EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 3, Article Number 6603
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Perfect 2-Distance Zero Forcing in Graphs

Javier A. Hassan1,2,∗, Erwan D. Hajim1, Angelica Mae L. Mahistrado3,
Akrimal M. Alayka
1Department of Mathematics, College of Arts and Sciences, MSU Tawi-Tawi College of
Technology and Oceanography, Bongao, Tawi-Tawi, Philippines
2Department of Mathematics, College of Science, Korea University, Seoul, South Korea
3Department of Mathematics, Ateneo de Davao University, Davao City, Philippines

Abstract. Let G be a graph. Then a 2-distance color change rule is defined as follows: If a
vertex x ∈ V (G) is colored and has exactly one hop neighbor y is uncolored, then y will become
colored. Moreover, let u, v, w ∈ V (G). If u 2-forces v and v 2-forces w, then we say that v and
w are perfectly 2-forced by u, and this process can extend to a chain of 2-forcing initiated by a
single vertex. In addition, a subset S of a vertex-set V (G) of G is called a perfect 2-distance zero
forcing set of G if there exists s ∈ S such that s perfectly 2-forces all other vertices outside S.
The minimum cardinality of a perfect 2-distance zero forcing set of G, denoted by Z2

p(G), is called
the perfect 2-distance zero forcing number of G. In this paper, this new parameter is introduced
and initially investigated on some classes of graphs and on the join of two graphs. A particular
variant of zero forcing called perfect co-zero forcing is defined to study the behavior of the perfect
2-distance zero forcing sets in the join of graphs. Characterizations of perfect 2-distance zero
forcing sets are formulated and subsequently used to obtain some formulas for solving the perfect
2-distance zero forcing numbers of the join of some graphs.

2020 Mathematics Subject Classifications: 05C69

Key Words and Phrases: Perfect 2-distance zero forcing set, perfect 2-distance zero forcing
number, perfect co-zero forcing

1. Introduction

The concept of zero forcing set was initially introduced in [1] as a bound for the mini-
mum rank problem. A zero forcing set in a graph is a subset of vertices with a particular
dynamic propagation property. The zero forcing process starts with a set of initially col-
ored vertices, typically with one color representing ”active” and another ”inactive” or

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6603

Email addresses:
javierhassan@msutawi-tawi.edu.ph (J. A. Hassan)
erwanhajim@msutawi-tawi.edu.ph (E. Hajim)
amlmahistrado@addu.edu.ph(A.M. Mahistrado)
akrimalalayka@msutawi-tawi.edu.ph (A. Alayka)

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)



J. A. Hassan et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6603 2 of 9

”unassigned.” Then, using certain propagation rules, the active vertices force neighboring
inactive vertices to become active. The goal is to determine the minimum size of a zero
forcing set required to force all vertices in the graph to become active. The concept of
zero forcing was further studied by many researchers, and these studies can be found in
[1–10].

In 2024, J. Hassan et al. [11], introduced another variant of zero forcing by changing
the distance to two (2) for a certain vertex to force another vertex in a graph. The said
parameter was investigated on some classes of graphs as well as examined its relationships
with other parameters such as standard zero forcing and hop domination.

In this paper, we introduce another variant of 2-distance zero forcing by adding a
certain property wherein the reinforcement is not allowed, and we call it a perfect 2-
distance zero forcing. That is, only one vertex in a set is needed to force all other vertices
outside the considered set, if any. This is far different compared to the standard 2-distance
zero forcing wherein the reinforcement is allowed. We believe, this new parameter and its
results would serve as reference to future researchers who will study on variants of zero
forcing, and would lead to an interesting topics of research in the future.

2. Terminology and Notation

Let G = (V (G), E(G)) be a simple and undirected graph. 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 vertex of a in
G is a hop neighbor of a vertex b in G if dG(a, b) = 2.

Let G be a graph and let x, y ∈ V (G). Then the 2-distance color change rule is if x is
colored (active) vertex and exactly one hop neighbor y of x is uncolored (inactive) , then
y will become colored (active). A 2-distance zero forcing set N of G is a subset of vertices
of G such that when the vertices in N are colored (active) and the remaining vertices are
uncolored(inactive) initially, repeated application of the 2-distance color change rule all
vertices of G will become colored (active). The minimum cardinality of a 2-distance zero
forcing set of G, denoted by Z2(G), is called the 2-distance zero forcing number of 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)}.

3. Results

We begin this section by introducing the concept of perfect 2-distance zero forcing in
a graph.

Definition 1. Let G be a graph. Then a 2-distance color change rule is defined as follows:
If a vertex x ∈ V (G) is colored and has exactly one hop neighbor y that is uncolored, then
y will become colored. In this case, we say that a vertex y is 2-forced by a vertex x in G.



J. A. Hassan et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6603 3 of 9

Moreover, let u, v, w ∈ V (G). If u 2-forces v and v 2-forces w, then we say that v and w
are perfectly 2-forced by u, and this process can extend to a chain of 2-forcing initiated
by a single vertex. In addition, a subset S of a vertex-set V (G) of G is called a perfect
2-distance zero forcing set of G if there exists s ∈ S such that s perfectly 2-forces all other
vertices outside S. The minimum cardinality of a perfect 2-distance zero forcing set of G,
denoted by Z2

p(G), is called the perfect 2-distance zero forcing number of G.

Example 1. Consider the graph P3 + P4 below. Let S = {v1, v2, v3, v5}. Then vertex v7
is 2-forced by vertex v5, vertex v4 is 2-forced by vertex v7, and v6 is 2-forced by vertex
v4. It follows that vertices v4, v6 and v7 are perfectly 2-forced by vertex v5. Therefore, S
is a perfect 2-distance zero forcing set of P3 + P4. It can easily be verified that a perfect
2-distance zero forcing number of P3 + P4 is 4, that is, Z2

p(P3 + P4) = 4.

v1

v4

v2

v5 v6 v7

v3

P3 + P4 :

Theorem 1. Let G be a graph. Then each of the following holds:

(i) Z2(G) ≤ Z2
P (G) ≤ |V (G)|.

(ii) Let G be a non-trivial graph. If G has a dominating vertex, then Z2
p(G) ≥ 2.

(iii) If every vertex of G is a dominating vertex, then Z2
P (G) = |V (G)|.

Proof. (i). Let G be a graph, and R be a minimum perfect 2-distance zero forc-
ing set of G. Then |R| = Z2

P (G) and R is a 2-distance zero forcing set of G. Thus,
Z2(G) ≤ |R| = Z2

P (G). The upper bound is clear since every perfect 2-distance zero
forcing set of G is always a subset of V (G).

(ii). Let x ∈ V (G) be a dominating vertex of G. Then dG(x, u) = 1 for all u ∈
V (G) \ {x}. Suppose that x /∈ S, where S is a minimum perfect 2-distance zero forcing
set of G. Then there must be a vertex w ∈ V (G) \ {x} such that dG(w, x) = 2, which is
a contradiction. Thus, x ∈ S. Since G is a non-trivial and x is a dominating vertex of G,
there exists y ∈ V (G) \ {x} such that y ∈ S. Therefore, Z2

P (G) ≥ 2.

(iii). Let V (G) = {v1, v2, · · · , vk}. Since v1 is a dominating vertex of G, dG(v1, vi) = 1
for all i ∈ {2, 3, · · · , k}. Applying the same argument in the proof of (ii), v1 ∈ Q, where
Q is a minimum perfect 2-distance zero forcing set of G. Now, since v2 ∈ V (G) \ {v1} is a
dominating vertex, v2 must be also in Q. Continuing in this manner, V (G) ⊆ Q, that is,
Q = V (G). Hence, Z2

p(G) = |V (G)|.



J. A. Hassan et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6603 4 of 9

Theorem 2. Let n be a positive integer.

Z2
p(Pn) =

{
n
2 + 1, if n is even
n+1
2 , if n is odd

Proof. Clearly, Z2
p(P1) = 1 and Z2

p(P2) = 2 = Z2
p(P3). Suppose that n ≥ 4 and

even, Let Pn = [v1, v2, ..., vn], and consider B = {v1, v2, v4, v6, . . . , vn}. Then vertices
v3, v5, . . . , vn−1 are perfectly 2-forced by v1. It follows that B is perfect 2-distance zero
forcing set of Pn. Notice that, if we remove vi from B for some i ∈ {2, 4, . . . , n}, then vi
will not be perfectly 2-forced by v1. Thus, B is a minimum perfect 2-distance zero forcing
set of Pn. Hence, Z

2
p(Pn) =

n
2 + 1 for all even integers n ≥ 2.

Now, assume that n ≥ 5 and odd. Let A = {v1, v2, v4, v6, . . . , vn−1}. Then vertices
v3, v5, . . . , vn are perfectly 2-forced by v1. Thus, A is a perfect 2-distance zero forcing set
of Pn. Moreover, if we remove vj from A for some j ∈ {2, 4, . . . , n−1}, then vj will not be
perfectly 2- forced by v1. This means that A is a minimum perfect 2-distance zero forcing
set of Pn. Hence, Z

2
p(Pn) =

n+1
2 for all odd integers n ≥ 5.

To study the behavior of perfect 2-distance zero forcing sets in the join of any two
graphs, the following concept shall be defined:

Definition 2. Let G be a graph. Then a co-color change rule is defined as follows: If a
vertex x ∈ V (G) is colored black and has exactly one non-neighbor y that is colored white,
then y will become black. In this case, we say that a vertex y is co-forced by a vertex x
in G. Moreover, let u, v, w ∈ V (G). If u co-forces v and v co-forces w, then we say that
v and w are perfectly co-forced by u, and this process can extend to a chain of co-forcing
initiated by a single vertex. In addition, a subset B of a vertex-set V (G) of G is called
a perfect co-zero forcing set of G if there exists u ∈ B such that u perfectly co-forces all
vertices outside B. The minimum cardinality of a perfect co-zero forcing set of G, denoted
by Zpco(G), is called the perfect co-zero forcing number of G.

Lemma 1. Let G be a non-trivial graph. If G has a dominating vertex, then Zpco(G) ≥ 2.

Proof. Let v ∈ V (G) be a dominating vertex of G, and let T be a minimum perfect
co-zero forcing set of G. Assume that v /∈ T . Then there must be a vertex w ∈ T \ {v}
such that w co-forces v, that is, dG(v, w) ≥ 2. Since v is a dominating vertex of G, it
follows that dG(v, u) = 1 for all u ∈ V (G)\{v}. Note that w ∈ V (G)\{v}, a contradiction.
Therefore, v ∈ T .

Moreover, since dG(v, u) = 1 for all u ∈ V (G) \ {v}, it follows that v cannot co-force
any vertex u ∈ V (G)\{v}. Thus, there must be another vertex t ∈ T such that t co-forces
u (possible t = u). Thus, T has at least two elements, that is, Zpco(G) ≥ 2.



J. A. Hassan et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6603 5 of 9

Theorem 3. Let m be a positive integer. Then

Zpco(Cm) =

{
3 , if m = 3, 4

m-3, if m ≥ 5.

Proof. Since C3 has dominating vertex, it follows that Zpco(C3) ≥ 2 by Lemma 1.
Assume that Zpco(C3) = 2, say, Q = {a, b} is a minimum perfect co-zero forcing set of
C3, where C3 = [a, b, c, a]. Then either vertex a or b must co-forced vertex c. That is,
dC3(a, c) ≥ 2 or dC3(b, c) ≥ 2. However, this is a contradiction to the fact that each pair
of vertices in C3 are adjacent. Therefore, Zpco(C3) = 2 is not possible. Since V (C3) is a
perfect co-zero forcing set of C3, it follows that Zpco(C3) = 3.

For m = 4, let C4 = [v1, v2, v3, v4, v1], and consider Q′ = {v1, v2, v3}. Then Q′ is a
perfect co-zero forcing set of C4. Hence, Zpco(C4) ≤ 3. Assume that Zpco(C4) = 2, say, a
minimum perfect co zero forcing set of C4 is R = {vi, vj}, where i, j ∈ {1, 2, 3, 4}. If vi and
vj are adjacent, then neither vi nor vj can co-force all the remaining vertices outside R,
a contradiction. If vi and vj are non-adjacent, then the remaining two vertices outside R
are both adjacent to vi and vj . That is, neither vi nor vj can co-force these two vertices.
Hence, Zpco(C4) = 2 is not possible. Since {v1, v2, v3} is a perfect co-zero forcing set of
C4, we have Zpco(C4) = 3.

Now, let n ≥ 5 and Cn = [x1, x2, · · · , xm, x1]. Consider X = {x1, x3, x4, · · · , xm−2}.
Then vertices xm−1, x2 and xm are perfectly co-forced by vertices x1. It follows that X is
a perfect co-zero forcing set of Cm. Thus, Zpco(Cm) ≤ m− 3 for all m ≥ 5. Assume that
Zpco(Cm) ≤ m− 4. Then there are at least four vertices xi, xj , xk, xl ∈ V (Cm) \N , where
N is a minimum perfect co-zero forcing set of Cm. Since the graph is cycle, at least two
vertices in {xi, xj , xk, xl} have distance of at least two to any vertex in N . That is, none
of the vertices in N can perfectly co-forces these vertices, which is a contradiction. In this
case, Zpco(Cm) ≤ m− 4 is not possible. Therefore, Zpco(Cm) = m− 3 for all m ≥ 5.

Theorem 4. Let n be a natural number. Then

Zpco(Pn) =


n , if n = 1, 2

2, if n = 3.

n-3, if n ≥ 4.

Proof. Clearly Zpco(P1) = 1 and Zpco(P2) = 2. For n = 3, let P3 = [a1, a2, a3]. Con-
sider S = {a1, a2}. Then a3 is co-forced by a vertex a1. Thus, S is a perfect co-zero
forcing set of P3, and so Zpco(P3) ≤ 2. Since a2 is a dominating vertex, if follows that
Zpco(P3) = 2 by Lemma 1.

Let n = 4, and suppose that P4 = [a1, a2, a3, a4]. Consider B = {a2}. Then a4, a1
and a3 are perfectly co-forced by vertex a2. Thus, B is a perfect co-zero forcing set of P4.



J. A. Hassan et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6603 6 of 9

Hence, Zpco(P4) = 1.

Now, for n ≥ 5, let Pn = [a1, a2, . . . , an]. Let B′ = {a2, a5, a6, . . . , an}. Then vertices
a4, a1 and a3 are perfectly co-forced by vertex a2. Thus, B

′ is a perfect co-zero forcing set
of Pn, and so Zpco(Pn) ≤ n− 3. Suppose that Zpco(Pn) ≤ n− 4. Then there exist at least
four vertices ai, aj , ak and al in V (G) such that ai, aj , ak, al /∈ Q, where Q is minimum
perfect co-zero forcing set of Pn. Since the graph is a path graph, at least two vertices in
{ai, aj , ak, al} have distance two to every vertex in Q. Thus, none of the vertices in Q can
perfectly co-forces vertices outside Q, a contradiction to the fact that Q is a perfect co-zero
forcing set of Pn. In this case, Zpco(Pn) ≤ n− 4 is not possible. Hence, Zpco(Pn) = n− 3
for all n ≥ 5.

We shall now characterize the perfect 2-distance zero forcing sets in the join of two
graphs as follows:

Theorem 5. Let G and H be any two graphs. Then P ⊆ V (G+H) is a perfect 2-distance
zero forcing if and only if one of the following conditions hold:

(i) P = PG ∪ V (H), where PG is a perfect co-zero forcing set of G.

(ii) P = V (G) ∪ PH such that PH is a perfect co-zero forcing set of H.

Proof. Suppose that P is a perfect 2-distance zero forcing set of G + H. Then
P = PG ∪ PH , where PG ⊆ V (G) and PH ⊆ V (H). If PG = ∅, then P = PH . However,
PH cannot 2-force any vertex in V (G), a contradiction. The same assertion follows when
we let PH = ∅. Thus, PG ̸= ∅ and PH ̸= ∅.

Now, if PG ⊂ V (G) and PH ⊂ V (H), that is, PG and PH are proper subsets of V (G)
and V (H), respectively. Then there exist x ∈ V (G) \ PG and y ∈ V (H) \ PH . Notice
that, vertex x can only be 2-forced by a vertex in PG ⊆ V (G) , and vertex y can only be
2-forced by a vertex in PH ⊆ V (H). Thus, none of the vertices in P can perfectly 2-forces
both x and y, a contradiction to our assumption that P is a perfect 2-distance zero forcing
set of G +H. Hence, the remaining cases are either P = PG ∪ V (H) or P = V (G) ∪ PH

where PG ⊆ V (G) and PH ⊆ V (H).

Assume that P = PG ∪ V (H). Suppose on the contrary that PG is not a perfect co-
zero forcing set of G. Then there exists y ∈ V (G) \ PG such that y cannot be perfectly
co-forced by any vertex in PG under the graph G. This means that y cannot be perfectly
2- forced by any vertex in PG under the graph G+H. However, this is a contradiction to
our assumption that P is a 2-distance zero forcing set of G +H. Hence, PG is a perfect
co-zero forcing set in G, and so (i) holds.



J. A. Hassan et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6603 7 of 9

Similarly, PH is a perfect co-zero forcing set in H when P = V (G) ∪ PH . That is, (ii)
holds.

Conversely, suppose that P = PG ∪ V (H), where PG is a perfect co-zero forcing set of
G. Then there exists x ∈ PG such that x perfectly co-forces other vertices outside PG in
G. This means that x can perfectly 2-forces other vertices outside P in G+H. It follows
that P = PG ∪V (H) is a perfect 2-distance zero forcing set of G+H. Similarly, the same
assertion follows when (ii) holds.

Theorem 6. Let G and H be any graphs. Then

Z2
p(G+H) = min{Zpco(G) + |V (H)|, |V (G)|+ Zpco(H)}.

Proof. Let P be a minimum perfect 2-distance zero forcing set of G + H. Then by
Theorem 4, either P = PG∪V (H) or P = V (G)∪PH where PG and PH are perfect co-zero
forcing sets of G and H, respectively. Hence, either

Z2
p(G+H) = |P | ≥ Zpco(G) + |V (H)| or Z2

p(G+H) = |P | ≥ |V (G)|+ Zpco(H).

Now, suppose that either P = PG ∪ V (H) or P = V (G) ∪ PH , where PG and PH

are minimum perfect co-zero forcing sets of G and H, respectively. Then P is a perfect
2-distance zero forcing set of G+H by Theorem 4. Thus, either

Z2
p(G+H) ≤ |P | = Zpco(G) + |V (H)| or Z2

p(G+H) ≤ |P | = Zpco(H) + |V (G)|.

Hence,

Z2
p(G+H) = Zpco(G) + |V (H)| or Z2

p(G+H) = Zpco(H) + |V (G)|.

Consequently,

Z2
p(G+H) = min{Zpco(G) + |V (H)|, |V (G)|+ Zpco(H)}.

Corollary 1. Let n and m be a natural numbers. Then each of the following holds:

(i) Z2
p(Sn) = Z2

p(K1 + K̄n) =

{
n+ 1 , if n = 1

n, if n ≥ 2.

(ii) Z2
p(Wn) = Z2

p(K1 + Cn) =

{
4 , if n = 3, 4

n− 2, if n ≥ 5.



J. A. Hassan et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6603 8 of 9

(iii) Z2
p(Fn) = Z2

p(K1 + Pn) =


n+ 1 , if n = 1, 2

3, if n = 3.

n− 2, if n ≥ 4.

(iv) Z2
p(Km,n) = Z2

p(K̄m + K̄n) =


2 , if m = n = 1

m+ n− 1, if 2 ≤ m ≤ n or 2 ≤ n ≤ m.

n, if m = 1 and n ≥ 2.

4. Conclusion

Perfect 2-distance zero forcing, a new variant of zero forcing has been introduced and
studied in this paper. It is observed that every graph admits perfect 2-distance zero forcing.
It is shown that the parameter for a perfect 2-distance zero forcing is always greater than
the parameter for standard 2-distance zero forcing on any simple and undirected graph.
Moreover, a certain variant of zero forcing called perfect co-zero forcing was defined to
solve the perfect 2-distance zero forcing number of the join of any two graphs.

Acknowledgements

The author would like to thank Mindanao State University - Tawi-Tawi College of
Technology and Oceanography, Korea University, and Ateneo de Davao University for
funding this research.

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