EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 17, No. 2, 2024, 1283-1293
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

2-Distance Zero Forcing Sets in Graphs

Javier A. Hassan∗, Lestlene T. Udtohan, Ladznar S. Laja

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

Abstract. In this paper, we introduce new concept in graph theory called 2-distance zero forc-
ing. We give some properties of this new parameter and investigate its connections with other
parameters such as zero forcing and hop domination. We show that 2-distance zero forcing and
hop domination(respectively, zero forcing parameter) are incomparable. Moreover, we characterize
2-distance zero forcing sets in some special graphs, and finally derive the exact values or bounds
of the parameter using these results.

2020 Mathematics Subject Classifications: 05C69

Key Words and Phrases: Zero forcing, 2-distance zero forcing, 2-distance zero forcing number

1. Introduction

Zero forcing is a propagation process in a graph that increases the number of blue
vertices given on initial set of blue vertices, with all other vertices white, and a color-
change rule. The color-change rule states that a blue vertex adjacent to a single white
neighbor can force its neighbor to blue. Formally, if u is a blue vertex and w is the only
white vertex in N(u), then u−→w will be used to denote that u forces w to blue. Given a
graph G, a zero forcing set B of G is a subset of vertices of V (G) such that B is initially
colored blue, and the remaining vertices in G are white, then iteratively applying applying
the color-change rule given B results in every vertex in G becoming blue. Zero forcing sets
have applications in control theory, network coding, and determining structural properties
of graphs. Some studies related to zero forcing sets and its variants can be found in [1–6].

Recently, J. Manditong et al.[16], introduced new variant of zero forcing in a graph
called zero forcing hop domination. They have established some properties of this param-
eter and determined its connections with other known parameters in graph theory. More-
over, they have obtained some exact values or bounds of the parameter on the generalized
graph, some families of graphs, and graphs under some operations via characterizations.
Some interesting studies related to zero forcing hop domination can be found in [7–15].

DOI: https://doi.org/10.29020/nybg.ejpam.v17i2.5046

Email addresses: javierhassan@msutawi-tawi.edu.ph (J. A. Hassan)
lestleneudtohan@msutawi-tawi.edu.ph (L. T.Udtohan)
ladznarlaja@msutawi-tawi.edu.ph (L.S. Laja)

https://www.ejpam.com 1283 © 2024 EJPAM All rights reserved.



J. A. Hassan, L. T. Udtohan, L. S. Laja / Eur. J. Pure Appl. Math, 17 (2) (2024), 1283-1293 1284

In this paper, we introduce the concept of 2-distance zero forcing sets in a graph. 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.
We study its connections with the standard zero forcing and hop domination parameter,
respectively. Moreover, we investigate this parameter on some families of graphs such as
complete, path, cycle, star, and complete bipartite graph. 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

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

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.

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 inG is the setNG[x] = NG(x)∪{x}. IfX ⊆ 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 vertex of a in G is a hop neighbor of a vertex b in G if dG(a, b) = 2. The set

N2
G(a) = {b ∈ V (G) : dG(a, b) = 2} is called the open hop neighborhood of a. The closed

hop neighborhood of a in G is given by N2
G[a] = N2

G(a) ∪ {a}. The open hop neighborhood

of S ⊆ V (G) is the set N2
G(S) =

⋃
a∈S

N2
G(a). The closed hop neighborhood of S in G is the

set N2
G[S] = N2

G(S) ∪ S.
A subset S of V (G) is a hop dominating of G if for every a ∈ V (G)\S, there exists

b ∈ S such that dG(a, b) = 2. The minimum cardinality among all hop dominating sets of
G, denoted by γh(G), is called the hop domination 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)}.



J. A. Hassan, L. T. Udtohan, L. S. Laja / Eur. J. Pure Appl. Math, 17 (2) (2024), 1283-1293 1285

The color change rule states that a blue vertex adjacent to a single white neighbor can
force its neighbor to blue. Formally, if u is a blue vertex and w is the only white vertex in
NG(u), then u−→w will be used to denote that u forces w blue. A zero forcing set for a
graph G is a subset of vertices in Z such that if initially the vertices in Z are colored blue
and the remaining vertices are colored white, the entire graph G May be colored blue by
repeatedly applying the color-change rule. Furthermore, the zero forcing number, Z(G), of
a graph G is the minimum cardinality of a set of blue vertices(whereas vertices in V (G)\S
are colored white) such that V (G) is turned blue after finitely many applications of ”the
color change rule”: a white vertex is converted to a blue vertices if it is the only white
neighbor of a blue vertex.

3. Results

We begin this section by introducing the concepts of 2-distance zero forcing set and
2-distance zero forcing number of a graph.

Definition 1. LetG 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). Formally, if x is a colored(active) vertex and y is the
only uncolored (inactive) vertex in N2

G(x), then x−→y will be used to denote that x 2-
forces y to be 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.

Example 1. Consider the graph G in Figure 1 and let N = {a, b, d}. Then vertex c is
2-forced by vertex a and vertex e is 2-forced by either vertex d or b. Thus, N is a 2-distance
zero forcing set of G. Moreover, Z2(G) = 3.

a

b

d

c e

G :

Figure 1: Graph G with Z2(G) = 3

Proposition 1. Let n be a positive integer. Then S is a 2-distance zero forcing set of Kn

if and only if S = V (Kn).

Proof. Let S be a 2-distance zero forcing set of Kn. Suppose that S ̸= V (Kn). Then
there exists x ∈ V (Kn) such that x /∈ S. However, dKn(x, y) = 1 for all y ∈ S. It follows



J. A. Hassan, L. T. Udtohan, L. S. Laja / Eur. J. Pure Appl. Math, 17 (2) (2024), 1283-1293 1286

that S cannot 2-forced x, a contradiction. Therefore, S = V (Kn).

The converse is clear.

Corollary 1. Let n be a positive integer. Then Z2(Kn) = n.

Theorem 1. Z2(G) = |V (G)| if and only if diam(H) ≤ 1 for each component H of G.

Proof. Suppose that Z2(G) = |V (G)|. Suppose further that diam(H) ≥ 2 for some
component H of G. Then there exist a, b ∈ V (H) such that dH(a, b) = 2 = dG(a, b). Let
N = V (G) \ {b}. Then N is a 2-distance zero forcing set of G. Thus, Z2(G) ≤ |V (G)|− 1,
a contradiction. Therefore, diam(H) ≤ 1 for each component H of G.

Conversely, suppose that diam(H) ≤ 1 for each component H of G. If G is connected,
then G = Kn. Thus, Z

2(G) = |V (G)| = n by Corollary 1. Suppose that G is disconnected.
Let H1, . . . ,Hk, k ≥ 2 be components of G. Since diam(Hi) ≤ 1, Z2(Hi) = |V (Hi)| for
each i ∈ {1, . . . , k}. Thus,

Z2(G) = Z2(H1) + · · ·+ Z2(Hk) = |V (H1)|+ · · ·+ |V (Hk)| = |V (G)|.

Corollary 2. Let n be a positive integer. Then, Z2(Kn) = n.

Proposition 2. Let G be a graph and let N be a 2-distance zero forcing set of G. Then
every dominating vertex v ∈ V (G), v ∈ N .

Proof. Let v ∈ V (G) be a dominating vertex of G. Then NG[v] = V (G), that is, v is
adjacent to every vertex u ∈ V (G) \ {v}. Suppose that v /∈ N . Then dG(v, w) = 2 for
some w ∈ N , a contradiction. Therefore, v ∈ N .

Proposition 3. Let n be a positive integer. Then,

Z2(Pn) =

{
1 , n = 1

2 , n ≥ 2

Proof. By Theorem 1, Z2(P1) = 1 and Z2(P2) = 2. Clearly, Z2(Pn) = 2 for n = 3, 4, 5
Suppose that n ≥ 6 Let V (Pn) = {v1, v2, ..., vn} and N = {v1, v2}. If n is odd, then
vertices v3, v5, . . . , vn are 2-forced by vertices v1, v3, . . . , vn−2, respectively, and vertices
v4, v6, . . . , vn−1 are 2-forced by vertices v2, v4, . . . , vn−3, respectively. If n is even, then
vertices v3, v5, . . . , vn−1 are 2-forced by vertices v1, v3, . . . , vn−3, respectively, and vertices
v4, v6, . . . , vn are 2-forced by vertices v2, v4, . . . , vn−2, respectively. Therefore, N is a 2-
distance zero forcing set of Pn. Since any singleton subset of V (Pn) is not a 2-distance
zero forcing set of Pn, it follows that N is a minimum 2-distance zero forcing set of Pn.
Consequently, Z2(Pn) = 2 for all n ≥ 2.



J. A. Hassan, L. T. Udtohan, L. S. Laja / Eur. J. Pure Appl. Math, 17 (2) (2024), 1283-1293 1287

Proposition 4. Let n be a positive integer. Then,

(i)

Z2(Cn) =


3 , n = 3

2 , n = 4 or n ≥ 5 and odd

4 , n ≥ 6 and even

(ii)

Z2(Sn) =

{
2 , n = 1

n , n ≥ 2

Proof. (i) By Theorem 1, Z2(C3) = 3. Clearly, Z2(C4) = 2. Suppose n ≥ 5 and odd.
Let V (Cn) = {v1, v2, ..., vn} and let N = {v1, v3}. Then N is a minimum 2-distance zero
forcing set of Cn. Thus, Z2(Cn) = 2 for all n ≥ 5 and odd. Next, suppose that n ≥ 6
and even. Let N ′ = {v1, v2, v3, v4}. Then N ′ is a minimum 2-distance zero forcing set Cn.
Thus, Z2(Cn) = 4 for all n ≥ 6 and even.

(ii) By Theorem 1, Z2(S1) = 2. Suppose that n ≥ 2. Let V (Sn) = {d, v1, ..., vn}, where
d is the dominating vertex of Sn. Consider M = {d, v1, ..., vn−1}. Then M is a minimum
2-distance zero forcing set of Sn. Therefore, Z

2(Sn) = n for all n ≥ 2.

Theorem 2. Let G be a graph. If H is a subgraph of G, then Z2(H) ≤ Z2(G) is not true
in general.

Proof. Consider the graph K2 + P3 below.

a b c

d e

K2 + P3 :

Let S1 = {a, b, d, e}. Then S1 is a minimum 2-distance zero forcing set of K2+P3. Hence,
Z2(K2 + P3) = 4. Now, consider the graph P3 + P4 below.

u1 u2 u3

u4 u5 u6 u7

P3 + P4 :



J. A. Hassan, L. T. Udtohan, L. S. Laja / Eur. J. Pure Appl. Math, 17 (2) (2024), 1283-1293 1288

Let S2 = {u1, u2, u6}. Then u3, u4, u7 and u5 are 2-forced by u1, u6, u4 and u7, respectively.
Thus, S2 is a 2-distance zero forcing set of P3+P4. It can be verified that Z2(P3+P4) = 3.
Consequently, the assertion follows.

Theorem 3. Let H be a graph. If K is a subgraph of H, then Z2(K) ≥ Z2(H) is not
true in general.

Proof. Consider the graph H below.

a
b

c

d

e

f

gH :

Let Q1 = {a, b, c, e, f}. Then Q1 is a minimum 2-distance zero forcing set of H. Thus,
Z2(H) = 5. Now, consider the subgraph K of H below.

a

b

c

d

g

K :

Let Q2 = {a, b, c}. Then Q2 is a minimum 2-distance zero forcing set of K. Thus,
Z2(K) = 3. Therefore, the assertion follows.



J. A. Hassan, L. T. Udtohan, L. S. Laja / Eur. J. Pure Appl. Math, 17 (2) (2024), 1283-1293 1289

Proposition 5. Let G be a graph. Then the 2-distance zero forcing Z2(G) and hop
domination parameters γh(G) of G are incomparable.

Proof. Consider the graph G below.

a1

a2

a3

a4

a5

a6

a7

a8

G :

Let Q = {a5, a6}. Then Q is a hop dominating set of G since N2
G[Q] = V (G). Since

N2
G[ai] ̸= V (G) for every i ∈ {1, 2, · · · , 8}, it follows that Q is a minimum hop dominating

set of G. Thus, γh(G) = 2. Now, let S = {a1, a2, a3, a6, a8}. Then S is a minimum
2-distance zero forcing set of G. Therefore, Z2(G) = 5.

On the other hand, consider the graph H below.

H :

a1

a2

a3

a4

a5

a6

a7

a8

a9

Let D = {a1, a2, a8}. Then vertices a3, a5 and a7 are 2-forced by the vertices a1, a3
and a5, respectively, and vertices a4, a6 and a9 are 2-forced by the vertices a2, a4 and
a6, respectively. This follows that D is a 2-distance zero forcing set of H. Moreover,
Z2(H) = 3. Next, let D′ = {a3, a4, a5, a6}. Then D′ is a minimum hop dominating set of
H. Consequently, γh(H) = 4.

Theorem 4. Let G be a graph. Then Z2(G) = γh(G) = |V (G)| if and only if every
component of G is complete.

Proof. Suppose that Z2(G) = |V (G)| = γh(G). Then V (G) is both the minimum 2-
distance zero forcing and minimum hop dominating set of G. Suppose there is a component
of G which is non-complete. Then there exist a, b ∈ V (Q) such that dG(a, b) = 2. Let
S = V (G) \ {a}. Then S is both a 2-distance zero forcing and a hop dominating set of G.
a contradiction. Therefore, every component of G is complete.



J. A. Hassan, L. T. Udtohan, L. S. Laja / Eur. J. Pure Appl. Math, 17 (2) (2024), 1283-1293 1290

Conversely, suppose that every component of G is complete. Then by Theorem 1,
Z2(G) = |V (G)|. Moreover, γh(G) = |V (G)|. Consequently,

Z2(G) = γh(G) = |V (G)|.

Proposition 6. Let G be a graph. Then zero forcing Z(G) and 2-distance zero forcing
Z2(G) parameter of G are incomparable.

Proof. Consider the graph P4 + P4 below.

a b c d

e f g h

P4 + P4 :

Let S1 = {b, f} and S2 = {a, b, c, d, e}. Then S1 and S2 are minimum 2-distance zero
forcing and zero forcing sets of P4 + P4, respectively. Thus, Z

2(P4 + P4) = 2 and
Z(P4 + P4) = 5.

Next, consider the graph G below.

u1

u2

u3 u4

u5

u6

u7

u8

u9

u10

u11

u12

u13

G :

Let D1 = {u1, u11, u12, u13} and D2 = {u1, u3, u11, u12, u13}. Then D1 and D2 are mini-
mum zero forcing and 2-distance zero forcing set of G, respectively. Therefore, Z(G) = 4
and Z2(G) = 5.



J. A. Hassan, L. T. Udtohan, L. S. Laja / Eur. J. Pure Appl. Math, 17 (2) (2024), 1283-1293 1291

Theorem 5. There exists a graph G such that Z2(G) = Z(G).

Proof. Consider the graph K2 + P4 below.

v1 v2 v3 v4

v5 v6

K2 + P4 :

LetD′ = {v1, v5, v6} andD′′ = {v3, v5, v6}. ThenD′ andD′′ are minimum zero forcing and
2-distance zero forcing sets of K2+P4, respectively. Hence, Z(K2+P4) = 3 = Z2(K2+P4).
Let G = K2 + P4. Then the assertion follows.

Theorem 6. Let G be a graph. Then Z2(G) = Z(G) = |V (G)| if and only if every
component of G is trivial.

Proof. Suppose that Z2(G) = Z(G) = |V (G)|. Suppose there is component K of G
which is non-trivial. Let V (K) = {k1, ..., kn}, k ≥ 2. Consider Q = V (G) \ {k1}. Then
Q is a zero forcing set of G. Thus, Z(G) ≤ |V (G)| − 1, a contradiction. Therefore, every
component of G is trivial.

The converse is clear.

4. Conclusion

The concept of 2-distance zero forcing in a graph has been introduced and investigated
in this paper. The 2-distance zero forcing numbers of some graphs are obtained. The
connections of the 2-distance zero forcing parameter with the standard zero forcing and
hop domination parameter have been presented. Interested researchers may study this
concept on graphs that were not considered in this study. Interested researchers may also
consider on providing an application of this parameter.

Acknowledgements

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



REFERENCES 1292

References

[1] F. Barioli, W. Barett, S. Fallat, H. Hall, L. Hogben, H. van der Holst, and B. Shader.
Zero forcing parameters and minimum rank problems. Linear Algebra Appl.,, 443:401–
411, 2010.

[2] K. Benson, D. Ferrero, M. Flagg, V. Furst, L. Hogben, V. Vasilevska, and B. Wiss-
man. Zero forcing and power domination for graphs products. Australas. J. Combin,
70:221–235, 2018.

[3] R. Davila, T. Kalinowski, and S. Sudeep. A lower bound on the zero forcing number.
Discrete Appl. Math., 250:363–367, 2018.

[4] S. M. Fallat and L. Hogben. Minimum rank, maximum nullity, and zero forcing
number of graphs. 2nd ed., in Handbook of Linear Algebra, CRC Press, Boca Raton,
FL, pages 775–810, 2013.

[5] M. Gentner, L. D. Penso, D. Rautenbach, , and U. S. Souza. Extremal values and
bounds for the zero forcing number. Discrete Appl. Math., 214:196–200, 2016.

[6] AIM Minimum Rank Special Graphs Work Group. Zero forcing sets and the minimum
rank of graphs,. Linear Algebra Appl.,, 428:1628–1648, 2008.

[7] J. Hassan, AR. Bakkang, and ASS. Sappari. j2-hop domination in graphs: Properties
and connections with other parameters. Eur. J. Pure Appl. Math., 16(4):2118–2131,
2023.

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

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

[10] J. Hassan and S. Canoy Jr. Grundy dominating and grundy hop dominating sequences
in graphs: Relationships and some structural properties. Eur. J. Pure Appl. Math.,
16(2):1154–1166, 2023.

[11] J. Hassan and S. Canoy Jr. Grundy total hop dominating sequences in graphs. Eur.
J. Pure Appl. Math., 16(4):2597–2612, 2023.

[12] 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., 16(3):1848–1861, 2023.

[13] S. Canoy Jr. and J. Hassan. Weakly convex hop dominating sets in graphs. Eur. J.
Pure Appl. Math., 15(4):1783–1796, 2022.



REFERENCES 1293

[14] S. Kaida, K.J. Maharajul, J. Hassan, L. S. Laja, A.B. Lintasan, and A.A. Pablo.
Certified hop independence: Properties and connections with other variants of inde-
pendence. Eur. J. Pure Appl. Math., 17(1):435–444, 2024.

[15] J. Manditong, J. Hassan, LS Laja, AA. Laja, NHM. Mohammad, and SU. Kam-
don. Connected outer-hop independent dominating sets in graphs under some binary
operations. Eur. J. Pure Appl. Math., 16(3):1817–1829, 2023.

[16] J. Manditong, A. Tapeing, J. Hassan, A.R. Bakkang, N.H. Mohammad, and S.U.
Kamdon. Some properties of zero forcing hop dominating sets in a graph. Eur. J.
Pure Appl. Math., 17(1):324–337, 2024.