EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 16, No. 4, 2023, 2106-2117
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Characterizations of J-Total Dominating Sets of Some
Graphs

Javier A. Hassan1, Jahiri U. Manditong1,∗, Alcyn Bakkang2, Sisteta U. Kamdon1,
Jeffrey Imer Salim1

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

Abstract. Let G be a graph with no isolated vertex. A subset M ⊆ V (G) is called a J-open set
if NG(a)\NG(b) ̸= ∅ and NG(b)\NG(a) ̸= ∅ ∀ a, b ∈ M, where a ̸= b. If in addition, M is a total
dominating in G, then we call M a J-total dominating set in G. The maximum cardinality among
all J-total dominating set in G, denoted by γJt(G), is called the J-total domination number of
G. In this paper, we characterize J-total dominating sets in some special graphs and join of two
graphs, and we use these results to obtain formulas for the parameters of these graphs. Moreover,
we determine its relationships with other known parameters in graph theory. Finally, we derive
the lower bound of the parameter for the corona of two graphs.

2020 Mathematics Subject Classifications: 05C69

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

1. Introduction

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.

Domination in a graph was introduced by Oystein Ore in 1962 in his book on graph
theory [10]. A subset D of vertices of a graph G is called a dominating of G if for every
x ∈ V (G) \ D, there exists y ∈ D such that xy ∈ E(G), that is, NG[D] = V (G). The
domination number of G, denoted by γ(G), is the minimum cardinality of a dominating set

DOI: https://doi.org/10.29020/nybg.ejpam.v16i4.4912

Email addresses: javierhassan@msutawi-tawi.edu.ph (J. Hassan),
jahirimanditong@msutawi-tawi.edu.ph (J. Manditong),
alcynbakkang@msutawi-tawi.edu.ph (A. Bakkang),
sistetakamdon@msutawi-tawi.edu.ph (S. Kamdon),
jeffreyimersalim@msutawi-tawi.edu.ph (J. Salim)

https://www.ejpam.com 2106 © 2023 EJPAM All rights reserved.



J.A. Hassan et al. / Eur. J. Pure Appl. Math, 16 (4) (2023), 2106-2117 2107

in G. A decade later, Cockayne and Hedetniemi [1] published a survey paper, in which the
notation γ(G) was first used for the domination number of a graph G. Since then, several
mathematicians had studied and introduced new domination parameters in graphs. Some
variants of domination were defined and further studied by researchers in [2–9, 11, 12].

In this paper, new variant of domination called J-total domination in a graph will be
introduced and investigated. We will characterize J-total dominating sets in some classes
of graphs and the join of two graphs, and we will use these results to determine the exact
value of each of these graphs. Moreover, we will determine the bound of the parameter
for the corona of two graphs. We believe that this new parameter would give additional
insights to researchers in the field and would help them for more research directions in the
future.

2. Terminology and Notation

Let G = (V (G), E(G)) be a graph. Two vertices a, b of G are said to be adjacent,
or neighbors, if ab is an edge of G. The open neighborhood of x in G is the set defined
by 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), then 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 path graph is a non-empty graph with vertex-set {x1, x2, . . . , xn} and edge-set
{x1x2, x2x3, . . . , xn−1xn}, where the x

′
is are all distinct. The path of order n is denoted

by Pn. If G is a graph and u and v are vertices of G, then a path from vertex u to vertex v
is sometimes called a u-v path. The cycle graph is the graph of order n ≥ 3 with vertex-set
{x1, x2, . . . , xn} and edge-set {x1x2, x2x3, . . . , xn−1xn, xnx1}. The cycle graph of order n
is denoted by Pn.

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.

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 S of V (G) is called a dominating of G if for every x ∈ V (G) \ S, there exists
y ∈ S such that xy ∈ E(G), that is, NG[S] = V (G). The domination number of G,
denoted by γ(G), is the minimum cardinality of a dominating set in G. Any dominating
set S with cardinality equal to γ(G) is called a γ-set of G.

A subset T of V (G) is called a total dominating of G if for every x ∈ V (G), there
exists y ∈ T such that xy ∈ E(G), that is, NG(T ) = V (G). The total domination number
of G, denoted by γt(G), is the minimum cardinality of a total dominating set in G. Any
total dominating set T with cardinality equal to γt(G) is called a γt-set of G.

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

A graph G is called a bipartite graph if its vertex-set V (G) can be partitioned into two



J.A. Hassan et al. / Eur. J. Pure Appl. Math, 16 (4) (2023), 2106-2117 2108

nonempty subsets V1 and V2 such that every edge of G has one end in V1 and one end in
V2. The sets V1 and V2 are called the partite sets of G. If each vertex in V1 is adjacent to
every vertex in V2, then G is called a complete bipartite graph. If |V1| = m and |V2| = n,
then the complete bipartite graph is denoted by Km,n. A star graph of order n+ 1 is the
complete bipartite graph K1,n.

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)}.

The fan Fn of order n+ 1, where n ≥ 1, is given by Fn = K1 + Pn.
The corona G and H, denoted by G ◦H, the graph obtained by taking one copy of G

and |V (G)| copies of H, and then Joining the ith vertex of G to every vertex of the ith
copy of H. We denote by Hv the copy of H in G ◦H corresponding to the vertex v ∈ G
and write v +Hv for ⟨{v}+Hv⟩.

3. Results

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

Definition 1. Let G be a graph with no isolated vertex. A subset M ⊆ V (G) is called
a J-open set in G if NG(a) \ NG(b) ̸= ∅ and NG(b) \ NG(a) ̸= ∅ ∀ a, b ∈ M,a ̸= b.
If in addition, M is a total dominating in G, then we call M a J-total dominating set
in G. The maximun cardinality among all J-total dominating sets in G, denoted by
γJt(G), is called the J-total domination number of G. Any J-total dominating set M with
| M |= γJt(G) (resp. |M | = γt(G)), is called a γJt-set or the maximum (resp. minimum)
J-total dominating set in G.

Remark 1. Let G be a graph with no isolated vertex. Then each of the following is true.

(i) A total dominating set T of G may not be a J-open set in G (hence not a J-total
dominating set).

(ii) A J-open set Q in G may not be a total dominating set in G (hence not a J-total
dominating set).

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

Proposition 1. Let G be a graph with no isolated vertex. Then

(i) γt(G) ≤ γJt(G).

(ii) 2 ≤ γJt(G) ≤| V (G) |.

Proof. (i) Let G be a graph with no isolated vertex and let M be a maximum J-
total dominating set of G. Then M is a total dominating set of G (by defintion). Since



J.A. Hassan et al. / Eur. J. Pure Appl. Math, 16 (4) (2023), 2106-2117 2109

γt(G) is the minimum cardinality among all total dominating sets in G, it follows that
γJt(G) =| M |≥ γt(G).

(ii) Since γt(G) ≥ 2 for any graph G with no isolated vertex, and so γJt(G) ≥ 2 by (i).
Since any J-total dominating set M is always a subset of a vertices V (G) of G, it follows
that γJt(G) ≤| V (G) |. Consequently, 2 ≤ γJt(G) ≤| V (G) |.

Remark 2. The bound given in Proposition 1 is tight. Moreover, strict inequality is
attainable.

For tightness, consider the graph G given in Figure 1 below.

G :

a b c d e

a′ b′ c′ d′ e′

Figure 1: A graph G with γt(G) = 5 = γJt(G)

Let S = {a, b, c, d, e}. Clearly, S is the minimum total dominating set of G. Thus,
γt(G) = 5. Observe that x′ ∈ NG(x) \NG(y) and y′ ∈ NG(y) \NG(x) for every x, y ∈ S,
where x ̸= y. It follows that NG(x) \ NG(y) ̸= ∅ and NG(y) \ NG(x) ̸= ∅ for every
x, y ∈ S, x ̸= y. Hence, S is a J-open set in G, showing that S is a J-total dominating set
of G. Notice that NG(a

′), NG(c
′) ⊆ NG(b), NG(b

′), NG(d
′) ⊆ NG(c) and NG(e

′) ⊆ NG(d)
and a, b, c, d, e must be in any total dominating set of G. Consequently, S is the maximum
J-total dominating set of G, and so γJt(G) = 5.

For strict inequality, consider the graph G′ given in Figure 2 below.

G′ :
u1

u2

u3 u4

u8

u5

u6

u7

Figure 2: A graph G′ with γt(G′) = 3 < 7 = γJt(G
′)

Let T1 = {u1, u2, u3, u4, u5, u6, u7} and T2 = {u3, u4, u5}. Then T2 is a minimum total
dominating set of G′. Hence, γt(G

′) = 3. Since T2 ⊆ T1, it follows that T1 is also a total



J.A. Hassan et al. / Eur. J. Pure Appl. Math, 16 (4) (2023), 2106-2117 2110

dominating set of G′. Observe that u2 ∈ NG′(u1)\NG′(ui) ∀ i ̸= 3, u3 ∈ NG′(u1)\NG′(u3),
u1 ∈ NG′(u2) \NG′(uj) ∀ j ̸= 3, u3 ∈ NG′(u2) \NG′(u3), u1 ∈ NG′(u3) \NG′(ur) ∀ r ̸= 2,
u2 ∈ NG′(u3) \NG′(u2), u8 ∈ NG′(u4) \NG′(uq) ∀ q ̸= 4, u7 ∈ NG′(u5) \NG′(us) ∀ s ̸= 6,
u6 ∈ NG′(u5) \ NG′(u6), u7 ∈ NG′(u6) \ NG′(ut) ∀ t ̸= 5, u5 ∈ NG′(u6) \ NG′(u5) and
u6 ∈ NG′(u7) \ NG′(um) ∀ m ̸= 5, u5 ∈ NG′(u7) \ NG′(u5). Thus, T1 is a J-open set
of G′, and so T1 is a J-total dominating set of G′. Hence, γJt(G

′) = 7. Consequently,
γJt(G

′) > γt(G
′).

Theorem 1. Let Kn be a complete graph of order n ≥ 2. Then M ⊆ V (Kn) is a J-total
dominating in Kn if and only if | M |≥ 2.

Proof. Let M ⊆ V (Kn) be a J-total dominating set in Kn. Then M is a total
dominating set in Kn. Since γt(Kn) = 2 for all n ≥ 2, it follows that | M |≥ γt(Kn) = 2.

Conversely, suppose thatM ⊆ V (Kn) with | M |≥ 2. Let V (Kn) = {v1, v2, . . . , vn} and
M = {v1, v2, . . . , vs} ⊆ V (Kn) where s ∈ {2, 3, . . . , n}. Observe that
vi ∈ NKn(vj) \NKn(vi) for all i ̸= j, i, j ∈ {1, 2, . . . , s}. Thus,

NKn(vj) \NKn(vi) ̸= ∅ for all i ̸= j,where i, j ∈ {1, 2, . . . , s},

showing that M is a J-open set in Kn ∀ n ≥ 2. Since any set {vi, vj}, i ̸= j, is a total
dominating in Kn, it follows that M is J-total dominating set in Kn ∀ n ≥ 2.

Corollary 1. Let n ≥ 2 be any positive integer. Then

γJt(Kn) = n.

Proof. LetM = V (Kn) = {a1, a2, ..., an}. Then by Theorem 1,M is J-total dominating
set in Kn. Thus, γJt(Kn) ≥ n. By Proposition 1, γJt(Kn) = n.

Theorem 2. Let m and n be positive integers with 2 ≤ m ≤ n. Then there exists a
connected graph H such that γt(H) = m and γJt(H) = n. That is, γJt(H)− γt(H) can be
made arbitrarily large.

Proof. For m = n, consider the graph H in Figure 3 below.



J.A. Hassan et al. / Eur. J. Pure Appl. Math, 16 (4) (2023), 2106-2117 2111

b3b2b1

c2c1 c3

a3a2a1

bm−1 bm

am−1 am

ym−1 ym

. . .H :

Figure 3: A graph H with γt(H) = m = γJt(H)

Let M = {a1, a2, ..., am−1, am}. Then M is a minimum total dominating set of H, and
so γt(H) = m. Since bi, ci ∈ NH(ai) \NH(aj) for every i ̸= j, i, j ∈ {1, 2, . . . ,m}, it follows
that M is a J-open set in H. Thus, M is a J-total dominating set of H. Now, observe
that NH(bi), NH(ci) ⊆ NH(ai+1), ∀ i ∈ {1, 2, . . . ,m − 1}, NH(bm), NH(cm) ⊆ NH(am−1)
and ai must be in any total dominating set of H for each i ∈ {1, 2, . . . ,m}. Therefore, M
is the maximum J-total dominating set of H, and so γJt(H) = m.

Suppose that m < n. Let s = n − m and consider the graph H ′ in Figure 4 below,
where ⟨{am, b1, b2, . . . , bs}⟩ induced a complete graph for all positive integer s ≥ 1.

x3x2x1

y2y1 y3

a3a2a1

xm−1 xm

am−1 am

ym−1 ym

b1

b2

bs

.

.
.

. . .H ′ :

Figure 4: A graph H′ with γt(H′) < γJt(H
′)

Let M1 = {a1, a2, ..., am} and M2 = {a1, a2, . . . , am, b1, b2, . . . , bs}. Then M1 is the
minimum total dominating set of H ′. Thus, γt(H

′) = m. Since M1 ⊆ M2, M2 is also a
total dominating set of H ′. Observe that M2 is a J-open set in H ′. Hence,M2 is a J-
total dominating set of H ′. Applying the same argument in the equality part and V (Ks)
is a J-open set in Ks, s ≥ 2 by Theorem 1, it follows that M2 is the maximum J-total



J.A. Hassan et al. / Eur. J. Pure Appl. Math, 16 (4) (2023), 2106-2117 2112

dominating set of H ′. Consequently, γJt(H
′) = s+m = n.

Theorem 3. Let G be a graph with no isolated vertex. Then

(i) M is a γt-set in G if and only if M is a minimum J-total dominating set in G.

(ii) If every component of G is non-trivial complete graph, then γJt(G) = |V (G)|. How-
ever, the converse is not true.

Proof. (i) Let M be a γt-set in G. Then M is the minimum total dominating set
in G. Suppose that M is not a J-open set in G. Then there exist a, b ∈ M such that
NG(a)\NG(b) = ∅ orNG(b)\NG(a) = ∅. It follows thatNG(a) ⊆ NG(b) orNG(b) ⊆ NG(a).
Assume that NG(a) ⊆ NG(b), then M\{a} is a total dominating set in G. However, this
is a contradiction to our assumption that M is the minimum total dominating set in G.
Hence, M is a J-open set in G, and so M is a minimum J-total dominating set in G.

The converse is clear.
(ii) Suppose that every component H of G is a non-trivial complete graph. Let

H1, . . . ,Hk, k ≥ 2 be components of G. Then by Corollary 1, γJt(Hi) =| V (Hi) | .
It follows that γJt(G) = γJt(Hi) + · · ·+ γJt(Hk)

=| V (Hi) | + · · ·+ | V (Hk)
=| V (G) |.

To see that the converse is not true, consider G = C5 below.

C5 :

a5

a3

a2
a1

a4

Figure 5: A graph C5 with γJt(C5) = 5

Let M = {a1, a2, ..., a5} = V (G). Observe that

a2 ∈ NG(a1) \NG(ai) ∀ i ̸= 3,

a5 ∈ NG(a1) \NG(aj) ∀ j ̸= 4,

a1 ∈ NG(a2) \NG(ak) ∀ k ̸= 5,

a3 ∈ NG(a2) \NG(ae) ∀ e ̸= 4,

a2 ∈ NG(a3) \NG(as) ∀ s ̸= 1,



J.A. Hassan et al. / Eur. J. Pure Appl. Math, 16 (4) (2023), 2106-2117 2113

a4 ∈ NG(a3) \NG(at) ∀ t ̸= 5,

a3 ∈ NG(a4) \NG(ar) ∀ r ̸= 2,

a5 ∈ NG(a4) \NG(aq) ∀ q ̸= 1,

a1 ∈ NG(a5) \NG(am) ∀ m ̸= 2, and

a4 ∈ NG(a5) \NG(an) ∀ n ̸= 3.

Thus, M = V (G) is a J-open set in G. Since NG(M) = V (G), it follows that M is a
J-total dominating set in G. Hence, γJt(G) = 5 =| V (G) | .

Theorem 4. Let Km,n be a complete bipartite graph where m,n ≥ 1. Then N ⊆ V (Km,n)
is a J-total dominating in Km,n if and only if N = {a, b} for some a ∈ V (Km) and
b ∈ V (Kn).

Proof. Let N ⊆ V (Km,n) be a J-total dominating in Km,n,m, n ≥ 1. Then N is
a total dominating set in Km,n. Thus, |N | ≥ 2. Let V (Km) = {u1, u2, . . . , um} and
V (Kn) = {v1, v2, . . . , vn}. Observe that Nkm,n(ui) = NKm,n(uj) ∀ i ̸= j,
i, j ∈ {1, 2, . . . ,m} and NKm,n(vr) = NKm,n(vq) ∀r ̸= q, r, q ∈ {1, 2, . . . , n}. This means
that there are m − 1 and n − 1 vertices of Km and Kn, respectively, cannot be in any
J-total dominating set of Km,n. Hence, | N |≤ 2, and so | N |= 2. Thus, N = {a, b} for
some a ∈ V (Km) and b ∈ V (Kn).

Conversely, let N = {a, b} for some a ∈ V (Km) and b ∈ V (Kn). Then
NKm,n(a) = V (Kn) and NKm,n(b) = V (Km). Hence, NKm,n(N) = V (Km,n), and
Nkm,n(a)\NKm,n(b) = V (Kn) ̸= ∅ and NKm,n(b)\NKm,n(a) = V (Km) ̸= ∅. Consequently,
N = {a, b} is a J-total dominating set in Km,n.

The following result follows immediately for Theorem 4.

Corollary 2. Let m,n ≥ 1 be positive integers. Then γJt(Km,n) = 2.

Theorem 5. Let m ≥ 2 be positive integer. Then

γJt(Pm) =


2 if m = 2, 3, 4

4 if m = 5

m− 2 if m ≥ 6.

Proof. Clearly, γJt(P2) = 2. For m = 3, let V (P3) = {v1, v2, v3} and let S = {v1, v2}.
Then v2 ∈ NP3(v1)\NP3(v2) and v1 ∈ NP3(v2)\NP3(v1). Thus, S is a J-open set in P3.
Since NP3(S) = V (P3), it follows that S is a J-total dominating set of P3. Notice that
NP3(v1) = NP3(v3). Hence, v1 and v3 cannot be both in any J-open set of P3. Therefore,
S = {v1, v2} is a maximum J-total dominating set in P3, showing that γJt(P3) = 2.

For m = 4, let V (P4) = {a1, a2, a3, a4} and S′ = {a2, a3}. Then

a1, a3 ∈ NP4(a2)\NP4(a3) and a2, a4 ∈ NP4(a3)\NP4(a2).



J.A. Hassan et al. / Eur. J. Pure Appl. Math, 16 (4) (2023), 2106-2117 2114

Thus, S′ is a J-open set in P4. Observe that NP4(S
′) = V (P4). Therefore, S

′ is a J-total
dominating set in P4. Notice that NP4(a1) ⊆ NP4(a3) and NP4(a4) ⊆ NP4(a2). This means
that a1 and a3 (resp. a2 and a4) cannot be both in any J-open set of P4. Consequently,
S′ = {v2, v3} is a maximum J-total dominating set of P4, and so γJt(P4) = 2.

For m = 5, let V (P5) = {u1, u2, u3, u4, u5} and consider C = {u1, u2, u4, u5}. Then
u1 ∈ NP5(u2) \NP5(ui) ∀ i ̸= 2, u2 ∈ NP5(u1)\NP5(uj) ∀j ̸= 1, u4 ∈ NP5(u5)\NP5(ur)
∀ r ̸= 5 and u5 ∈ NP5(u4)\NP5(uq) ∀ q ̸= 4. Hence, C is a J-open set in P5. Since
NP5(C) = V (5), it follows that C is a J-total dominating set of P5. Notice that
NP5(u1) ⊆ NP5(u3). Thus, u1 and u3 cannot be both in any J-open set of P5. Conse-
quently, C is a maximum J-total domianting set of P5, and so γJt(P5) = 4.

Next, suppose that m ≥ 6. Let V (Pm) = {w1, w2, . . . , wm} and consider
C ′ = {w2, w3, ..., wm−2, wm−1}. Notice that wi−1 ∈ NPm(wi) \ NPm(wj) and
wj+1 ∈ NPm(wj) \NPm(wi) ∀i < j, i, j ∈ {2, 3, . . . ,m− 1}. It follows that

NPm(wi) \NPm(wj) ̸= ∅ ∀ i ̸= j, i, j ∈ {2, 3, ...,m− 1}.

Thus, C ′ is a J-open set in Pm for allm ≥ 6. SinceNPm(C
′) = V (Pm), C ′ is a J-total domi-

nating set of Pm. Now, observe that NPm(w1) ⊆ NPm(w3) and
NPm(wm) ⊆ NPm(wm−2). Hence, w1 and w3 (resp. wm−2 and wm) cannot be both in
any J-open set of Pm. Therefore, C ′ is a maximum J-total dominating set of Pm, and so
γJt(Pm) = m− 2 for all m ≥ 6.

Theorem 6. Let n be any positive integer. Then

γJt(Fn) =


2 if n = 1

3 if n = 2, 3, 4

5 if n = 5

n− 1 if n ≥ 6

Proof. Since F1 and F2 are complete graphs, γJt(F1) = 2 and γJt(F2) = 3 by Corollary
1. For n = 3, let V (F3) = {v0, v1, v2, v3}, where v0 is the dominating vertex of F3.
Consider M = {v0, v1, v2}. Then vi ∈ NF3(v0) \ NF3(vi) and v0 ∈ NF3(vi) \ NF3(v0)
∀ i ̸= 0, and v2 ∈ NF3(v1)\NF3(v2) and v1 ∈ NF3(v2)\NF3(v1). Thus, M is a J-open
set in F3. Since NFn(M) = V (F3), it follows that M is a J-total dominating set of F3.
Since NF3(v1) = NF3(v3), v1 and v3 cannot be both in any J-open set of F3. Therefore,
M is a maximum J-total dominating set of F3, and so γJt(F3) = 3. Similarly, if n = 4,
then γJt(F4) = 3.

For n = 5, let V (F5) = {v0, v1, v2, V3, v4, v5}, where v0 is the dominating vertex of F5.
Let M ′ = {v0, v1, v2, , v4, v5}. Then vj ∈ NF5(v0) \ NF5(vj) and v0 ∈ NF5(vj) \ NF5(v0)
∀ j ̸= 0. Since {v1, v2, v4, v5} is a J-open set in P5 by Theorem 5, it follows that M ′ is a
J-open set in F5. Notice that NF5(M

′) = V (F5). Thus, M
′ is a J-total dominating set of

F5. Since NF5(v1) ⊆ NF5(v3), it follows that v1 and v3 cannot be both in any J-open set
of F5. Therefore, M

′ is a maximum J-total dominating set of F5, and so γJt(F5) = 5.



J.A. Hassan et al. / Eur. J. Pure Appl. Math, 16 (4) (2023), 2106-2117 2115

Next, suppose that n ≥ 6. Let V (Fn) = {u0, u1, . . . , un}, where u0 is the dominating
vertex of Fn. Let C = {u0, u2, u3, . . . , un−1}. Observe that ur ∈ NFn(uo) \ NFn(ur) and
u0 ∈ NFn(ur) \NFn(uo) ∀ r ̸= 0. Since {u2, u3, ..., un−1} is a J-open set in Pn by Theorem
5, it follows that C is a J-open set in Fn. Observe further that NFn(C) = V (Fn). Hence,
C is a J-total dominating set of Fn. Since NFn(u1) ⊆ NFn(u3) and NFn(un) ⊆ NFn(un−2),
u1 and u3 (resp. un−2 and un) cannot be both in any J-open set of Fn. Therefore, C is a
maximum J-total dominating set of Fn, showing that γJt(Fn) = n− 1 for all n ≥ 6.

Theorem 7. Let G and H be two graphs with no isolated vertices. A subset M of vertices
of G+H is a J-total dominating set of G+H if and only if one of the following conditions
holds:

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

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

(iii) M = MG ∪MH , where MG and MH are J-open sets in G and H, respectively.

Proof. Let M be a J-total dominating set of G + H. If MH = ∅, then M = MG is
a J-total dominating set in G. Thus, (i) holds. If MG = ∅, then M = MH is a J-total
dominating set in H, and hence (ii) holds. Next, assume that MG and MH are both
non-empty. Suppose on the contrary that MG is not a J-open set in G. Then there exist
a, b ∈ MG ⊆ M such that either NG(a) \ NG(b) = ∅ or NG(b) \ NG(a) = ∅. Thus,
NG+H(a) \NG+H(b) = ∅ or NG+H(a) \NG+H(b) = ∅, a contradiction to the fact that M
is a J-open set in G+H. Therefore, DG is a J-open set in G. Similarly, MH is a J-open
set in H. Consequently, (iii) holds.

Conversely, if (i) or (ii) holds, then the assertion follows. Next, suppose that (iii) holds.
SinceMG andMH are both non-empty, it follows thatM is a total dominating set inG+H.
Let a, b ∈ M . Suppose that a, b ∈ MG ⊆ M . Since MG is a J-open set in G, we have
NG(a) \NG(b) ̸= ∅ and NG(b) \NG(a) ̸= ∅. It follows that NG+H(a) \NG+H(b) ̸= ∅ and
NG+H(b)\NG+H(a) ̸= ∅. Hence M is a J-open set in G+H. Similarly, if a, b ∈ MH ⊆ M ,
then M is a J-open set in G + H. Now, assume that a ∈ MG and b ∈ MH . If a is a
dominating vertex of G, then we are done. Similarly, if b a dominating vertex of H.
Suppose that a and b are not dominating vertices of G and H, respectively. 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, M is a J-open set in G+H. Consequently, D is a J-total
dominating set in G+H.

The following result follows immediately from Theorem 7.

Corollary 3. Let G and H be two graphs with no isolated vertices. Then

γJt(G+H) = γJt(G) + γJt(H).



J.A. Hassan et al. / Eur. J. Pure Appl. Math, 16 (4) (2023), 2106-2117 2116

Theorem 8. Let G be a connected non-trivial graph and H be a graph with no isolated
vertex. If M = V (G) ∪ (

⋃
v∈V (G)Mv), where Mv is a J-total dominating set in Hv for

each v ∈ V (G), then M is a J-total dominating set in G ◦H. Moreover,

γJt(G ◦H) ≥ |V (G)|+ γJt(H) · |V (G)|.

Proof. Let M = V (G) ∪ (
⋃

v∈V (G)Mv), where Mv is a J-total dominating set in Hv

for each v ∈ V (G). Since G is connected, it follows that V (G) is a total dominating set
in G. Moreover, since Mv is a total dominating set in Hv for each v ∈ V (G), M is a
total dominating set in G ◦H. Now, let a, b ∈ M . If a, b ∈ Mu for some u ∈ V (G), then
NH(a) \NH(b) ̸= ∅ and NH(b) \NH(a) ̸= ∅. It follows that NG◦H(a) \NG◦H(b) ̸= ∅ and
NG◦H(b)\NG◦H(a) ̸= ∅. Thus, M is a J-open set in G◦H. Similarly, if a, b ∈ V (G), then
M is a J-open set in G ◦H. Assume that a ∈ Ms and b ∈ Mt for some s, t ∈ V (G), s ̸= t.
Then s ∈ NG◦H(a) \ NG◦H(b) and t ∈ NG◦H(b) \ NG◦H(a), hence we are done. Suppose
that a ∈ Mw for some w ∈ V (G) and b ∈ V (G). If w = b, then b ∈ NG◦H(a) \ NG◦H(b)
and a ∈ NG◦H(b) \ NG◦H(a), and we are done. Suppose w ̸= b. Since H is graph with
no isolated vertex, there exists q ∈ Hw such that q ∈ NG◦H(a) \ NG◦H(b). Clearly,
NG◦H(b) \NG◦H(a) = Mb ̸= ∅. Therefore, M is a J-open set in G ◦H, showing that M
is a J-total dominating set in G ◦H. Consequently,

γJt(G ◦H) ≥ |V (G)|+ γJt(H) · |V (G)|.

4. Conclusion

The concept of J-total domination has been introduced and investigated in this study.
Characterizations of J-total dominating sets in some graphs and join of two graphs are
formulated and were used to solve exact values of the parameters of these graphs. Some
bounds and relationships of this newly defined parameter have been established. Other
graphs that were not considered in this study could be an interesting topic to consider by
researchers for further investigation of the concept. They may also consider the bounds
of the parameter with respect to other well known parameters in graph theory.

Acknowledgements

The authors would like to thank Mindanao State University- Tawi-Tawi College of
Technology and Oceanography for funding this research. Moreover, the authors would
like to thank the referees for their invaluable comments and suggestions that contributed
a lot for the improvement of this paper.



REFERENCES 2117

References

[1] E.J. Cockayne and S.T. Hedetniemi. Towards a theory of domination in graphs.
Networks,, 7:247–261, 1977.

[2] B. Gayathri and S. Kaspar. Connected co-independent domination of a graph. Intl.
J. Contemp. Math. Sciences, 9(6):423–429, 2011.

[3] J. Hassan and S. Canoy Jr. Connected grundy hop dominating sequences in graphs.
Eur. J. Pure Appl. Math., 16(2):1212–1227, 2023.

[4] J. Hassan, S. Canoy Jr., and Chrisley Jade Saromines. Convex hop domination in
graphs. Eur. J. Pure Appl. Math., 16(1):319–335, 2023.

[5] J. Hassan, A. Lintasan, and N.H. Mohammad. Some properties and realization prob-
lems involving connected outer-hop independent hop domination in graphs. Eur. J.
Pure Appl. Math., 6(3):1848–1861, 2023.

[6] M. Henning and A. Yeo. Total domination in graphs. In springer monographs in
Mathematics; Springer New York, NY, USA,, 2013.

[7] S. Canoy Jr and J. Hassan. Weakly convex hop dominating sets in graphs. Eur. J.
Pure Appl. Math., 16(2):1196–1211, 2023.

[8] A. Klobucar. Total domination numbers of cartesian products. Math. Commun.,
1:35–44, 2014.

[9] J. Manditong, J. Hassan, L. Laja, A. Laja, N.H. Mohammad, and S. Kamdon. Con-
nected outer-hop independent dominating sets in graphs under some binary opera-
tions,. Eur. J. Pure Appl. Math., 16(3):1817–1829, 2023.

[10] O. Ore. Theory of graphs. Amer Math. Soc. Colloq. Publ., 38 (Amer. Math. Soc.,
Providence, RI), 1962.

[11] J. Sigarreta. Total domination on some graph operators. mathematics, 9(241):1–9,
2021.

[12] A. Sugumaran and E. Jayachandran. Domination number of some graphs. Intl. Jour.
of Scientific, Development and Research (IJSDR), 3(11):2455–2631, 2018.