Global outer connected domination number of a graph
In this paper we obtain some bounds for outer connected domination numbers and global outer connected domination numbers of graphs.
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irk-123456789-1883442023-02-24T01:27:24Z Global outer connected domination number of a graph Alishahi, M. Mojdeh, D.A. In this paper we obtain some bounds for outer connected domination numbers and global outer connected domination numbers of graphs. 2018 Article Global outer connected domination number of a graph / M. Alishahi, D.A. Mojdeh // Algebra and Discrete Mathematics. — 2018. — Vol. 25, № 1. — С. 18-26. — Бібліогр.: 10 назв. — англ. 1726-3255 2010 MSC: 05C69. http://dspace.nbuv.gov.ua/handle/123456789/188344 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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In this paper we obtain some bounds for outer connected domination numbers and global outer connected domination numbers of graphs. |
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Alishahi, M. Mojdeh, D.A. |
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Alishahi, M. Mojdeh, D.A. Global outer connected domination number of a graph Algebra and Discrete Mathematics |
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Alishahi, M. Mojdeh, D.A. |
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Alishahi, M. |
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Global outer connected domination number of a graph |
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Global outer connected domination number of a graph |
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Global outer connected domination number of a graph |
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Global outer connected domination number of a graph |
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Global outer connected domination number of a graph |
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global outer connected domination number of a graph |
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Global outer connected domination number of a graph / M. Alishahi, D.A. Mojdeh // Algebra and Discrete Mathematics. — 2018. — Vol. 25, № 1. — С. 18-26. — Бібліогр.: 10 назв. — англ. |
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Algebra and Discrete Mathematics |
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2025-07-16T10:21:53Z |
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2025-07-16T10:21:53Z |
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1837798589957406720 |
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“adm-n1” — 2018/4/2 — 12:46 — page 18 — #20
Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 25 (2018). Number 1, pp. 18–26
c© Journal “Algebra and Discrete Mathematics”
Global outer connected domination number
of a graph
Morteza Alishahi and Doost Ali Mojdeh∗
Communicated by D. Simson
Abstract. For a given graph G = (V,E), a dominating
set D ⊆ V (G) is said to be an outer connected dominating set if
D = V (G) or G−D is connected. The outer connected domination
number of a graph G, denoted by γ̃c(G), is the cardinality of a
minimum outer connected dominating set of G. A set S ⊆ V (G) is
said to be a global outer connected dominating set of a graph G if S
is an outer connected dominating set of G and G. The global outer
connected domination number of a graph G, denoted by γ̃gc(G), is
the cardinality of a minimum global outer connected dominating set
of G. In this paper we obtain some bounds for outer connected dom-
ination numbers and global outer connected domination numbers
of graphs. In particular, we show that for connected graph G 6= K1,
max{n− m+1
2
, 5n+2m−n
2
−2
4
} 6 γ̃gc(G) 6 min{m(G),m(G)}. Fi-
nally, under the conditions, we show the equality of global outer
connected domination numbers and outer connected domination
numbers for family of trees.
Introduction
In this paper the number of vertices of graph G denoted by n(G) and
the number of edges of graph G denoted by m(G). The neighborhood of
∗Corresponding author.
2010 MSC: 05C69.
Key words and phrases: global domination, outer connected domination, global
outer connected domination, trees.
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M. Alishahi, D. A. Mojdeh 19
vertex u ∈ V (G) is denoted by NG(u) = {v ∈ V (G) : uv ∈ E(G)}. The
degree of the vertex u ∈ V (G) denoted by dG(u) and dG(u) = |NG(u)|.
δ(G) = minu∈V (G) dG(u) and ∆(G) = maxu∈V (G) dG(u). u ∈ V (G) is an
end-vertex of G, if dG(u) = 1 and the vertex that is not an end-vertex
but is adjacent to an end-vertex is named a support vertex. For every
u ∈ V (G), delete all the end-vertices of N(u) except one, the remaining
graph is called the pruned of G and denoted by Gp.
The complement of a graph G, denoted by G, is a graph with |V (G)|
vertices and two vertices in G are adjacent if and only if they are not
adjacent in G. Every maximal connected subgraph of a graph G named
a component of G and the number of components of graph G denoted
by c(G). For connected graphs we have c(G) = 1. For every connected
graph G, a set S ⊆ V (G) is a vertex-cut if G− S is disconnected. If S is
a vertex-cut of connected graph G and G1 is a component of G− S, then
the subgraph 〈S ∪ V (G1)〉 named a S-lob of graph G. A subdivision of an
edge uv is obtained by inserting a new vertex w and replacing the edge
uv with the edges uw and wv. A Spider is a tree obtained from a star by
subdividing all of its edges. A wounded spider is a tree obtained from a
spider by removing at least one end-vertex. For more notation and graph
theory terminology not defined herein, we refer the reader to [10]. A set
S ⊆ V (G) is a dominating set of G if every vertex of V (G)−S is adjacent
to at least one vertex of S. The cardinality of a smallest dominating set of
G, denoted by γ(G), is called the domination number of G. A dominating
set of cardinality γ(G) is called a γ-set of G. A set S ⊆ V (G) is a global
dominating set of G if S is a dominating set of G and G. The cardinality
of a smallest global dominating set of G, denoted by γg(G), is called the
global domination number of G, for more see [1, 4, 5, 8, 9].
One of many applications of global domination have been given in [1],
which relates to a communication network modeled by a graph G, where
subnetworks are defined by some matching Mi of cardinality k. The neces-
sity of these subnetworks could be due for reason of security, redundancy or
limitation of recipients for different classes of messages. For this practical
case, the global domination number represents the minimum number of
master stations needed such that a message issued simultaneously from
all masters reaches all desired recipients after traveling over only one
communication link. We note that Carrington [2] gave two other appli-
cations of global dominating sets for graph partitioning commonly used
in the implementation of parallel algorithms. With these applications we
understand the value of global dominating set.
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20 Global outer connected domination number
A set S ⊆ V (G) is called an outer connected dominating set of G if S
is a dominating set of G and S = V (G) or 〈V (G)− S〉 is connected. The
cardinality of a smallest outer connected dominating set of G, denoted by
γ̃c(G), is called the outer connected domination number of G. An outer
connected dominating set of cardinality γ̃c(G) is called a γ̃c(G)-set of G.
For more details on outer domination see [3, 6]. We give a new definition
as follows:
Definition 1. A set S ⊆ V (G) is a global outer connected dominating set
of G if S is an outer connected dominating set of G and G. The cardinality
of a smallest global outer connected dominating set of G, denoted by
γ̃gc(G), is called the global outer connected domination number of G. A
global outer connected dominating set of cardinality γ̃gc(G) is called a
γ̃gc-set of G.
1. Bounds on (global) outer connected domination
In this section we find upper and lower bounds for global outer domi-
nation number of graphs. First, we state some results from [3].
Theorem 2. ([3], Theorem 5) If G is a connected graph, then γ̃c(G) 6
n(G)− δ(G).
The following observation has a straightforward proof.
Observation 3. The only connected graphs without any P4 as a subgraph,
are P1, C3 and K1,t, t > 1.
Theorem 4. Let G be a connected graph. If γ̃c(G) = n(G) − 1, then
G = K1,t, t > 1.
Proof. For G = K1,t, t > 1, we have γ̃c(G) = n(G)− 1. We show that for
every graph G 6= K1,t, γ̃c(G) 6= n(G)− 1. If G has a P4 as a subgraph like
u0u1u2u3, then the set V (G)−{u1, u2} is an outer connected dominating
set of G, so γ̃c(G) 6 n(G) − 2. For G = P1 and G = C3 we have
γ̃c(G) 6= n(G)− 1. By Observation 3 the desired result holds.
Theorem 5. If G 6= K1,t, t > 1 is a connected graph and S is a γ̃c(G)-set,
then S contains all of the pendant vertices.
Proof. Let u be an end-vertex of G and let v be its support vertex. If u /∈ S,
then v ∈ S and because of connectivity of V −S we have S = V (G)−{u},
so γ̃c(G) = n(G)− 1. By Theorem 4, G = K1,t, that is contradiction.
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M. Alishahi, D. A. Mojdeh 21
Theorem 6. If G is a connected graph and S is a γ̃c-set of G and Sp is
a γ̃c-set of Gp, then |V (G)− S| = |V (Gp)− Sp|.
Proof. Let P (G) and P (Gp) be the set of end-vertices (pendant vertices) of
G and Gp respectively. If G = K1,t, t > 1, then Gp = P2, so |V (G)−S| =
|V (Gp) − Sp| = 1. If G 6= K1,t, t > 1, then by Theorem 5 we have
P (G) ⊆ S and P (Gp) ⊆ Sp and 〈G−NG[P (G)]〉 = 〈Gp −NGp
[P (Gp)]〉,
hence |V (G)− S| = |V (Gp)− Sp|.
Theorem 7. If S is an outer connected dominating set of G and A ⊆ S
is a vertex-cut of G and c(G−A) = t, then S contains all the vertices of
t− 1 numbers of S-lobs.
Proof. Let B1, B2 be two S-lobs of G, u ∈ V (B1), v ∈ V (B2) and u, v ∈
V (G) − S. Then there does not exist any path between u and v in
V (G)− S.
Since global outer connected dominating set is an outer connected
dominating set, we have the following.
Observation 8. For any graph G, γ̃gc(G) > γ̃c(G) and γ̃gc(G) > n(G)
∆(G)+1
Observation 9. Let G 6= K1 be a graph. If G,G are connected, then
γ̃gc(G) 6 min{m(G),m(G)}.
Proof. For every vertex u ∈ V (G), the set V (G)−{u} is an outer connected
dominating set of G. Connectivity of G implies that |V (G) − {u}| =
n(G)− 1 6 m(G). Using this reasons for the complement graph G, the
desired result holds.
Theorem 10. Let G be a graph with n vertices and m edges. Then
γ̃gc(G) > max{n− m+1
2 , 5n+2m−n2
−2
4 }
Proof. Let S be a γ̃gc(G)-set. Since S is a dominating set of G and
〈V (G)−S〉 is connected we count the minimum edges. Since m > (n−|S|−
1)+ (n− |S|) = 2n− 2|S| − 1, hence we have |S| > n− m+1
2 . Furthermore
S is a dominating set of G, and 〈V (G)− S〉 is connected, so we have
|S| > n− m(G)+1
2 = n−
(
n(n−1)
2
−m)+1
2 , therefore |S| > 5n+2m−n2
−2
4 .
Theorem 11. For every graph G, γ̃gc(G) /∈ {n(G)− 2, n(G)− 3}
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22 Global outer connected domination number
Proof. Let S be a γ̃gc(G)-set. If |S| = n − 2, then V (G) − S = {x, y},
where x and y are adjacent in G, so they are not adjacent in G, that is a
contradiction. If |S| = n−3, then V (G)−S = {x, y, z}, where 〈V (G)−S〉
is P3 or C3 in G. In both of cases 〈V (G)− S〉 is disconnected in G, that
is also a contradiction.
Theorem 12. For every graph G, γ̃gc(G) = n(G) − 4 if and only if
〈V (G)− S〉 = P4.
Proof. Let S be a γ̃gc(G)-set with cardinality |S| = n−4. Let V (G)−S =
{x, y, z, t}. If 〈V (G) − S〉 has less than tree edges then 〈V (G) − S〉 is
disconnected. If 〈V (G)− S〉 has more than tree edges then 〈V (G)− S〉 is
disconnected. Therefore 〈V (G)− S〉 has tree edges, and 〈V (G)− S〉 is P4
or K1,3 or K3 + P1. If 〈V (G)− S〉 = K1,3 or K3 + P1, then 〈V (G)− S〉
or 〈V (G)− S〉 is disconnected, so 〈V (G)− S〉 = P4.
Theorem 13. Let G be a graph with at most 5 vertices. If G 6= Kn and
G 6= Kn, then γ̃gc(G) = n(G)− 1.
Proof. Let S be a γ̃gc(G)-set. Every global dominating set of G has at least
two vertices, so |V (G)−S| 6 3. By Theorem 11 we have γ̃gc(G) = n−1.
Theorem 14. Let G 6= K6 and G 6= K6. If n(G) = 6 and m(G) 6= 7, 8,
then γ̃gc(G) = 5.
Proof. Let S be a γ̃gc(G)-set. Since S is a global dominating set of G we
have |S| > 2, so |V (G)−S| 6 4, by Theorem 11, we have |V (G)−S| = 1 or
|V (G)−S| = 4. Let |V (G)−S| = 4 and S = {x, y}. Then by Theorem 12,
〈V (G)− S〉 = P4 and every vertex of V (G)− S is adjacent to exact one
of the vertices x or y. Hence the number of edges of G equals to 8 if x and
y are adjacent, and equals to 7 if not adjacent, therefore γ̃gc(G) = 5.
2. Global outer connected domination number of trees
In this section we study the global outer connected dominating set of
Trees. We start by a theorem from [3].
Theorem 15. ([3], Theorem 6) If T is a tree and n(T ) > 3, then γ̃c(T ) >
∆(T ). Furthermore γ̃c(T ) = ∆(T ) if and only if T is a wounded spider.
Theorem 16. If G is a tree, then γ̃c(Gp) 6 n(Gp)−∆(Gp).
“adm-n1” — 2018/4/2 — 12:46 — page 23 — #25
M. Alishahi, D. A. Mojdeh 23
Proof. The inequality holds for graphs without vertices. Let G has at least
on edge. Let dGp
(u) = ∆(Gp). If u is a support vertex and v ∈ NGp
(u) is
an end-vertex of Gp, then consider S = V (Gp)−NGp
[u] ∪ {v}. If u is not
a support vertex, then consider S = V (Gp)−NGp
[u] ∪ {w}, that w is an
arbitrary vertex of NGp
(u). The set S is an outer connected dominating
set of Gp with cardinality n(Gp)−∆(Gp).
Theorem 17. ([3], Theorem 7) If T is a tree of order n(T ) > 3, then
γ̃c(T ) > ⌈n2 ⌉.
Theorem 18. Let T be a tree, S be a γ̃c(T )-set and |S| 6 n(T )−2. Then
S is a dominating set of T .
Proof. Since |S| 6 n(T )− 2, so V (T )− S has at least two vertices. Let
u be a vertex of V (T )− S which be adjacent to all vertices of S. Every
vertex v ∈ V (T )− S − {u} is adjacent to at least one vertex of S like w.
Since 〈V (G)− S〉 is connected there exists a path between vertices u and
v in 〈V (T )− S〉 like P . The path P together with the edges uw and vw
forms a cycle in T , that is a contradiction.
It is well known that if diam(G) > 3, then diam(G) 6 3, and therefore
G is connected. Now we have the following.
Theorem 19. Let T be a tree and S be an outer connected dominating
set of T . If diam(〈V (T ) − S〉) > 3, then S is a global outer connected
dominating set of T .
Proof. By Theorem 18, the result holds.
Lemma 20. Let T be a tree. Then T is disconnected if and only if
T = K1,t, t > 1.
Proof. It is clear that K1,t is disconnected. If T = P1 then T is connected.
If T 6= P1,K1,t, then diam(T ) > 3, so T is connected.
3. Trees T with γ̃gc(T ) = γ̃c(T )
In this section we verify the conditions that for a family trees tree T ,
γ̃gc(T ) = γ̃c(T ).
As an immediate result from Lemma 20 we have.
Corollary 21. Let T be a tree and S be a γ̃c(T )-set. If 〈V (T ) − S〉 6=
K1, K1,t, then γ̃gc(T ) = γ̃c(T ).
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24 Global outer connected domination number
Theorem 22. Let T be a tree. If γ̃c(T ) < n−∆(T ), then γ̃gc(T ) = γ̃c(T ).
Proof. Let S be a γ̃c(T )-set and |S| < n − ∆(G). Then |V (T ) − S| >
∆(T ). If 〈V (T ) − S〉 = K1,t, and u ∈ V (T ) − S, d(u) = t, then since
N(u) ∩ S 6= ∅, we have |V (T )− S| 6 d(u) 6 ∆(G), that is contradiction,
therefore 〈V (T ) − S〉 is not K1 or K1,t. Now Corollary 21 implies that
γ̃gc(T ) = γ̃c(T ).
Theorem 23. Let T be a tree. If there exist two adjacent vertices u, v ∈
V (Tp) such that dTp
(u) > 3, dTp
(v) > 3 and dTp
(u) + dTp
(v) > ∆(Tp) + 2,
then γ̃gc(T ) = γ̃c(T ).
Proof. If u is a support vertex, let u1 ∈ NTp
(u) be the end-vertex in
Tp and if u is not a support vertex, let u1 be an arbitrary vertex of
NTp
(u)− {v}. Consider the vertex v1 with the same definition about u1.
The set S = V (Tp)− (NTp
(u) ∪NTp
(v)) ∪ {u1, v1} is an outer connected
dominating set of Tp such that diam(〈V (Tp) − S〉) = 3 and |V (Tp) −
S| > dTp
(u) + dTp
(v)− 2 > ∆(T ), therefore |S| < n−∆(T ). Now using
Theorem 22 we have γ̃gc(T ) = γ̃c(T ).
Theorem 24. Let T be a pruned tree. If T has a subtree as path Pk =
u1u2 . . . uk, k > 4 such that d(u1) > 2 , d(uk) > 2, d(ui) > 3, i =
2, 3, . . . , k − 1, and Σk
i=1d(ui)− 2(k − 1) > ∆(T ), then γ̃gc(T ) = γ̃c(T ).
Proof. Let U = {u1, u2, . . . , uk} and for i ∈ {1, 2, . . . , k} let Tui
be the
maximal subtree of T containing ui and not containing any vertex in
U −{ui}. Let Si be a subset of V (Tui
) as follows. If ui is a support vertex
and ti is the leaf adjacent to ui, then ti ∈ Si, else if ui is not a support
vertex and ti is an arbitrary vertex of N(ui) in Tui
, then ti ∈ Si. Let Si
contains all the vertices of Tui
such that they are at distance at least two
from ui. Let S = ∪k
i=1Si. The set S is an outer connected dominating set of
T . Since |V (T )−S| = Σk
i=1d(ui)−2(k−1), so |S| < n(T )−∆(T ), therefore
γ̃c(T ) < n(T )−∆(T ). By Theorem 22 we have γ̃gc(T ) = γ̃c(T ).
The converse of the Theorem 24 is not true. For counterexample
consider Figure 1. The set V (T ) − {a, b, c, d} is a γ̃gc-set and the sets
V (T ) − {a, b, c, d} and V (T ) − {e, f, g, h} are γ̃c-sets, so γ̃gc(T ) = γ̃c(T )
but there is not any path with the properties mentioned in theorem.
We have the following result from [7].
Theorem 25. ([7] Theorem) Let G be a graph with diam(G) > 5 and
δ(G) > 3, then γ̃gc(G) 6 n(G)− diam(G)− 1.
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M. Alishahi, D. A. Mojdeh 25
❡ ❡ ❡
❡
❡
❡
❡ ❡ ❡
❡
❡
❡
❡ ❡
a b c d e f g
h
Figure 1.
Theorem 26. Let T be a tree and P (T ) be the set of leaves of T . If
diam(T ) > 5 and d(u) > 3, for every u ∈ V (T ) − P (T ), then γ̃gc(T ) 6
n(G)− diam(T ) + 1.
Proof. Let k = diam(T ) and d(x, y) = k and P = xu0u1 . . . uk−2y be
the longest path in T . It is clear that d(x) = d(y) = 1 and d(ui) > 3,
i = 0, 1, . . . , k−2. We show that S = V (T )−{u0, u1, . . . , uk−2} is a global
outer connected dominating set of size n(T )−diam(T )+ 1. Since P is the
shortest path between x and y, and d(ui) > 3, i = 0, 1, . . . , k− 2, so S is a
dominating set of T . Since diam(T ) > 5 so V (T )−S has at least 4 vertices,
by Theorem 18, S is a dominating set of T . Since 〈V (T )− S〉 6= Pt, t > 4,
so 〈V (T )− S〉 and 〈V (T )− S〉 are connected.
Conclusion
A graph G is said to be a unicyclic graph, if it is connected and has
one and only one cycle. For example any cycle is a unicyclic graph. Or if
we add an edge to a tree we obtain a unicyclic graph. We now make a
future research works as follows.
Let F be a family of unicyclic graphs. Does there exist some conditions
such that, we have the equality of outer connected domination numbers
and the global outer connected domination numbers for this family?
Let G be a 2-cyclic graph, that is G has exact two cycles. If we delete
an edge from one of cycles, then the new graph will be changed to a
unicyclic graph. Do we extend the result on unicyclic graphs to 2-cyclic
graphs?
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26 Global outer connected domination number
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Contact information
M. Alishahi Department of Mathematics, University of
Tafresh, Tafresh, Iran
E-Mail(s): morteza.alishahi@gmail.com
Web-page(s): www.tafreshu.ac.ir
D. A. Mojdeh Department of Mathematics, University of
Mazandaran, Babolsar, Iran
E-Mail(s): damojdeh@umz.ac.ir
Web-page(s): www.umz.ac.ir
Received by the editors: 11.12.2015.
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