EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 12, No. 4, 2019, 1410-1425
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

On Semitotal Domination in Graphs

Imelda S. Aniversario1,2, Sergio R. Canoy Jr.1,2, Ferdinand P. Jamil1,2,∗

1 Department of Mathematics and Statistics, College of Science and Mathematics, Mindanao
State University-Iligan Institute of Technology, 9200 Iligan City, Philippines
2 Center for Graph Theory, Algebra and Analysis, Premier Research Institute of Science
and Mathematics (PRISM), Mindanao State University-Iligan Institute of Technology, 9200
Iligan City, Philippines

Abstract. A set S of vertices of a connected graph G is a semitotal dominating set if every vertex
in V (G)\S is adjacent to a vertex in S, and every vertex in S is of distance at most 2 from another
vertex in S. A semitotal dominating set S in G is a secure semitotal dominating set if for every
v ∈ V (G) \ S, there is a vertex x ∈ S such that x is adjacent to v and that (S \ {x}) ∪ {v} is a
semitotal dominating set in G.
In this paper, we characterize the semitotal dominating sets and the secure semitotal dominating
sets in the join, corona and lexicographic product of graphs and determine their corresponding
semitotal domination and secure semitotal domination numbers.

2010 Mathematics Subject Classifications: 05C69

Key Words and Phrases: Semitotal dominating set, secure semitotal dominating set, semitotal
domination number, secure semitotal domination number

1. Introduction

The concept of semitotal domination was introduced by W. Goddard, M. Henning
and C. McPil (see [13]) in 2014. It is further studied by M.Henning and A. Marcon (see
[17, 18]) in 2014 and 2016, and by G. Hao and W. Zhuang (see [14]) in 2018. Accordingly,
this parameter is a strengthening of domination but a relaxation of both total domination
and weakly connected domination [13].

In this paper, we investigate semitotal domination in the join, corona and lexicographic
product of graphs. We also introduce the secure semitotal domination and investigate the
concept in these classes of graphs.

All graphs considered in this study are finite and undirected. We refer to [7] for the
basic graph terminologies used here. The symbols V (G) and E(G) denote the vertex set

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v12i4.3501

Email addresses: imelda.aniversario@g.msuiit.edu.ph (I.S. Aniversario),
sergio.canoy@g.msuiit.edu.ph (S.R.Jr. Canoy), ferdinand.jamil@g.msuiit.edu.ph (F. Jamil)

http://www.ejpam.com 1410 c© 2019 EJPAM All rights reserved.



I. S. Aniversario, S. R. Jr. Canoy, F.P. Jamil / Eur. J. Pure Appl. Math, 12 (4) (2019), 1410-1425 1411

and edge set, respectively, of G. For S ⊆ V (G), |S| is the cardinality of S. In particular,
|V (G)| is the order of G.

Given two graphs G and H with disjoint vertex sets, the join of G and H is the graph
G+H with V (G+H) = V (G)∪V (H) and E(G+H) = E(G)∪E(H)∪{uv : u ∈ V (G), v ∈
V (H)}. The corona of G and H is the graph G ◦ H obtained by taking one copy of G
and |V (G)| copies of H, and then joining the ith vertex of G to every vertex in the ith

copy of H. The lexicographic product or composition G[H] of G and H is the graph with
V (G[H]) = V (G)× V (H) and (u, v)(u′, v′) ∈ E(G[H]) if and only if either uu′ ∈ E(G) or
u = u′ and vv′ ∈ E(H). In any of these graphs, G and H are referred to as their basic
component graphs.

For v ∈ V (G), the neighborhood NG(v) of v refers to the set of all vertices of G that are
adjacent to v. The closed neighborhood of v is the set NG[v] = NG(v)∪{v}. For S ⊆ V (G),
NG(S) = ∪v∈SNG(v) and NG[S] = S ∪NG(S). A set S ⊆ V (G) is a dominating set in G
if NG[S] = V (G). Thus, S is a dominating set in G if and only if for each v ∈ V (G) \ S
there exists u ∈ S such that uv ∈ E(G). The minimum cardinality of a dominating set
in G, denoted by γ(G), is the domination number of G. Provided that G has no isolated
vertices, a set S ⊆ V (G) is a total dominating set in G if for every v ∈ V (G) there exists
u ∈ S such that uv ∈ E(G). The minimum cardinality of a total dominating set in G,
denoted by γt(G), is the total domination number of G. We refer to [1–3, 6, 8, 9, 11, 15, 16]
for the fundamentals and recent developments and applications of domination theory in
graphs.

A set S ⊆ V (G) is said to be nearly dominating in G if for every v ∈ V (G) \ NG[S],
S ∪ {v} is a dominating set in G. The symbol γη(G) denotes the minimum cardinality of
a nearly dominating set in G. Clearly, γη(G) = 0 if and only if G is a complete graph, and
since dominating sets are nearly dominating sets, γη(G) ≤ γ(G).

A secure (total) dominating set is a (total) dominating set S having the property that
for each v ∈ V (G) \ S, there exists u ∈ S ∩ NG(v) such that (S \ {u}) ∪ {v} is a (total)
dominating set inG. The minimum cardinality γs(G) (resp. γst(G)) of a secure dominating
set (resp. secure total dominating set) in G is the secure domination number (resp. secure
total domination number) of G. A secure dominating set of cardinality γs(G) is called
a γs-set. Secure domination and secure total domination in graphs have been studied in
[4, 5, 10, 12, 19].

Let G be a graph without isolated vertices. A set S ⊆ V (G) is a semitotal dominating
set in G if S is a dominating set in G such that for every x ∈ S there exists y ∈ S\{x} such
that dG(x, y) ≤ 2. The smallest cardinality of a semitotal dominating set in G, denoted
by γt2(G), is called the semitotal domination number of G. A semitotal dominating set
in G with cardinality γt2(G) is called a γt2-set. It is worth noting that since a semitotal
dominating set is a dominating set and total dominating sets are semitotal dominating
sets, max{2, γ(G)} ≤ γt2(G) ≤ γt(G) for graphs G without isolated vertices. For all
connected graphs G on n ≥ 4 vertices, γt2(G) ≤ n

2 [13]. In the referred paper, the authors
characterized those trees and graphs of minimum degree 2 achieving this bound. Other
excellent exposition on semitotal domination are found in [17] and in [18].



I. S. Aniversario, S. R. Jr. Canoy, F.P. Jamil / Eur. J. Pure Appl. Math, 12 (4) (2019), 1410-1425 1412

2. Secure semitotal domination

A semitotal dominating set S ⊆ V (G) is a secure semitotal dominating set if for
each u ∈ V (G) \ S, there exists v ∈ S ∩ NG(u) such that (S \ {v}) ∪ {u} is a semitotal
dominating set in G. The smallest cardinality of a secure semitotal dominating set in G is
called the secure semitotal domination number of G and is denoted by γst2(G). A secure
semitotal dominating set with cardinality γst2(G) is called a γst2-set. Secure semitotal
dominating sets are both semitotal dominating sets and secure dominating sets. On the
other hand, secure total dominating sets are secure semitotal dominating sets. Thus,
max{γt2(G), γs(G)} ≤ γst2(G) ≤ γst(G) for all graphs G without isolated vertices.

Example 1. (1) For n ≥ 2, γst2(Pn) =


dn2 e, if n 6= 2, 6

2, n = 2

4, n = 6.

(2) For n ≥ 3, γst2(Cn) =
⌈
n
2

⌉
.

(3) For m,n ≥ 2, γst2(Km,n) = min{m,n, 4}.

For v ∈ V (G), we write N2
G(v) = {u ∈ V (G) \ {v} : dG(u, v) ≤ 2}, and for S ⊆ V (G),

we write N2
G(S) = ∪v∈SN2

G(v). Precisely, S is a semitotal dominating set if and only if
V (G) \ S ⊆ NG(S) and S ⊆ N2

G(S).

Theorem 1. Let G be a connected graph of order n ≥ 2. Then γst2(G) = 2 if and only if
there exists a dominating set {x, y} in G satisfying the following properties:

(i) dG(x, y) ≤ 2;

(ii) N2
G(x) = V (G) \ {x} and N2

G(y) = V (G) \ {y}; and

(iii) {x, z} and {u, y} are dominating sets in G for all z ∈ NG(y) \ {x} and for all
u ∈ NG(x) \ {y}.

Proof. Suppose that γst2(G) = 2, and let S = {x, y} be a γst2-set of G. Then S is a
dominating set in G and dG(x, y) ≤ 2. Suppose that, in the contrary, N2

G(x) 6= V (G)\{x},
and let z ∈ V (G)\N2

G(x) with z 6= x. Then z /∈ S. Since S is a secure semitotal dominating
set in G and z /∈ N2

G(x), y ∈ NG(z) and (S \ {y})∪{z} = {x, z} is a semitotal dominating
set in G, a contradiction since dG(x, z) > 2. Thus, N2

G(x) = V (G) \ {x}. Similarly,
N2
G(y)\{y} = V (G)\{y}. Now let z ∈ V (G)\S. Since S is a secure semitotal dominating

set, there exists w ∈ S ∩NG(z) such that T = (S \ {w}) ∪ {z} is a semitotal dominating
set, and hence a dominating set in G. If w = y, then T = {x, z} and if x = w, then
T = {y, z}.

Conversely, let S = {x, y} be a dominating set in G satisfying the properties (i), (ii)
and (iii). By property (i), S is a semitotal dominating set in G. Let z ∈ V (G) \ S.
Then x ∈ S ∩ NG(z) or y ∈ S ∩ NG(z). Assume that x ∈ S ∩ NG(z). Note that, by
properties (ii) and (iii), (S \ {x})∪{z} = {y, z} is a semitotal dominating set in G. Thus,
γst2(G) = |S| = 2.



I. S. Aniversario, S. R. Jr. Canoy, F.P. Jamil / Eur. J. Pure Appl. Math, 12 (4) (2019), 1410-1425 1413

3. In the join of graphs

For any graph G, γt2(G + K1) = 2. More specifically, a semitotal dominating set in
G+K1 is either of the form V (K1) ∪ S for some nonempty S ⊆ V (G), or a nonsingleton
dominating set in G in case G is nontrivial.

Theorem 2. Let G and H be nontrivial graphs, and S ⊆ V (G+H). Then S is a semitotal
dominating set in G+H if and only if one of the following holds:

(i) S ⊆ V (G) is a nonsingleton dominating set in G;

(ii) S ⊆ V (H) is a nonsingleton dominating set in H;

(iii) S ∩ V (G) 6= ∅ and S ∩ V (H) 6= ∅.

Proof. Suppose that S ⊆ V (G) is a nonsingleton dominating set in G. Then S is a
dominating set in G + H. Let v ∈ S. Since S is nonsingleton, we may take u ∈ S with
u 6= v. Note that dG+H(u, v) ≤ 2. Since v is arbitrary, S is a semitotal dominating set in
G+H. Similarly, if S ⊆ V (H) is a nonsingleton dominating set in H, then S is a semitotal
dominating set in G + H. Suppose that S intersects both V (G) and V (H). Then S is a
total dominating set, hence a semitotal dominating set, in G+H.

Conversely, suppose that S is a semitotal dominating set in G + H. Then S is a
dominating set in G+H and |S| ≥ 2. If S ⊆ V (G) (resp. S ⊆ V (H)), then (i) (resp. (ii))
holds. Otherwise, property (iii) holds.

Corollary 1. For all graphs G and H, γt2(G+H) = 2.

Let G be any graph and Kp the complete graph of order p ≥ 2. Note that for any
x, y ∈ V (Kp), {x, y} is a dominating set in G + Kp satisfying the properties (i), (ii) and
(iii) of Theorem 1. Thus, γst2(G+Kp) = 2.

Proposition 1. For noncomplete graphs G and H,

2 ≤ γst2(G+H) ≤ 4.

Proof. Let S = {x, y, u, v}, where x, y ∈ V (G) and u, v ∈ V (H). Then S is a semitotal
dominating set in G+H by Theorem 2. For each w ∈ V (G)\S (resp. each w ∈ V (H)\S),
wv ∈ E(G + H) (resp. wx ∈ E(G + H)) and, by Theorem 2, (S \ {v}) ∪ {w} (resp.
(S \ {x}) ∪ {w}) is a semitotal dominating set in G + H. Thus, S is a secure semitotal
dominating set in G+H. Consequently, 2 ≤ γst2(G+H) ≤ |S| = 4.

Corollary 2. Let G and H be noncomplete graphs of orders m and n, respectively. Then
γst2(G+H) = 2 if and only if at least one of the following is true:

(i) γs(G) = 2;

(ii) γs(H) = 2;



I. S. Aniversario, S. R. Jr. Canoy, F.P. Jamil / Eur. J. Pure Appl. Math, 12 (4) (2019), 1410-1425 1414

(iii) there exist x ∈ V (G) and y ∈ V (H) such that {x} and {y} are nearly dominating
sets in G and H, respectively.

Proof. Suppose that γst2(G+H) = 2, and let S = {x, y} be a dominating set in G+H
satisfying properties (i), (ii) and (iii) in Theorem 1. First, suppose that S ⊆ V (G).
Then S is a dominating set in G. Let z ∈ V (G) \ S. Assume xz ∈ E(G). By Theorem
1(iii), {z, y} = (S \ {x}) ∪ {z} is a dominating set in G + H, hence in G. Thus, S is a
secure dominating set in G. Since G in not complete, γs(G) = 2. Similarly, if S ⊆ V (H),
then γs(H) = 2. Suppose that x ∈ V (G) and y ∈ V (H). Let z ∈ V (G) \ NG[x]. Then
z ∈ NG+H(y) \ {x}. By Theorem 1(iii), {x, z} is a dominating set in G + H, hence a
dominating set in G. Similarly, {w, y} is a dominating set in H for all w ∈ V (H) \NG[y].

Conversely, suppose that γs(G) = 2, and let S = {x, y} be a γs-set of G. By Theorem
2, S is a semitotal dominating set in G + H. Let z ∈ V (G + H) \ S. Suppose that
z ∈ V (H). In particular, xz ∈ E(G + H) and (S \ {x}) ∪ {z} = {z, y}, which is a
semitotal dominating set in G + H by Theorem 2. Suppose that z ∈ V (G). Since S is a
secure dominating set, either xz ∈ E(G) and {y, z} is a dominating set in G or zy ∈ E(G)
and {x, z} is a dominating set in G. In either case, S is a secure semitotal dominating set
in G + H by Theorem 2, so that γst2(G + H) = |S| = 2. Similarly, if γs(H) = 2, then
γst2(G+H) = 2. Finally, suppose that x and y satisfy property (iii), and put S = {x, y}.
By Theorem 2, S is a semitotal dominating set in G+H. Let z ∈ V (G)\S. Suppose that
xz ∈ E(G). Then x ∈ S ∩NG+H(z) and (S \ {x})∪ {z} = {z, y}, which by Theorem 2, is
a semitotal dominating set in G + H. Suppose that z /∈ NG[x]. By property (iii), {x, z}
is a dominating set in G, and hence a semitotal dominating set in G+H by Theorem 2.
Now, zy ∈ E(G+H) and (S \ {y}) ∪ {z} = {x, z}. Similarly, if z ∈ V (H) \ S, then there
exists w ∈ S ∩NG+H(z) such that (S \ {w})∪{z} is a semitotal dominating set in G+H.
Thus, S is a secure semitotal dominating set in G+H. Therefore, γst2(G+H) = 2.

In view of Corollary 2(iii), γst2(K1,n +K1,m) = 2 = γst2(C4 + C4).

Corollary 3. Let G and H be noncomplete graphs of orders m and n, respectively, and
suppose that γst2(G + H) 6= 2. Then γst2(G + H) = 3 if and only if at least one of the
following is true:

(i) γs(G) = 3;

(ii) γs(H) = 3;

(iii) there exist x, y ∈ V (G) such that {x, y} is a nearly dominating set in G;

(iv) there exist x, y ∈ V (H) such that {x, y} is a nearly dominating set in H.

Proof. Suppose that γst2(G+H) = 3, and let S ⊆ V (G+H) be a γst2-set of G+H.
Suppose that S ⊆ V (G). By Theorem 2, S is a dominating set in G. Let w ∈ V (G) \ S.
There exists x ∈ S ∩NG+H(w) such that T = (S \ {x}) ∪ {w} is a semitotal dominating
set in G + H. Since T ⊆ V (G), T is a dominating set in G by Theorem 2, showing that



I. S. Aniversario, S. R. Jr. Canoy, F.P. Jamil / Eur. J. Pure Appl. Math, 12 (4) (2019), 1410-1425 1415

S is a secure dominating set in G. Hence, γs(G) ≤ |S| = 3. In view of Corollary 2,
since γst2(G + H) 6= 2, γs(G) = 3. Similarly, if S ⊆ V (H), then γs(H) = 3. Now, let
S = {x, y, z}, and assume that T = {x, y} ⊆ V (G) and z ∈ V (H). If T is a dominating
set in G, then property (iii) holds. Suppose not, and let u ∈ V (G) \ NG[T ]. Since S
is a secure semitotal dominating set in G + H, (S \ {z}) ∪ {u} = {x, y, u} is a semitotal
dominating set in G + H. Thus, {x, y, u} is a dominating set in G. Accordingly, T is a
nearly dominating set in G. Property (iv) is proved similarly.

Conversely, suppose that γs(G) = 3, and S = {x, y, z} ⊆ V (G) is a γs-set of G. By
Theorem 2, S is a semitotal dominating set in G+H. Let w ∈ V (G+H)\S. If w ∈ V (H),
then in particular, xw ∈ E(G+H) and (S \ {x}) ∪ {w} = {y, z, w}, which is a semitotal
dominating set in G + H by Theorem 2. Suppose that w ∈ V (G). Since S is a secure
dominating set in G, there exists t ∈ S ∩ NG(w) such that T = (S \ {t}) ∪ {w} is a
dominating set in G. Hence T is a semitotal dominating in G + H. Thus, S is a secure
semitotal dominating set in G + H. Since γst2(G + H) 6= 2, γst2(G + H) = 3 = |S|.
Similarly, if γs(H) = 3, then γst2(G + H) = 3. Suppose that property (iii) holds, and
let {x, y} be a nearly dominating set in G. Pick any z ∈ V (H), and put S = {x, y, z}.
Then S is a semitotal dominating set in G + H. Let w ∈ V (G + H) \ S. If w ∈ V (H),
then wx ∈ E(G + H) and (S \ {x}) ∪ {w} = {y, z, w} is a semitotal dominating set
in G + H by Theorem 2. Suppose that w ∈ V (G), and suppose that w ∈ NG[{x, y}],
say wx ∈ E(G). Then x ∈ S ∩ NG+H(w) and (S \ {x}) ∪ {w} = {w, y, z} is a semitotal
dominating set in G+H. Suppose that w /∈ NG[{x, y}]. Here we note that wz ∈ E(G+H)
and (S \ {z}) ∪ {w} = {x, y, w}. Since {x, y} is a nearly dominating set, {x, y, w} is a
dominating set in G, and therefore, is a semitotal dominating set in G + H by Theorem
2. All these imply that S is a secure semitotal dominating set, and since γst2(G+H) 6= 2,
γst2(G+H) = |S| = 3. Similarly, if property (iv) holds, then γst2(G+H) = 3.

4. In the corona of graphs

The following lemma is used in the succeeding proposition.

Lemma 1. [8] Let G be any connected graph and H any graph. Then S ⊆ V (G ◦H) is a
dominating set in G ◦H if and only if S ∩ V (Hv + v) is a dominating set in Hv + v for
all v ∈ V (G).

Proposition 2. Let G be a nontrivial connected graph and H any nontrivial graph, and
S ⊆ V (G ◦H). Then S is a semitotal dominating set in G ◦H if and only if the following
hold:

(i) S ∩ V (Hv + v) is a dominating set in Hv + v for all v ∈ V (G); and

(ii) |S ∩ V (Hv)| ≥ 2 for each v ∈ V (G) \ S with NG(v) ∩ S = ∅;

Proof. Suppose that S is a semitotal dominating set in G◦H. Then S is a dominating
set in G ◦H so that property (i) follows immediately from Lemma 1. Let v ∈ V (G) \ S



I. S. Aniversario, S. R. Jr. Canoy, F.P. Jamil / Eur. J. Pure Appl. Math, 12 (4) (2019), 1410-1425 1416

such that NG(v)∩S = ∅. Then S ∩V (Hv + v) = S ∩V (Hv). By property (i), S ∩V (Hv)
is a dominating set in Hv + v, and consequently in Hv. Let u ∈ S ∩ V (Hv). Since S is
a semitotal dominating set in G ◦H, there exists w ∈ S \ {u} such that dG◦H(u,w) ≤ 2.
Since S ∩NG[v] = ∅, w ∈ V (Hv). This proves that property (ii) holds.

Conversely, suppose that all properties hold for S. By property (i), S is a dominating
set in G ◦ H. Let u ∈ S, and let v ∈ V (G) such that u ∈ V (Hv + v). Suppose that
u ∈ V (Hv). If v ∈ S, then v is the required vertex in S for which dG◦H(u, v) ≤ 2.
Suppose that v /∈ S. If NG(v) ∩ S 6= ∅, say w ∈ NG(v) ∩ S, then dG◦H(u,w) = 2.
Suppose that NG(v) ∩ S = ∅. By property (ii), we may pick w ∈ S ∩ V (Hv) \ {u}. Then
dG◦H(u,w) ≤ 2. Finally, suppose that u = v. Since G is a nontrivial connected graph, we
may pick w ∈ V (G) such that vw ∈ E(G). Since S ∩ V (Hw + w) is a dominating set in
Hw +w, S ∩ V (Hw +w) 6= ∅. For any z ∈ S ∩ V (Hw +w), dG◦H(v, z) ≤ 2. Accordingly,
S is a semitotal dominating set in G ◦H.

Corollary 4. Let G be a nontrivial connected graph and H any nontrivial graph, and
S ⊆ V (G ◦H). Then S is a semitotal dominating set in G ◦H if and only if

S = A ∪ [∪v∈ASv] ∪ [∪u∈V (G)\ADu],

where

(i) A ⊆ V (G);

(ii) Sv ⊆ V (Hv) for each v ∈ A;

(iii) Du is a dominating set in Hu for each u ∈ V (G) \A; and

(iv) |Du| ≥ 2 for each u ∈ V (G) \A with NG(u) ∩A = ∅.

Corollary 5. For all nontrivial connected graphs G and any graph H,

γt2(G ◦H) = |V (G)|.

For nontrivial connected graphs G, V (G) is a secure semitotal dominating set in the
corona G◦Kp for any integer p ≥ 1. This, together with Corollary 5, yields γst2(G◦Kp) =
|V (G)|.

In what follows, we consider G ◦H, where H is noncomplete.

Theorem 3. Let G be a nontrivial connected graph and H be any noncomplete graph
without isolated vertices, and let S ⊆ V (G ◦H). Then S is a secure semitotal dominating
set if and only if S is a semitotal dominating set in G◦H satisfying the following properties:

(i) S ∩ V (Hv) is a secure dominating set in Hv for each v ∈ V (G) \ S; and

(ii) S ∩ V (Hv) is a nearly dominating set in Hv for all v ∈ S ∩ V (G).



I. S. Aniversario, S. R. Jr. Canoy, F.P. Jamil / Eur. J. Pure Appl. Math, 12 (4) (2019), 1410-1425 1417

Proof. For each v ∈ V (G), we write Sv = S ∩ V (Hv). Suppose that S is a secure
semitotal dominating set in G ◦H. Then S is a semitotal dominating set in G ◦H. Let
v ∈ V (G) \S. By Proposition 2, Sv is a dominating set in Hv. Let x ∈ V (Hv) \Sv. Since
S is a secure semitotal dominating set in G ◦H, there exists y ∈ S ∩NG◦H(x) such that
S∗ = (S \ {y}) ∪ {x} is a semitotal dominating set in G ◦H. Clearly, y ∈ Sv ∩ NHv(x).
Write

S∗ =
(
∪u∈V (G)\{v}S ∩ V (Hu + u)

)
∪ (Sv \ {y}) ∪ {x}. (1)

Since S∗ is a dominating set in G◦H, (Sv \ {y})∪{x} is a dominating set in Hv by Lemma
1. Thus, Sv is a secure dominating set in Hv. This proves property (i). To prove (ii),
let v ∈ S ∩ V (G). If Sv is a dominating set in Hv, then we are done. Suppose that Sv is
not a dominating set in Hv, and let x ∈ V (Hv) \NHv [Sv]. Since S is a secure semitotal
dominating set in G ◦H and x ∈ V (G ◦H) \ S, there exists u ∈ S ∩NG◦H(x) such that
(S \ {u})∪{x} is a semitotal dominating set in G ◦H. Necessarily, u = v so that Sv ∪{x}
is a semitotal dominating set in Hv + v. By Theorem 2, Sv ∪ {x} is a dominating set in
Hv. Thus, Sv is nearly dominating in Hv.

Conversely, suppose that all the properties hold for a semitotal dominating set S in
G ◦H. Let x ∈ V (G ◦H) \ S, and let v ∈ V (G) such that x ∈ V (Hv + v). We consider
two cases:

Case 1: x = v.
Pick any y ∈ Sx. Since x ∈ V (G), (Sx \ {y})∪ {x} is a dominating set in Hx + x. Put

S∗ = (S \ {y}) ∪ {x}. Then S ∩ V (Hu + u) = S∗ ∩ V (Hu + u) for all u ∈ V (G) \ {x},
and S∗ ∩ V (Hx + x) = (Sx \ {y}) ∪ {x}. Thus, S∗ satisfies property (i) of Proposition
2. Let u ∈ V (G) \ S∗ with NG(u) ∩ S∗ = ∅. Then u 6= x so that u ∈ V (G) \ S and
NG(u)∩S = ∅. Since S is a semitotal dominating set in G◦H, |S∗u| = |Su| ≥ 2. Since u is
arbitrary, property (ii) of Proposition 2 holds for S∗. Thus, S∗ is a semitotal dominating
set in G ◦H.

Case 2: x 6= v.
In this case, x ∈ V (Hv)\Sv. First, suppose that v /∈ S. By property (i), Sv is a secure

dominating set in Hv. Thus, there exists y ∈ Sv ∩ NHv(x) for which (Sv \ {y}) ∪ {x} is
a dominating set in Hv, and consequently in Hv + v as well. Put S∗ = (S \ {y}) ∪ {x}.
Then S ∩ V (Hu + u) = S∗ ∩ V (Hu + u) for all u ∈ V (G) \ {v}, and S∗ ∩ V (Hv + v) =
(Sv \ {y})∪ {x}. Thus, S∗ satisfies property (i) of Proposition 2. Let u ∈ V (G) \ S∗ with
NG(u) ∩ S∗ = ∅. Since S and S∗ differ only by their respective Sv and S∗v , u ∈ V (G) \ S
and NG(u) ∩ S = ∅. If u 6= v, then |S∗u| = |Su| ≥ 2. If u = v, then |Su \ {y}| ≥ 1 so that
|S∗u| = | (Su \ {y}) ∪ {x}| ≥ 2. This shows that S∗ satisfies property (ii) of Proposition 2.
Thus, S∗ is a semitotal dominating set in G ◦H.

Next, suppose that v ∈ S. By property (ii), Sv is a nearly dominating set in Hv.
Suppose that x ∈ NHv [Sv], and y ∈ Sv for which xy ∈ E(Hv). Put S∗ = (S \ {y}) ∪ {x}.
Since v ∈ S∗ ∩ V (Hv + v), S∗ ∩ V (Hv + v) is a dominating set in Hv + v, and S∗ satisfies
property (i) of Proposition 2. Let u ∈ V (G) \ S∗ with NG(u) ∩ S∗ = ∅. Note also
in here that S∗ and S differ only by their respective S∗v and Sv. Thus u ∈ V (G) \ S



I. S. Aniversario, S. R. Jr. Canoy, F.P. Jamil / Eur. J. Pure Appl. Math, 12 (4) (2019), 1410-1425 1418

and NG(u) ∩ S = ∅. Since u 6= v, |S∗u| = |Su| ≥ 2, and S∗ satisfies property (ii) of
Proposition 2. Accordingly, S∗ is a semitotal dominating set in G ◦H. Finally, suppose
that x /∈ NHv [Sv]. Put S∗ = (S \ {v})∪{x}. Note in here that S∗∩V (Hv+v) = Sv∪{x},
which is a dominating set in Hv because Sv is a nearly dominating set in Hv. As argued
previously, S∗ satisfies property (i) of Proposition 2. By Lemma 1, S∗ is a dominating
set in G ◦ H. Since Hv is a noncomplete graph and Sv is a nearly dominating set in
Hv, Sv 6= ∅. Clearly, dG◦H(x, y) ≤ 2 for all y ∈ S∗v \ {x$.Supposethatthereexistsz∈ S∗
such that dG◦H(z, y) > 2 for all y ∈ S∗ \ {z}. In view of the preceding arguments,
z ∈ S \ V (Hv + v). Suppose that z ∈ V (G). By property (ii), Sz is nearly dominating in
Hz. But by the definition of z, Sz = ∅, which is impossible. Suppose that z ∈ V (Hw) for
some w ∈ V (G). Then w /∈ S. By property (i), S ∩ V (Hw) is a secure dominating set.
Since H is a noncomplete graph, |S ∩ V (Hw)| ≥ 2, which is impossible. This shows that
S∗ is a semitotal dominating set in G ◦H. Therefore, S is a secure semitotal dominating
set.

Corollary 6. Let G be a nontrivial connected graph and H be any noncomplete graph
without isolated vertices, and let S ⊆ V (G ◦H). Then S is a secure semitotal dominating
set if and only if

S = A ∪ [∪v∈ASv] ∪ [∪u∈V (G)\ADu],

satisfying the following properties:

(i) A ⊆ V (G);

(ii) Sv is a nearly dominating set in Hv for each v ∈ A;

(iii Du is a secure dominating set in Hv for each u ∈ V (G) \A; and

(iv) |Du| ≥ 2 for each u ∈ V (G) \A with NG(u) ∩A = ∅.

Corollary 7. Let G be a nontrivial connected graph and H be any noncomplete graph
without isolated vertices.

(i) If γη(H) = γs(H), then γst2(G ◦H) = |V (G)|γη(H).

(ii) If γη(H) < γs(H), then γst2(G ◦H) = |V (G)| (1 + γη(H)).

5. In the lexicographic product of grahs

Theorem 4. [2] Let G and H be nontrivial connected graphs. Then C = ∪x∈S ({x} × Tx)
is a dominating set if and only if one of the following holds:

(i) S is a total dominating set in G;

(ii) S is a dominating set in G and for each x ∈ S \NG(S), Tx is a dominating set in
H.



I. S. Aniversario, S. R. Jr. Canoy, F.P. Jamil / Eur. J. Pure Appl. Math, 12 (4) (2019), 1410-1425 1419

The next theorem follows immediately from Theorem 4.

Theorem 5. [3] Let G and H be nontrivial connected graphs. Then C = ∪x∈S ({x} × Tx)
is a total dominating set if and only if one of the following holds:

(i) S is a total dominating set in G;

(ii) S is a dominating set in G and for each x ∈ S \NG(S), Tx is a total dominating set
in H.

Theorem 6. Let G and H be nontrivial connected graphs, and let C = ∪x∈S ({x} × Tx) ⊆
V (G[H]). Then C is a semitotal dominating set in G[H] if and only if one of the following
holds:

(i) S is a total dominating set in G;

(ii) S is semitotal dominating set in G and for each x ∈ S \NG(S), Tx is a dominating
set in H;

(iii) S is a dominating set in G such that Tx is a dominating set in H for each x ∈
S \NG(S), and |Tx| ≥ 2 for each x ∈ S \N2

G(S).

Proof. By Theorem 4, each of the conditions (i), (ii) and (iii) implies that C is a
dominating set in G[H]. If condition (i) holds, then by Theorem 5, C is a total dominating
set, hence a semitotal dominating set in G[H]. Suppose that condition (ii) holds, and let
(x, y) ∈ C. Since S is a semitotal dominating set in G, there exists u ∈ S such that
1 ≤ dG(x, u) ≤ 2. Pick v ∈ Tu. Then (u, v) ∈ C and 1 ≤ dG[H]((x, y), (u, v)) ≤ 2.
Thus, C is a semitotal dominating set in G[H]. Suppose that condition (iii) holds, and
let (x, y) ∈ C. If x ∈ N2

G(S), then there exists u ∈ S such that 1 ≤ dG(x, u) ≤ 2. Pick
v ∈ Tu. Then 1 ≤ dG[H]((x, y), (u, v)) ≤ 2. Suppose that x /∈ N2

G(S). Pick z ∈ Tx \ {y}.
Then (x, z) ∈ C and dG[H]((x, y), (x, z)) ≤ 2.

Suppose that C is a semitotal dominating set in G[H]. Then S is a dominating set
in G by Theorem 4. If S is a total dominating set in G, then (i) holds. Suppose that
S is not a total dominating set in G. By Theorem 4, Tx is a dominating set in H for
each x ∈ S \ NG(S). If S is a semitotal dominating set in G, then (ii) holds. Suppose
that S is not a semitotal dominating set in G. Let x ∈ S \ N2

G(S), and let u ∈ Tx.
Since C is a semitotal dominating set in G[H], there exists (a, b) ∈ C such such that
1 ≤ dG[H]((x, u), (a, b)) ≤ 2. Since x ∈ S \N2

G(S), a = x and |Tx| ≥ 2.

The following well-known lemma is essential for the desired results.

Lemma 2. [3] Let G be a nontrivial connected graph and S ⊆ V (G) a dominating set in
G. Then

γt(G) ≤ |S ∩NG(S)|+ 2|S \NG(S)|.

Following the usual proof also establishes the next lemma.



I. S. Aniversario, S. R. Jr. Canoy, F.P. Jamil / Eur. J. Pure Appl. Math, 12 (4) (2019), 1410-1425 1420

Lemma 3. If G is a nontrival connected graph and S ⊆ V (G) is a dominating set in G,
then

γt2(G) ≤ 2|S \N2
G(S)|+ |S ∩N2

G(S)|.

Corollary 8. If G and H are nontrivial connected graphs with γ(H) = 1, then γt2(G[H]) =
γt2(G).

Proof. Let v ∈ V (H) be such that {v} is a dominating set in H. Let S ⊆ V (G) be a
semitotal dominating set in G. By Theorem 6, S × {v} is a semitotal dominating set in
G[H]. Thus, γt2(G[H]) ≤ |S|. Since S is arbitrary, γt2(G[H]) ≤ γt2(G).

Let C = ∪x∈S ({x} × Tx) ⊆ V (G[H]) be a semitotal dominating set in G[H]. By
Theorem 6, S is a dominating set in G. If S satisfies (i) or (ii) in Theorem 6, then S is a
semitotal dominating set in G, and

γt2(G) ≤ |S| ≤
∑
x∈S
|Tx| = |C|.

Suppose that S is not a semitotal dominating set inG. Let S1 = S\N2
G(S), S2 = S∩N2

G(S).
By Theorem 6,

C = (∪x∈S1 ({x} × Tx)) ∪ (∪x∈S2 ({x} × Tx)) ,

where |Tx| ≥ 2 for all x ∈ S1. Thus,

|C| =
∑
x∈S1

|Tx|+
∑
x∈S2

|Tx|

≥ 2|S1|+ |S2|
= 2|S \N2

G(S)|+ |S ∩N2
G(S)|.

By Lemma 3, γt2(G) ≤ |C|. Since C is arbitrary, γt2(G) ≤ γt2(G[H]).

Corollary 9. Let G and H be nontrivial connected graphs with γ(H) = 2, and let C =
∪x∈S ({x} × Tx) ⊆ V (G[H]). Then C is a γt2-set of G[H] if and only if one of the following
holds:

(i) S is a γt-set of G and |Tx| = 1 for each x ∈ S;

(ii) S is a semitotal dominating set in G such that γt(G) = 2|S \NG(S)|+ |S ∩NG(S)|.
Further, |Tx| = 1 for each x ∈ S ∩ NG(S) and Tx is a γ-set of H for each x ∈
S \NG(S);

(iii) S is a dominating set in G such that γt(G) = 2|S \NG(S)|+ |S ∩NG(S)| and where
|Tx| = 1 for each x ∈ S ∩ NG(S), Tx is a γ-set of H (thus, |Tx| = 2) for each
x ∈ S ∩

(
N2
G(S) \NG(S)

)
, and |Tx| = 2 for each x ∈ S \N2

G(S).

Proof. Let C be a γt2-set of G[H]. First, suppose that S is a total dominating set in
G. We claim that |Tx| = 1 for each x ∈ S. Suppose that |Tx| ≥ 2 for some x ∈ S. Let



I. S. Aniversario, S. R. Jr. Canoy, F.P. Jamil / Eur. J. Pure Appl. Math, 12 (4) (2019), 1410-1425 1421

C∗ = S × {v} = ∪x∈S ({x} × {v}), where v ∈ V (H). Then C∗ is a semitotal dominating
set in G[H] by Theorem 6, and |C∗| = |S| < |C|, which is impossible since C is a γt2-set.
Necessarily, S is a γt-set of G. In this case, (i) holds.

Next, suppose that S is a semitotal dominating set in G and Tx is a dominating set
for each x ∈ S \NG(S). Invoking Lemma 2,

|C| =
∑

x∈S∩NG(S)

|Tx|+
∑

x∈S\NG(S)

|Tx|

≥ 2|S \NG(S)|+ |S ∩NG(S)|
≥ γt(G).

Suppose that γt(G) < 2|S \ NG(S)| + |S ∩ NG(S)|. Take a γt-set S∗ ⊆ V (G) of G,
and let u ∈ V (H). Then C∗ = S∗ × {u} is a semitotal dominating set in G[H] with
|C∗| = γt(G) < |C|, a contradiction. Thus

γt(G) = 2|S \NG(S)|+ |S ∩NG(S)|.

Suppose that |Tx| ≥ 2 for some x ∈ S ∩NG(S) or |Tx| ≥ 3 for some x ∈ S \NG(S). Let
{u, v} be a γ-set of H. By Theorem 6, C∗ = ((S \NG(S))× {u, v})∪((S ∩NG(S))× {u})
is a semitotal dominating set in G[H]. Moreover,

|C∗| = 2|S \NG(S)|+ |S ∩NG(S)| <
∑

x∈S\NG(S)

|Tx|+
∑

x∈S∩NG(S)

|Tx| = |C|,

a contradiction. Thus, |Tx| = 1 for all x ∈ S ∩NG(S) and |Tx| = 2, hence Tx is a γ-set of
H, for all x ∈ S \NG(S). In this case, (ii) holds.

Now, suppose that S is not a semitotal dominating set in G. By Theorem 6, S is a
dominating set in G such that Tx is a dominating set in H for each x ∈ S \ NG(S), and
|Tx| ≥ 2 for each x ∈ S \N2

G(S). Invoking Lemma 2 and the assumptions that γ(H) = 2,

|C| =
∑

x∈S\N2
G(S)

|Tx|+
∑

x∈S∩(N2
G(S)\NG(S))

|Tx|+
∑

x∈S∩NG(S)

|Tx|

≥ 2|S \N2
G(S)|+ 2|S ∩

(
N2
G(S) \NG(S)

)
|+ |S ∩NG(S)|

= 2|S \NG(S)|+ |S ∩NG(S)|
≥ γt(G).

As done previously, γt(G) = 2|S \ NG(S)| + |S ∩ NG(S)|. Suppose that |Tx| ≥ 2 for
some x ∈ S ∩ NG(S), |Tx| > 2 for some x ∈ S ∩

(
N2
G(S) \NG(S)

)
, or |Tx| > 2 for some

x ∈ S \N2
G(S). Let T = {u, v} ⊆ V (H) be a γ-set of H, and put

C∗ = ((S \NG(S))× T ) ∪ ((S ∩NG(S))× {u}) .

Then C∗ is a semitotal dominating set in G[H] by Theorem 6. However,

|C| > |C∗| = 2|S \NG(S)|+ |S ∩NG(S)| = γt(G),



I. S. Aniversario, S. R. Jr. Canoy, F.P. Jamil / Eur. J. Pure Appl. Math, 12 (4) (2019), 1410-1425 1422

a contradiction. In this case, (iii) holds.
To prove the converse, we only show the case where S satisfies (i). The other two

cases may follow similar arguments. By Theorem 6, C is a semitotal dominating set in
G[H]. Let C∗ = ∪x∈S∗ ({x} × Tx) ⊆ V (G[H]) be a γt2-set of G[H]. Then as shown in the
necessity part of the proof,

2|S∗ \NG(S∗)|+ |S∗ ∩NG(S∗)| = γt(G).

Thus, |C∗| ≥ γt(G) = |S| = |C|, and C is a γt2-set of G[H].

Corollary 10. Let G and H be nontrivial connected graphs with γ(H) ≥ 2, and let
C = ∪x∈S ({x} × Tx) ⊆ V (G[H]). Then C is a γt2-set of G[H]) if and only if S is a γt-set
of G and |Tx| = 1 for all x ∈ S.

Proof. Suppose that C is a γt2-set of G[H]. Suppose that S is not a total dominating
set in G. By Theorem 6 and Lemma 2

|C| ≥ 2|S \N2
G(S)|+ γ(H)|S ∩

(
N2
G(S) \NG(S)

)
|+ |S ∩NG(S)|

> 2|S \NG(S)|+ |S ∩NG(S)|
≥ γt(G).

Now, take any γt-set S∗ ⊆ V (G) of G and any u ∈ V (H). Then C∗ = S∗ × {u} is
a semitotal dominating set in G[H] by Theorem 6. Further, |C∗| = γt(G) < |C|, a
contradiction. Therefore, S is a total dominating set in G. In view of Theorem 6, since C
is a γt2-set, |S| = γt(G) and |Tx| = 1 for all x ∈ S.

At this point, the converse is a routine.

Combining Corollary 9 and Corollary 10 yields the following:

Corollary 11. Let G and H be nontrivial connected graphs with γ(H) ≥ 2. Then
γt2(G[H]) = γt(G).

Theorem 7. Let G be a nontrivial connected graph, and let n ≥ 2. Then C = ∪x∈S({x}×
Tx) ⊆ V (G[Kn]) is a secure semitotal dominating set in G[H] if and only if one of the
following holds:

(i) S is a secure semitotal dominating set in G.

(ii) S is a semitotal dominating set in G or S is a dominating set in G with |Tu| ≥ 2 for
all u ∈ S \N2

G(S). In any case, for each u ∈ V (G) \ S, there exists x ∈ S ∩NG(u)
such that either

(a) |Tx| ≥ 2 or

(b) |Tx| = 1, (S \{x})∪{u} is a dominating set in G, u ∈ N2
G(S \{x}) and |Tz| ≥ 2

for all z ∈ (S \ {x}) \N2
G(S \ {x}) ∪ {u}).



I. S. Aniversario, S. R. Jr. Canoy, F.P. Jamil / Eur. J. Pure Appl. Math, 12 (4) (2019), 1410-1425 1423

Proof. Suppose that C is a secure semitotal dominating set in G[Kn]. Then C is a
semitotal dominating set in G[Kn]. If S is a secure semitotal dominating set in G, then
(i) holds. Suppose that S is a not a secure semitotal dominating set in G. By Theorem
6, S is a semitotal dominating set in G or S is a dominating set in G with |Tu| ≥ 2 for all
u ∈ S \N2

G(S). Let u ∈ V (G) \S. Pick any v ∈ V (Kn). Then there exists (x, y) ∈ C such
that (x, y)(u, v) ∈ E(G[Kn]) and C∗ = (C \ {(x, y)}) ∪ {(u, v)} is a semitotal dominating
set in G[Kn]. Write C∗ = ∪a∈S∗({a}×T ∗a ). Then S∗ is a dominating set in G. If |Tx| ≥ 2,
then (ii)(a) holds. Suppose that Tx = {y}. Since C∗ is a semitotal dominating set,
S∗ = (S \ {x}) ∪ {u} is a dominating set in G, and since |T ∗u | = 1, u ∈ N2

G(S \ {x}). Let
z ∈ (S\{x})\N2

G(S∗). Suppose that |Tz| = 1, say Tz = {w}. Then dG[Kn]((z, w), (a, b)) > 2
for all (a, b) ∈ C∗ \ {(z, w)}, a contradiction. This shows that |Tz| ≥ 2, and (ii)(b) holds.

Conversely, suppose that (i) holds. Since S is a semitotal dominating set, C is a
semitotal dominating set in G[Kn] by Theorem 6. Let (u, v) ∈ V (G[Kn])\C. Suppose that
u ∈ S. Pick a ∈ Tu. Then (u, a) ∈ C and (u, v)(u, a) ∈ E(G[Kn]). Write (C \ {(u, a)}) ∪
{(u, v)} = ∪a∈S∗({a}×T ∗a ). Then S∗ = S and, therefore, S∗ is a semitotal dominating set
inG. Consequently, (C\{(u, a)})∪{(u, v)} is a semitotal dominating set inG[Kn]. Suppose
that u /∈ S. Since S is a secure semitotal dominating set, there exists x ∈ S ∩NG(u) such
that (S \ {x}) ∪ {u} is a semitotal dominating set in G. Pick y ∈ Tx. Then (x, y) ∈ C
and (x, y)(u, v) ∈ E(G[Kn]). Write (C \ {(x, y)}) ∪ {(u, v)} = ∪a∈S∗({a} × T ∗a ). Either
S∗ = S ∪ {u} or S∗ = (S \ {x}) ∪ {u}. In either case, S∗ is a semitotal dominating set in
G. Thus, (C \ {(x, y)}) ∪ {(u, v)} is a dominating set in G[Kn]. This shows that C is a
secure semitotal dominating set in G[Kn]. Suppose that (ii) holds. By Theorem 6, C is
a semitotal dominating set in G[Kn]. Let (u, v) ∈ V (G[Kn]) \ C, and let x ∈ S ∩ NG(u)
be such that |Tx| ≥ 2. Pick y ∈ Tx. Then (x, y) ∈ C and (x, y)(u, v) ∈ E(G[Kn]). Write
C∗ = (C \ {(x, y)}) ∪ {(u, v)} = ∪a∈S∗({a} × T ∗a ). Then S∗ = S ∪ {u}. If S is a semitotal
dominating set in G, then so is S∗ and consequently, C∗ is a semitotal dominating set
in G[Kn]. Suppose, on the other hand that S is a dominating set in G with |Tz| ≥ 2
for all z ∈ S \ N2

G(S). Since u ∈ N2
G(S∗) and S \ N2

G(S∗) ⊆ S \ N2
G(S), |T ∗z | ≥ 2

for all z ∈ S∗ \ N2
G(S∗). Thus, C∗ is a semitotal dominating set in G[Kn]. Now, let

x ∈ S ∩NG(u) be such that |Tx| = 1, S∗ = (S \ {x}) ∪ {u} is a dominating set in G, and
|Tz| ≥ 2 for all z ∈ (S \ {x}) \ N2

G(S∗). Pick y ∈ Tx. Then (x, y)(u, v) ∈ E(G[Kn]) and
C∗ = (C \{(x, y)}∪{(u, v)} = ∪a∈S∗({a}×T ∗a ). Since |T ∗z | ≥ 2 for all z ∈ S∗ \N2

G(S∗), C∗

is a dominating set by Theorem 6. Let (z, w) ∈ C∗. If z ∈ N2
G(S∗), then pick a ∈ S∗ \ {z}

such that dG(z, a) ≤ 2. For any b ∈ T ∗a , (a, b) ∈ C∗ and dG[Kn]((z, w), (a, b)) ≤ 2. Suppose
that z /∈ N2

G(S∗). Then z 6= u and |Tz| ≥ 2, say w, t ∈ Tz. Then (z, t), (z, w) ∈ C∗ and
dG[Kn]((z, w), (z, t)) ≤ 2. Thus C∗ is a semitotal dominating set in G[Kn].

Corollary 12. Let G be a nontrivial connected graph, and let n ≥ 2. Then γst2(G[Kn]) ≤
min{γst2(G), 2γt2(G)}.



REFERENCES 1424

Acknowledgements

This research is fully supported by the Office of the Vice Chancellor for Research and
Extension, MSU-Iigan Institute of Technology, Philippines, through the Premier Research
Institute of Science and Mathematics (PRISM).

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