n for all n ≥ 1.



REFERENCES 2094

(ii) γJ(Wn) = γJ(K1 + Cn) = n for all n ≥ 4.

(iii) γJ(Fn) = γJ(K1 + Pn) = n− 2 for all n ≥ 3.

Proof. Let D = DG ∪DH be a γJ -set of G+H. Then by Theorem 10, either DH = ∅
and DG is J-dominating set in G or DG = ∅ and DH is J-dominating set in H. It follows
that

γJ(G+H) = |D| = |DG|+ |DH | = |DG| ≤ γJ(G)

or
γJ(G+H) = |D| = |DG|+ |DH | = |DH | ≤ γJ(DH) by Proposition 1.

Thus,

γJ(G+H) ≤ max{γJ(G), γJ(DH)}.

Since G is complete, it follows that γJ(G+H) ≤ γJ(DH)}.
On the other hand, suppose that D = DG ∪ DH , where either DH = ∅ and

DG = D is γJ - set in G or DG = ∅ and DH = D is γJ -set in H. Then by Theorem
10, D is a J-dominating set in G + H. It follows that γJ(G) = |DG| ≤ γJ(G + H) or
γJ(H) = |DH | ≤ γJ(G +H) by Proposition 1. Thus, max{γJ(G), γJ(H)} ≤ γJ(G +H).
Since G is complete, we have γJ(H) ≤ γJ(G + H). Consequently, γJ(G + H) = γJ(H).
Moreover, particular cases follow from Theorem 6 and Proposition 2.

4. Conclusion

The concept of J-domination has been introduced and investigated in this study. Its
bounds with respect to the order of a graph and other parameters have been determined.
It was shown that any graph G admits a J-domination and was found out that a graph G
satisfying γJ(G) = |V (G)| is not unique. Moreover, characterizations of J-dominating sets
in some graphs have been used to determine exact values of parameter of some graphs.
Some graphs that were not considered in this study could be an interesting to investi-
gate further for the concept. In addition, researchers may consider the complexity and
algorithm, and real life application of the concept.

Acknowledgements

The authors would like to thank Mindanao State University - Tawi-Tawi College of
Technology and Oceanography for funding this research.

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