EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 17, No. 1, 2024, 124-134
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

J2-Independence Parameters of Some Graphs

Javier A. Hassan1, Aziz B. Tapeing1,∗, Hounam B. Copel1, Alcyn Bakkang2,
Sharifa Dianne A. Aming3

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
3 Office of the Chancellor, MSU Tawi-Tawi College of Technology and Oceanography,
Bongao, Tawi-Tawi, Philippines

Abstract. Let G be a graph. A subset I ′ of a vertex-set V (G) of G is called a J2-independent in G
if for every pair of distinct vertices a, b ∈ I ′, dG(a, b) ̸= 1, N2

G[a]\N2
G[b] ̸= ∅ and N2

G[b]\N2
G[a] ̸= ∅.

The maximum cardinality among all J2-independent sets in G, denoted by αJ2(G), is called the
J2-independence number of G. Any J2-independent set I ′ satisfying |I ′| = αJ2(G) is called the
maximum J2-independent set of G or an αJ2-set of G. In this paper, we establish some bounds
of this parameter on a generalized graph, join and corona of two graphs. We characterize J2-
independent sets in some families of graphs, and we use these results to derive the exact values of
parameters of these graphs. Moreover, we investigate the connections of this new parameter with
other variants of independence parameters. In fact, we show that the J2-independence number of
a graph is always less than or equal to the standard independence number.

2020 Mathematics Subject Classifications: 05C69

Key Words and Phrases: independent set, J2-independent set, J2-independence number

1. Introduction

The independent set in graph has been studied excessively and one of the topics in
Graph Theory which has been growing rapidly. Moreover, the problem of finding the
maximum independent set in graphs is a fundamental problem not just in Graph Theory
but also in Theoretical Computer Science. A subset V ′ of the vertex-set V (G) of a graph
G is said to be an independent if no two vertices in V ′ are adjacent. An independent set is

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v17i1.4946

Email addresses: javierhassan@msutawi-tawi.edu.ph (J. Hassan)
aziztapeing@msutawi-tawi.edu.ph (A. Tapeing),
hounamcopel@msutawi-tawi.edu.ph (H. Copel)
alcynbakkang@msutawi-tawi.edu.ph (A. Bakkang),
sharifadianneaming@msutawi-tawi.edu.ph (S. Aming)

https://www.ejpam.com 124 © 2024 EJPAM All rights reserved.



A. Tapeing et al. / Eur. J. Pure Appl. Math, 17 (1) (2024), 124-134 125

called maximum if it is of largest cardinality, that is, if V ′ ∪{v} is not an independent set
for any v ∈ V (G)\V ′, and it is denoted by i(G) to be the number of maximal independent
sets of G.

In 1992, Jiuqiang Liu[8] developed new properties for the number of maximal indepen-
dent sets i(G) and the number of maximum independent sets im(G), as well as determine
the largest number of maximal and maximum independent sets possible in a k-connected
graph of order n(with n large) and characterize the respective extremal graphs. In [1],
established an upper bound as a tool to prove that the disjoint union of complete bipar-
tite graphs Kd,d maximises the number of independent sets of a d-regular graph. Some
variants of representing the independent sets in graphs were studied by some researchers
(see[1–4, 6, 7, 9–11]).

In 2022, J. Hassan et al. [6] introduced the hop independent sets in graphs. A subset
S of V (G) is called a hop independent if for every pair of distinct vertices x, y ∈ S,
dG(x, y) ̸= 2. The maximum cardinality of a hop independent set in G, denoted by
αh(G), is called the hop independence number of G. Any hop independent set S with
cardinality equal to αh(G) is called an αh-set of G. They have shown that every maximum
hop independent set in a graph is a hop dominating set, that is, the hop independence
number of a graph G is always greater than or equal to the hop domination number of a
graph. They have characterized this type of set in graphs under some binary operations
such join, corona, lexicographic product and Cartesian product of two graphs. These
characterizations had been used to derive some formulas of a hop independence numbers
of these graphs.

Recently, J. Hassan et al. [5] introduced and investigated new concept called J2-hop
domination. A subset T = {v1, v2, · · · , vm} of vertices of a graph G is called a J2-set if
N2

G[vi]\N2
G[vj ] ̸= ∅ for every i ̸= j, where i, j ∈ {1, 2, . . . ,m}. A J2-set T is called a J2-hop

dominating in G if for every a ∈ V (G) \ T , there exists b ∈ T such that dG(a, b) = 2. The
J2-hop domination number of G, denoted by γJ2h(G), is the maximum cardinality among
all J2-hop dominating sets in G. They have shown that every maximum hop independent
set is a J2-hop dominating, hence, this parameter is always greater or equal compare to
the hop independence parameter on any graph. Moreover, they derived some lower and
upper bounds of the parameter for a generalized graph, join and corona of two graphs,
respectively.

In this paper, we initiate the study of new variant of independence called
J2-independence. A certain subset S of a vertex-set V (G) of G is called a J2-independent
if S is both a J2-set and an independent set of a graph G. We investigate its proper-
ties and its relationships with other variants of independence. Further, we characterize
J2-independent sets in some classes of graphs and we use these results to determine the
J2-independence numbers of these graphs. Furthermore, we present some lower bounds
of the parameter on the join and corona of two graphs.

We believe that the results of this study would give additional insights to researchers
in the field and would help them for more research directions in the future.



A. Tapeing et al. / Eur. J. Pure Appl. Math, 17 (1) (2024), 124-134 126

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 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 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 Cn = [x1, x2, . . . , xn, x1] is the graph of
order n ≥ 3 with vertex-set {x1, x2, . . . , xn} and edge-set {x1x2, x2x3, . . . , xn−1xn, xnx1}.

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)}.

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⟩.

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 (resp. hop independent) if for every pair
of distinct vertices x, y ∈ I, dG(x, y) ̸= 1 (resp. dG(x, y) ̸= 2). The maximum cardinality
of an independent set (resp. hop independent set) in G, denoted by α(G)(resp. αh(G)), is
called the independence (resp. hop independence) number of G. Any independent (resp.
hop independent) set I with cardinality equal to α(G) (resp. αh) is called an α-set (resp.
αh-set) of G.

3. Results

We begin this section by introducing the concept of J2-independence in a graph.

Definition 1. Let G be a simple graph. A subset I ′ of V (G) is called a J2-independent



A. Tapeing et al. / Eur. J. Pure Appl. Math, 17 (1) (2024), 124-134 127

in G if for every pair of distinct vertices a, b ∈ I ′, dG(a, b) ̸= 1, N2
G[a]\N2

G[b] ̸= ∅ and
N2

G[b]\N2
G[a] ̸= ∅. The maximum cardinality among all J2-independent sets in G, de-

noted by αJ2(G), is called the J2-independence number of G. Any J2-independent set I ′

satisfying |I ′| = αJ2(G) is called the maximum J2-independent set of G or an αJ2-set of
G.

Example 1. Consider the graph K in Figure 1. Let I = {a, d, g, h}. Clearly I is a
maximum independent set of K. Notice that N2

K [a] = {a, d, e}, N2
K [d] = {a, d, c, f, g},

N2
K [g] = {c, d, g, h}, and N2

K [h] = {e, g, h}. Thus, N2
K [a]\N2

K [d] = {e},
N2

K [a]\N2
K [g] = {a, e}, N2

K [a]\N2
K [h] = {a, d}, N2

K [d]\N2
K [a] = {c, f, g},

N2
K [d]\N2

K [g] = {a, f}, N2
K [d]\N2

K [h] = {a, c, d, f}, N2
K [g]\N2

K [a] = {c, g, h},
N2

K [g]\N2
K [d] = {h}, N2

K [g]\N2
K [h] = {c, d}, N2

K [h]\N2
K [a] = {g, h},

N2
K [h]\N2

K [d] = {e, h}, N2
K [h]\N2

K [g] = {e}. Therefore, I is a maximum J2-independent
set of K, and so αJ2(K) = 4.

b d

e

f

g

h

ca

K :

Figure 1: Graph K with αJ2(K) = 4

Remark 1. Let G be a Graph. Then

(i) any singleton set {x}, where x ∈ V (G), is a J2-indeppendent set of G; and

(ii) an independent set I may not be a J2-independent in G.

Proposition 1. Let G be a graph. Then

(i) αJ2(G) ≤ α(G); and

(ii) 1 ≤ αJ2(G) ≤ |V (G)| .

Proof. (i) Let G be a graph and let I be a maximum J2-independent set of G. Then
I is an independent set in G. Since αG is the maximum cardinality of an independent set
in G, it follows that αJ2(G) = |I| ≤ α(G).

(ii) Since every singleton set {x}, where x ∈ V (G), is a J2-independent, we have
αJ2(G) ≥ 1. Morever, since any J2-independent set I of G is always a subset of V (G), it
follows that αJ2(G) ≤ |V (G)| . Therefore, 1 ≤ αJ2(G) ≤ |V (G)| .



A. Tapeing et al. / Eur. J. Pure Appl. Math, 17 (1) (2024), 124-134 128

Remark 2. Let G be a graph. Then the difference α(G)−αJ2(G) can be arbitrarily large.

To see this, let m be any positive integer and consider the graph G in Figure 2. Let
I = {v1, v2, . . . , vm+1} and I ′ = {u}. Then I is a maximum independent set of G. Hence,
α(G) = m+ 1. Now, clearly I ′ is a J2-independent set of G. Since dG(u, vi) = 1 for each
i ∈ {1, 2, . . . ,m+1}, and N2

G[vs] = N2
G[vt] ∀ s ̸= t, where s, t ∈ {1, 2, . . . ,m+1}, it follows

that I ′ is a maximum J2-independent set of G. Consequently,

α(G)− αJ2(G) = m+ 1− 1 = m.

Since m can be made arbitrarily large, the assertion follows.

v2

v3v4

v1

vm+1

v6

v5
u

G :

. . .

Figure 2: Graph G with α(G)− αJ2(G) = m

Theorem 1. Let G be a graph. Then αJ2(G) = |V (G)| if and only if every component of
G is trivial.

Proof. Suppose that αJ2(G) = |V (G)| , say that I = V (G) is the maximum J2-
independent set of G. Since I is an independent set of G, da(a, b) ̸= 1 ∀ a, b ∈ V (G). Sup-
pose there is a component K of G which is non-trivial. Then there exist
x, y ∈ V (K) ⊆ V (G) such that dK(x, y) = dG(x, y) = 1, a contradiction. Hence,
every component of G is trivial.

Conversely, suppose that every componentK ofG is trivial. Let V (G) = {a1, a2, . . . , am},
m ∈ N. Then dG(ai, aj) ̸= 1 and ai ∈ N2

G[ai]\N2
G[aj ] ∀ i ̸= j, where i, j ∈ {1, 2, . . . ,m}.

Thus, N2
G[ai]\N2

G[aj ] ̸= ∅ ∀ i ̸= j, i, j ∈ {1, 2, . . . ,m}. Therefore, V (G) is a J2− indepen-
dent set of G, and so αJ2(G) = |V (G)| .

Theorem 2. Let G be a graph. If G is complete, then αJ2(G) = 1. However, the converse
is not true.



A. Tapeing et al. / Eur. J. Pure Appl. Math, 17 (1) (2024), 124-134 129

Proof. Let G be a complete graph. Then α(G) = 1. Hence αJ2(G) = 1 by Proposition
1. To see that the converse is not true, consider P3 which is not complete grph. Let
V (P3) = {u1, u2, u3}. Observe that N2

P3
[u1] = N2

P3
[u3]. Thus, u1 and u3 cannot be both in

any J2-independent set I of G. Since dP3(u1, u2) = 1 = dP3(u2, u3), either {u1}, {u2} or
{u3} is a maximum J2-independent set of P3. Therefore, in either case, αJ2(P3) = 1, and
so the assertion follows.

Theorem 3. Let G be a graph. Then αJ2(G) = α(G) if and only if G has an α-set Q
such that Q forms a J2-set in G.

Proof. Suppose that αJ2(G) = α(G) = k, say Q = {w1, w2, . . . , wk} is a maximum
J2-independent set of G. Then Q is an independent set of G. Since αJ2(G) = α(G), it
follows that Q is an α-set of G. Since Q is a J2-independent set of G, Q is a J2-set of G.

Conversely, suppose G has an α-set Q of G. Then Q is a maximum independent set
of G. Since Q forms a J2-set in G, it follows that Q is a maximum J2-independent set of
G. Hence, α(G) = |Q| = αJ2(G).

Theorem 4. Let q be a positive integer. Then αJ2(Cq) =


1, q = 3, 4

2, q = 5, 6

α(Cq), q ≥ 7.

Proof. Clearly, αJ2(C3) = 1. For q = 4, let V (C4) = {a1, a2, a3, a4} and
L = {a1}. Then, L is a J2-independent set of Cq. Since dC4(a1, a2) = 1 = dC4(a1, a4)
and N2

C4
[a1] = N2

C4
[a3], it follows that L = {a1} is a maximum J2-independent set of C4.

Thus, αJ2(C4) = 1. For q = 5, let V (C5) = {a1, a2, a3, a4, a5}. Consider N = {a1, a3}.
Then N is a maximum independent set of C5. Note that N2

C5
[a1] = {a1, a3, a4} and

N2
C3
[a3] = {a1, a3, a5}. Thus, N2

C5
[a1]\N2

C5
[a3] = {a4} ≠ ∅ and N2

C3
[a3]\N2

C5
[a1] = {a5} ≠

∅. Hence, N is a maximum J2-independent set in C5, and so αJ2(C5) = 2. Similarly,
αJ2(C6) = 2. Suppose that q ≥ 7. Let V (Cq) = {v1, v2, . . . , vq}, and consider the following
two cases:

Case 1. q is odd
Let Q = {v1, v3, . . . , vn−4, vn−2}. Then Q is a maximum independent set of Cq, and

so α(Cq) = |Q| . Observe that vn−1 ∈ N2
Cq
[v1]\N2

Cq
[vj ] ∀ j ̸= 1, vr−2 ∈ N2

Cq
[vr]\N2

Cq
[vq]

∀ r < q, where r, q ∈ {3, 5, . . . , n − 2}, vs+2 ∈ N2
Cq
[vs]\N2

Cq
[vt] ∀ s < t, where

s, t ∈ {3, 5, . . . , n − 2}. Thus, N2
Cq
[vi]\N2

Cq
[vj ] ̸= ∅ ∀ i ̸= j, where

i, j ∈ {1, 3, . . . , n − 4, n − 2}, showing that Q is a J2-set in Cq. Hence, Q is a maxi-
mum J2-independent set of Cq, and so αJ2(Cq) = |Q| = α(Cq).

Case 2. q is even
Let R = {v1, v3, ..., vn−3, vn−1}. R is a maximum independent set of Cq, and so

α(Cq) = |R| .Notice that vn−1 ∈ N2
Cq
[v1]\N2

Cq
[vi] ∀ i ̸= n−3, n−1, v1 ∈ N2

Cq
[v1]\N2

Cq
[vn−3],

v3 ∈ N2
Cq
[v1]\N2

Cq
[vn−1], vj−2 ∈ N2

Cq
[vj ]\N2

Cq
[vi]. ∀ i > j, i ̸= n− 1 vt+2 ∈ N2

Cq
[vt]\N2

Cq
[vs]

∀ s < t, s ̸= 1, t ̸= n−1, vs ∈ N2
Cq
[vs]\N2

Cq
[vn−1] ∀ s ̸= n−3, vn−5 ∈ N2

Cq
[vn−3]\N2

Cq
[vn−1],



A. Tapeing et al. / Eur. J. Pure Appl. Math, 17 (1) (2024), 124-134 130

vn−1 ∈ N2
Cq
[vn−1]\N2

Cq
[vm] ∀ m ̸= 1, n − 3, vn−3 ∈ N2

Cq
[vn−1]\N2

Cq
[v1] and

v1 ∈ N2
Cq
[vn−1]\N2

Cq
[vn−3]. Thus, N

2
Cq
[vi]\N2

Cq
[vj ] ∀ i ̸= j, i, j ∈ {1, 3, . . . , n − 3, n − 1}

and so R is a J2-set in Cq. Consequently, αJ2(Cq) = |R| = α(Cq) for all n ≥ 7.

Theorem 5. Let m and n be positive integers. Then αJ2(Km,n) = 1.

Proof. Let V (Km,n) = {u1, u2, . . . , um, v1, v2, . . . , vn}, where V (Km) = {u1, u2, . . . , um}
and V (Kn) = {v1, v2, . . . , vn}. Consider M = {u1}. Then M is a J2-independent set
of Km,n. Observe that N2

Km,n
[ui] = N2

Km,n
[uj ] ∀ i ̸= j, where i, j ∈ {1, 2, . . . ,m} and

N2
Km,n

[vs] = N2
Km,n

[vt] ∀ i ̸= j, i, j ∈ {1, 2, . . . , n}. Since ur and vq are adjacent for all

r ∈ {1, 2, . . . ,m} and q ∈ {1, 2, . . . , n}, it follows that M is a maximum J2-independent
set of Km,n. Therefore, αJ2(Km,n) = 1 ∀ m,n ≥ 1.

Theorem 6. Let S and T be two connected graphs. A subset L of vertices of S + T is a
J2-independent set of S + T if one of the following holds;

(i) L is a J2-independent set in S

(ii) L is a J2-independent set in T

Proof. Suppose that L is a J2-independent set in S. Then L is an independent set
in S. Let a, b ∈ L. Then dS(a, b) ̸= 1. If dS(a, b) = 2, then dS+T (a, b) = 2 ̸= 1, and
we are done. If ds(a, b) ≥ 3, then dS+T (a, b) = 2 ̸= 1. Therefore, L is an independent
set of S + T. It suffices to show that L is a J2-set in S + T. Let x, y ∈ L. Since L is a
J2-independent set is S, it follows N2

S [x]\N2
S [y] ̸= ∅ and N2

S [y]\N2
S [x] ̸= ∅. Assume that

ds(x, y) = 2. Since L is a J2-independent set in S, there exist w, z ∈ V (S) such that
w ∈ N2

S [x]\N2
S [y] and z ∈ N2

S [y]\N2
S [x]. Let s ∈ NS(w) ∩ NS(x) and t ∈ NS(z) ∩ NS(y).

Then s ∈ N2
S+T [y]\N2

S+T [x] and t ∈ N2
S+T [x]\N2

S+T [y]. Thus, L is a J2- set in S + T.
Next, suppose that dS(x, y) ≥ 3. Let u ∈ NS(x) and v ∈ NS(y), then

u ∈ N2
S+T [y]\N2

S+T [x] and v ∈ N2
S+T [x]\N2

S+T [y]. Hence, L is a J2-set in S + T, showing
that L is a J2-independent set in S + T. Similarly, if L is a J2-independent set in T, then
L is a J2-independent set in S + T.

Corollary 1. Let S and T be two connected graphs. Then

αJ2(S + T ) ≥ max {αJ2(S), αJ2(T )}.

Proof. Let L be a maximum J2-independent set of S. Then by Theorem 6, L is a
J2-independent set of S + T. Since αJ2(S + T ) is the maximum cardinality among all
J2-independent sets of S + T, it follows that

αJ2(S + T ) ≥ |L| = αJ2(S).

Similarly, if L′ is a maximum J2-independent set of T , then

αJ2(S + T ) ≥
∣∣L′∣∣ = αJ2(T ).



A. Tapeing et al. / Eur. J. Pure Appl. Math, 17 (1) (2024), 124-134 131

Consequently,
αJ2(S + T ) ≥ max {αJ2(S), αJ2(T )}.

Remark 3. The Theorem 6 does not hold if either S or T is disconnected.

Consider the graph K3 + P3 in Figure 3, where S = K3 is disconnected and T = P3.
Let S′ = {d, e}. Then dS(d, e) ̸= 1, d ∈ N2

S [d] \NS [e] and e ∈ N2
S [e] \NS [d]. Thus, S′ is

a J2-independent set in S. However, N2
S+T [d] = N2

S+T [e] = {d, e, f}. Hence, S′ is not a
J2-set in S + T . Consequently, S′ is not a J2-independent set in S + T .

a b c

fed

G1 :

Figure 3: Graph K3 + P3

Theorem 7. Let S and T be connected graphs. If W =
⋃

a∈V (S)

Ta, where Ta is a J2-

independent set of T for each a ∈ V (S), then W is a J2-independent set of S ◦T. Moreover,

αJ2(S ◦ T ) ≥ αJ2(T ) · |V (S)| .

Proof. Let W =
⋃

a∈V (S)

Ta, where Ta is a J2-independent set of T for each a ∈ V (S).

Let x, y ∈ w. If x, y ∈ Tc for some c ∈ V (S), then dS◦T (x, y) ̸= 1 because Tc is an
independent set of T .

Claim: N2
S◦T [x]\N2

S◦T [y] ̸= ∅ and N2
S◦T [y]\N2

S◦T [x] ̸= ∅.
Since N2

T [x]\N2
T [y] ̸= ∅, there exists w ∈ V (T ) such that dT (x,w) = 2 and

dT (y, w) ̸= 2. If dT (x, y) = 2,, then dT (y, w) ̸= 1. Thus, dT (y, w) ≥ 3. Let
t ∈ NT (x) ∩NT (w). Then t ∈ N2

S◦T [y]\N2
S◦T [x]. Hence, N

2
S◦T [y]\N2

S◦T [x] ̸= ∅.
Assume that dT (x, y) ≥ 3. Suppose that dT (y, w) = 1. Let v ∈ NT (x) ∩NT (w). Then

v ∈ N2
S◦T [y]\N2

S◦T [x]. Thus, N
2
S◦T [y]\N2

S◦T [x] ̸= ∅. If dT (y, w) ≥ 3, then by preceding argu-
ment, N2

S◦T [y]\N2
S◦T [x] ̸= ∅. Similarly, if N2

T [y]\N2
T [x] ̸= ∅, then



A. Tapeing et al. / Eur. J. Pure Appl. Math, 17 (1) (2024), 124-134 132

N2
S◦T [x]\N2

S◦T [y] ̸= ∅. Therefore, W is a J2-independent set of S ◦ T . Consequently,

αJ2(S ◦ T ) ≥ αJ2(T ) · |V (S)| .

Remark 4. The Theorem 7 does not hold if T is disconnected.

Consider the graph S ◦ T in Figure 4, where T is disconnected. Let B = {u1, u3}.
Then dT (u1, u3) ̸= 1. Hence, B is an independent set of T . Observe that N2

T [u1] = {u1}
and N2

T [u3] = {u3}. Thus, N2
T [u1]\N2

T [u3] = {u1} ̸= ∅ and N2
T [u3]\N2

T [u1] = {u3} ̸= ∅.
Therefore, B is a J2-independent set of T . Now, notice that

N2
T+S [u1] = {u1, u3, y} ⊆ {u1, u2, u3, y} = N2

T+S [u3].

It follows that B is not a J2-set of S + T . Consequently, B is not a J2-independent set of
S + T .

S : T :

x y z

v1

v2
v3 v4

v5
v6 v7

v8
v9

S ◦ T :

Figure 4: Graph S ◦ T

Theorem 8. Let G be a graph. Then the hop independence and J2-independence param-
eters are incomparable.

Proof. Consider the graph G in Figure 5. Let Q = {a, e}, Then Q is an independent set
of G. Observe that N2

G[a] = {a, h} and N2
G[e] = {d, e, g}. Thus, N2

G[a]\N2
G[e] = {a, h} ≠ ∅

and N2
G[e]\N2

G[a] = {d, e, g} ≠ ∅ and so Q is a J2 independent set of G. Since, dG(a, b) =
dG(a, d) = dG(a, c) = 1, N2

G[a] ⊆ N2
G[h], dG(e, f) = 1 = dG(e, h) and N2

G[e] = N2
G[g], it

follows that Q is a maximum J2-independent set of G. Hence, αJ2(G) = 2. Now, let
Q′ = {a, b, c, d}. Then Q′ is a maximum hop independent set of G. Therefore, αh(G) = 4.



A. Tapeing et al. / Eur. J. Pure Appl. Math, 17 (1) (2024), 124-134 133

a b e f

g

hdc

G :

Figure 5: Graph G with αh(G) = 4 and α2
J(G) = 2

Next consider the graph K2 + P13 in Figure 6. Let R = {a, d, f, h, j,m}. Then, R is a
maximum J2-independent set of K2 + P13, and so α2

J(K2 + P13) = 6.
Now, let R′ = {a, b, x, y}. Then, R′ is a maximum hop independent set K2 + P13.

Hence, αh(K2 + P13) = 4.

a b c d e f g h i j k l m

K2 + P13 :

x y

Figure 6: Graph K2 + P13 with αh(K2 + P13) = 4 and α2
J(K2 + P13) = 6

4. Conclusion

The concept of J2-independence 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 J2-independence. Moreover, characterizations of
J2-independent sets in some classes of graphs have been presented and used to determine
the exact values of the parameter. Some graphs that were not considered in this study
could be an interesting topic to consider for further investigation of the concept.



REFERENCES 134

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 led to the
improvement of the paper.

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