Some Parameters of Domination on the Neighborhood Graph

2017·0.2 MB·

by  M. H. Akhbari

International J.Math. Combin. Vol.4(2017), 138-150
Some Parameters of Domination on the Neighborhood Graph
M. H. Akhbari
(InstituteofMathematicsandMechanics,KazanFederalUniversity,18KremlyovskayaStr.,Kazan420008,Russia)
F. Movahedi
(DepartmentofMathematics,GolestanUniversity,Gorgan,Iran)
S. V. R. Kulli
(DepartmentofMathematics,GulbargaUniversity,Gulbarga585106,India)
E-mail: [email protected],[email protected],[email protected]
Abstract: Let G= (V,E) be a simple graph. The neighborhood graph N(G) of a graph
G is the graph with the vertex set V ∪S where S is the set of all open neighborhood sets
ofGandwith verticesu,v∈V(N(G))adjacent ifu∈V andv isan openneighborhood set
containingu. Inthispaper,weobtainthedominationnumber,thetotaldominationnumber
and the independent domination number in the neighborhood graph. We also investigate
these parameters of domination on thejoin and thecorona of two neighborhood graphs.
Key Words: Neighborhoodgraph,dominationnumber,Smarandachelydominatingk-set,
total domination, independentdomination, join graph, corona graph.
AMS(2010): 05C69, 05C72.
§1. Introduction
Let G=(V,E) be a simple graph with V(G) =n vertices and E(G) =m edges. The neigh-
| | | |
borhood of a vertex u is denoted by N (u) and its degree N (u) by deg (u). The minimum
G G G
| |
and maximum degree of a graph G are denoted by δ =δ(G) and ∆=∆(G), respectively. The
open neighborhood of a set S G is the set N(S) = N(v), and the closed neighbor-
⊆ v V(G)
∈
hood of S is the set N[S]= N(S) S. A cut-vertex of a graph G is any vertex u V(G) for
∪ S ∈
whichinducedsubgraphG u hasmorecomponentsthanG. Avertexwithdegree1iscalled
\{ }
an end-vertex [1].
Adominatingsetis asetD ofverticesofGsuchthateveryvertexoutside D isdominated
by somevertexofD. The dominationnumber ofG, denotedbyγ(G), is the minimumsize ofa
dominating set of G, and generally, a vertex set Dk of G is a Smarandachely dominating k-set
S
if each vertex of G is dominated by at least k vertices of S. Clearly, if k =1, such a set Dk is
S
nothing else but a dominating set of G. A dominating set D is a total dominating set of G if
everyvertexofthe graphis adjacentto atleastone vertexin D. The totaldominationnumber
of G, denoted by γ (G) is the minimum size of a total dominating set of G. A dominating
t
1ReceivedDecember26,2016,Accepted November27,2017.
SomeParametersofDominationontheNeighborhoodGraph 139
set D is called an independent dominating set if D is an independent set. The independent
domination number of G denoted by γ (G) is the minimum size of an independent dominating
i
set of G [1].
The join of two graphs G and G , denoted by G +G , is the graph with vertex set
1 2 1 2
V(G ) V(G ) and edge set E(G ) E(G ) uv u V(G )andv V(G ) . The corona
1 2 1 2 1 2
∪ ∪ ∪{ | ∈ ∈ }
of two graphs G and G is the graph G = G G formed from one copy of G and V(G )
1 2 1 2 1 1
◦ | |
copies of G , where the ith vertex of G is adjacent to every vertex in the ith copy of G . For
2 1 2
every v V(G ), Gv is the copy of G whose vertices are attached one by one to the vertex
∈ 1 2 2
v. The coronaG K , in particular,is the graphconstructed from a copy of G, where for each
1
◦
vertex v V(G), a new vertex v and a pendant edge vv are added [2].
′ ′
∈
We use K , C and P to denote a complete graph, a cycle and a path of the order n,
n n n
respectively. A complete bipartite graphdenotes by K and the graphK of order n+1 is
m,n 1,n
a star graph with one vertex of degree n and n end-vertices.
The neighborhoodgraphN(G) of a graph G is the graphwith the vertex set V S where
∪
S is the set of all open neighborhood sets of G and two vertices u and v in N(G) are adjacent
if u V and v is an open neighborhood set containing u. In Figure 1, a graph G and its
∈
neighborhood graph are shown. The open neighborhood sets in graph G are N(1)= 2,3,4 ,
{ }
N(2)= 1,3 ,N(3)= 1,2 and N(4)= 1 [3].
{ } { } { }
Figure 1 The graph G and the neighborhood graph of G.
In this paper, we determine the domination number, total domination number and inde-
pendent domination number for the neighborhood graph of a graph G. Also, we consider the
join graph and the corona graph of two neighborhoodgraphs and investigate some parameters
of domination of these graphs.
§2. Lemma and Preliminaries
In the text follows we recall some results that establish the domination number, the total
domination number and the independent domination number for graphs, that are of interest
140 M.H.Akhbari,F.MovahediandS.V.R.Kulli
for our work.
Lemma 2.1([3]) If G be a graph without isolated vertex of order n and the size of m, then
N(G) is a bipartite graph with 2n vertices and 2m edges.
Lemma 2.2([3]) If T be a tree with n 2, then N(T)=2T.
≥
Lemma 2.3 ([3]) For a cycle C with n 3 vertices,
n
≥
2C if n is even,
n
N(C )=
n 
C if n is odd.
 2n

Lemma 2.4 ([3])
(i) For 1 m n, N(K )=2K ;
m,n m,n
≤ ≤
(ii) For n 1, N(K¯ )=K¯ ;
n n
≥
(iii) A graph G is a r-regular if and only if N(G) is a r-regular graph.
Lemma 2.5 ([1]) Let γ(G) be the domination number of a graph G, then
(i) For n 3, γ(C )=γ(P )= n ;
≥ n n ⌈3⌉
(ii) γ(K )=γ(K )=1;
n 1,n
(iii) γ(K )=2;
m,n
(iv) γ(K¯ )=n.
n
Lemma 2.6 ([4]) If T be a tree of order n and l end-vertices, then
n l+2
γ(T) − .
≥ 3
Lemma 2.7 ([5]) Let G be a r-regular graph of order n. Then
n
γ(G) .
≥ r+1
Lemma 2.8 ([6]) Let γ be the total domination number of G. Then
t
(i) γ (K )=γ (K )=2;
t n t n,m
n if n 0 (mod4),
2 ≡
(ii) γt(Pn)=γt(Cn)= nn++221 iofthner≡wi2se(.mod4),
2
(iiii) Let T be a nontrivial tree of order n and l end-vertices, then
n l+2
γ (T) − ;
t
≥ 2
(iv) Let G be a graph , then γt(G)≥1+ |C2|, where C is the set of cut-vertices of G.
SomeParametersofDominationontheNeighborhoodGraph 141
Lemma 2.9 ([7]) Let γ be the independent domination number of G. Then
i
(i) γ (P )=γ (C )= n ;
i n i n 3
(ii) γi(Kn,m)=min n(cid:6),m(cid:7) ;
{ }
(iii) For a graph G with n vertices and the maximum degree ∆,
n
γ (G) n ∆.
i
1+∆ ≤ ≤ −
l m
(iv) If G is a bipartite graph of order n without isolated vertex, then
n
γ (G) ;
i
≤ 2
(v) For any tree T with n vertices and l end-vertices,
n+l
γ (T) .
i
≤ 3
Lemma 2.10 ([8]) For any graph G, χ(G) (G)+1 where χ(G) is the chromatic number
≤ △
of G.
Lemma 2.11 ([9]) For any graph G, κ(G) δ(G), where κ(G) is the connectivity of G.
≤
§3. The Domination Number, the Total Domination Number and the
Independent Domination Number on N(G)
In this section, we propose the obtained results of some parameters of domination on a neigh-
borhood graph.
Theorem 3.1 Let the neighborhood graph of G be N(G), then
(i) γ(N(P ))=2 n ;
n 3
(cid:6) (cid:7)2 n if n is even,
(ii) γ(N(C ))= ⌈3⌉
n  2n if n is odd.
 3
(iii) γ(N(K1,n))=γ((cid:6)N((cid:7)Kn))=2;

(iv) For 2 n m, γ(N(K ))=4;
n,m
≤ ≤
(v) For n 2, γ(N(K¯ ))=2n.
n
≥
Proof (i) Using Lemma 2.2, for n 2, N(P ) = 2P . So, it is sufficient to consider a
n n
≥
dominating set of P . By Lemma 2.5(i), γ(P )= n . Therefore,
n n 3
(cid:6) (cid:7) n
γ(N(P ))=2γ(P )=2 .
n n
3
(cid:6) (cid:7)
(ii) If n is even then by Lemma 2.3, N(C )=2C . So, we consider a cycle C of order n
n n n
142 M.H.Akhbari,F.MovahediandS.V.R.Kulli
and using Lemma 2.5(i), we have
n
γ(N(C ))=2γ(C )=2 .
n n
3
(cid:6) (cid:7)
If n is odd, then since N(C ) is a cycle of order 2n so, γ(N(C ))=γ(C )= 2n .
n n 2n 3
The segments on (iii), (iv) and (v) can be obtained similarly by apply(cid:6)ing(cid:7)Lemma 2.1,
Lemma 2.4 and Lemma 2.5.
2
Theorem 3.2 Let T be a tree of order n with l end-vertices. Then
2
(n l+2) γ(N(T)) n.
3 − ≤ ≤
Proof Using Lemma 2.2, for every tree T, N(T) = 2T. So, we consider a tree T to
investigate its domination number. Thus, by Lemma 2.6, for every tree T of order n with l
end-vertices,
n l+2
γ(T) − .
≥ 3
Therefore,
n l+2
γ(N(T))=2γ(T) 2 − .
≥ 3
SinceT iswithoutisolatedverticesso,N(T)isa(cid:0)graphwith(cid:1)outanyisolatedvertex. There-
fore, V(T) V(N(T)) is a dominating set of N(T). Thus, γ(N(T)) n. It completes the
⊆ ≤
result.
2
Theorem 3.3 Let G be a r-regular graph. Then,
2n
γ(N(G)) .
≥ r+1
Proof Using Lemma 2.5(iii), since G is an r-regular graph so, N(G) is a r-regular graph
too. According to Lemma 2.1 and Lemma 2.7, we have
2n
γ(N(G)) .
≥ r+1 2
Theorem 3.4 Let N(G) be a neighborhood graph of G. Then for every vertex x V(G),
∈
deg (x) is equal with deg (x).
G N(G)
Proof Assume x V(G) and deg (x) = k. So, the neighborhood set of x is N(x) =
G
∈
y , ,y where y V(G). In graph N(G), x is adjacent to a vertex such as N(u) that
1 k i
{ ··· } ∈
consists x. Then, x is adjacent to N(y ) for every 1 i k. Thus, degree of x is k in N(G).
i
≤ ≤
Therefore, deg (x)=deg (x).
G N(G)
2
Theorem 3.5 Let γ(N(G)) be the domination number of N(G). For any graph G of order n
SomeParametersofDominationontheNeighborhoodGraph 143
with the maximum degree ∆(G),
2n
γ(N(G)) 2n ∆(G).
1+∆(G) ≤ ≤ −
l m
Proof Let D be a dominating set of N(G). Each vertex of D can dominate at most itself
and ∆(N(G)) other vertices. Since by Theorem 3.4, ∆(N(G))=∆(G) so,
2n
γ(N(G))= D .
| |≥ 1+∆(G)
l m
Now,letv be avertexwiththe maximumdegree∆(N(G)) andN[v]be a closedneighbor-
hood set of v in N(G). Then v dominates N[v] and the vertices in V(N(G)) N[v] dominate
\
themselves.
Hence, V(N(G)) N[v] is the dominating set of cardinality 2n ∆(N(G)). So,
\ −
γ(N(G)) 2n ∆(N(G))=2n ∆(G).
≤ − − 2
WeestablisharelationbetweenthedominationnumberofN(G)andthechromaticnumber
χ(G) of the graph G.
Theorem 3.6 For any graph G,
γ(N(G))+χ(G) 2n+1.
≤
Proof By Theorem 3.5, γ(N(G)) 2n ∆(G) and by Lemma 2.10, χ(G) ∆(G)+1.
≤ − ≤
Thus,
γ(N(G))+χ(G) 2n+1.
≤ 2
We obtain a relation between the domination number of N(G) and the connectivity κ(G)
of G following.
Theorem 3.7 For any graph G,
γ(N(G))+κ(G) 2n.
≤
Proof By Theorem 3.5, γ(N(G)) 2n ∆(G) and by Lemma 2.11, κ(G) δ(G).
≤ − ≤
Therefore,
γ(N(G))+κ(G) 2n ∆(G)+δ(G),
≤ −
since, δ(G) ∆(G) so,
≤
γ(N(G))+κ(G) 2n.
≤ 2
The following theorem is an easy consequence of the definition of N(G), Lemmas 2.2–2.4
144 M.H.Akhbari,F.MovahediandS.V.R.Kulli
and Lemma 2.8.
Theorem 3.8 Let the neighborhood graph of G be N(G) and γ (N(G)) be the total domination
t
number of N(G). Then
n if n 0 (mod4),
≡
(i) γt(N(Pn))=γt(N(Cn))= n+2 if n 2 (mod4),
 n+1 other≡wise,
(ii) For every n,m≥1, γt(N(Km,n))=4;
(iii) For n 2, γ (N(K ))=4.
t n
≥
Theorem 3.9 Let G be a graph of order n without isolated vertices and with the maximum
degree ∆. Then,
2n
γ (N(G)) .
t
≥ ∆
Proof Let D be a total dominating set of N(G). Then, every vertex of V(N(G)) is
adjacent to some vertices of D. Since, every v D can have at most ∆(N(G)) neighborhood,
∈
it followsthat∆(N(G))γ (N(G)) V(N(G)) =2n. By Theorem3.4,∆(N(G))=∆(G)=∆
t
≥| |
so, ∆γ (N(G)) 2n. Therefore,
t
≥ 2n
γ (N(G)) .
t
≥ ∆ 2
Theorem 3.10 Let T be a nontrivial tree of order n and l end-vertices. Then,
γ (N(T)) n+2 l.
t
≥ −
Proof Using Lemma 2.2, N(T)=2T and so, γ (N(T))=2γ (T). By Lemma 2.8(iv),
t t
n+2 l
γ (T) − .
t
≥ 2
Therefore,
n+2 l
γ (N(T))=2γ (T) 2 − =n+2 l.
t t
≥ 2 − 2
(cid:16) (cid:17)
Theorem 3.11 Let G be a graph with x cut-vertices. Then,
γ (N(G)) 1+x.
t
≥
Proof Let C be the set of cut-vertices of N(G). Since for every cut-vertex u of G, u and
N(u) are both cut-vertices in N(G) so, |C| = 2x. By Lemma 2.8(iv), γt(N(G)) ≥ 1+ |C2|.
Therefore, we have
C 2x
γ (N(G)) 1+ | | =1+ =1+x.
t
≥ 2 2 2
Theorem 3.12 Let γ (G) be the independent domination number of G. Then
i
SomeParametersofDominationontheNeighborhoodGraph 145
(i) γ (N(K ))=2m;
i n,m
(ii) γ (N(K ))=2;
i 1,n
(iii) γ (N(K¯ ))=2n;
i n
(iv) γ (N(P ))=2 n ;
i n ⌈3⌉
2 n if n is even,
(v) γ (N(C ))= 3
i n  (cid:6)2n(cid:7) if n is odd.
 3
(cid:6) (cid:7)
Proof The theorem easily proves using Lemma 2.3, Lemma 2.4(i,ii), Lemma 2.5 and
Lemma 2.9(i,ii).
2
Theorem 3.13 For a graph G with n vertices and the maximum degree ∆,
2n
γ (N(G)) 2n ∆.
i
1+∆ ≤ ≤ −
l m
Proof It is easy to see that N(G) is a graphof order 2n and the maximum degree ∆. So,
using Lemma 2.9(iii) we have the result.
2
We establish a relation between the independent domination number of N(G) and the
chromatic number χ(G) of G.
Theorem 3.14 For any graph G,
γ (N(G))+χ(G) 2n+1.
i
≤
Proof By Theorem 3.13, γ (N(G)) 2n ∆(G) and by Lemma 2.10, χ(G) ∆(G)+1.
i
≤ − ≤
So,
γ (N(G))+χ(G) 2n+1.
i
≤ 2
ThefollowingtheoremistherelationbetweentheindependentdominationnumberofN(G)
and the connectivity κ(G) of G.
Theorem 3.15 For any graph G,
γ (N(G))+κ(G) 2n.
i
≤
Proof By Theorem 3.13, γ (N(G)) 2n ∆(G) and by Lemma 2.11, κ(G) δ(G). So,
i
≤ − ≤
γ (N(G))+κ(G) 2n ∆(G)+δ(G) 2n.
i
≤ − ≤ 2
Theorem 3.16 Let G be a simple graph of order n and without any isolated vertex. Then
γ (N(G)) n.
i
≤
146 M.H.Akhbari,F.MovahediandS.V.R.Kulli
Proof For every graph G with n vertices, N(G) is a bipartite graph of order 2n. Since G
doesn’t haveany isolatedvertex so,N(G) is a graphwithout isolatedvertex. Thus,by Lemma
2.9(iv) we have
2n
γ (N(G)) =n.
i
≤ 2 2
Theorem 3.17 Let T be a tree with n vertices and l end-vertices without isolated vertices.
Then
2
γ (N(T)) (n+2l).
i
≤ 3
Proof ForeverytreeT,N(T)=2T. Letvbeanend-vertexofG. Then,thecorresponding
vertices of v and N(v) are end-vertices in N(G). Thus, if T has l end-vertices then 2l end-
vertices are in N(T). So, by Lemma 2.9(v) we have
n+2l
γ (N(T))=2γ (T) 2 .
i i
≤ 3 2
(cid:16) (cid:17)
§4. The Results of the Combination of Neighborhood Graphs
In this section, we consider two graphs G and G and study the join and the corona of their
1 2
neighborhoodgraphsintwocases. InSection4.1,weconsidertwocasesforthejoinofgraphs: i)
theneighborhoodgraphofG +G thatdenotesbyN(G +G ),ii)thejoinoftwographsN(G )
1 2 1 2 1
and N(G ). So, the domination number, the total domination number and the independent
2
domination number of these graphs are obtained. In Section 4.2, we study the domination
number, the total domination number and the independent domination number on two cases
of the corona graphs: i) N(G G ) and ii) N(G ) N(G ).
1 2 1 2
◦ ◦
4.1 The Join of Neighborhood Graphs
LetG be a simple graphofordern with m edges andG be asimple graphwithn vertices
1 1 1 2 2
and m edges. By the definition of the join of two graphs, G +G has n +n vertices and
2 1 2 1 2
m +m +m m edges. So, the neighborhood graph of G +G has 2(n +n ) vertices and
1 2 1 2 1 2 1 2
2m edges where m = m +m +m m . For every x V(G +G ) that x V(G ), we have
1 2 1 2 1 2 1
∈ ∈
deg (x) = deg (x)+n . Also, if y V(G +G ) and y V(G ) then deg (y) =
G1+G2 G1 2 ∈ 1 2 ∈ 2 G1+G2
deg (y)+n . On the other hand, using Theorem 3.4 we know that deg (x) = deg (x).
G2 1 G N(G)
So,deg (x)=deg (x). Thus,ifx V(G ),thendeg (x)=deg (x)+n
G1+G2 N(G1+G2) ∈ 1 N(G1+G2) G1 2
and if y V(G ) then deg (y)=deg (y)+n .
∈ 2 N(G1+G2) G2 1
Now,letG andG besimplegraphswithoutanyisolatedvertex. Thus,thejoinofN(G )
1 2 1
and N(G ) denotes N(G )+N(G ) of order 2(n +n ). Also, N(G +G ) has 2m +2m +
2 1 2 1 2 1 2 1 2
4m m edges. Therefore,E(N(G )+N(G ))=E(N(G +G ))+2m m . Also,wecanobtain
1 2 1 2 1 2 1 2
for every x V(N(G )), deg (x) = deg (x)+2n and for every y V(G ),
∈ 1 N(G1)+N(G2) N(G1) 2 ∈ 2
deg (y)=deg (y)+2n .
N(G1)+N(G2) N(G2) 1
SomeParametersofDominationontheNeighborhoodGraph 147
Theorem 4.1 Let G and G be simple graphs without isolated vertex. If order of G is n
1 2 1 1
and ∆(G ) n 1, then
1 1
≥ −
γ(N(G +G ))=γ (N(G +G ))=2.
1 2 i 1 2
Proof Letx V(G )beavertexwiththemaximumdegreeatleastn 1. So,xdominates
1 1
∈ −
n 1 vertices of G . Let D = x,N (x) and N (x) be the open neighborhood set
1− 1 { G1+G2 } G1+G2
ofx inG +G . Since, everyvertexofG is adjacentto allofverticesofG inG +G so,the
1 2 1 2 1 2
degree of x in G +G is n +n 1 and x dominates n +n 1 in N(G +G ). Similarity,
1 2 1 2 1 2 1 2
− −
N (x) dominates n +n 1 vertices of N(G +G ). So, γ(N(G +G ))= D =2.
G1+G2 1 2− 1 2 1 2 | |
Since, x and N (x) are not adjacent in N(G +G ). Thus, D is an independent
G1+G2 1 2
dominating set in N(G +G ). Therefore, γ (N(G +G ))=2.
1 2 i 1 2
2
Theorem 4.2 Let G and G be simple graphs without isolated vertices. Then
1 2
2 γ(N(G +G )) 4.
1 2
≤ ≤
Proof It is clearly to obtain γ(N(G +G )) 2. Let S = x,N (x),y,N (y)
1 2 ≥ { G1+G2 G1+G2 }
where x V(G ) and y V(G ). Then, x dominates all of vertices of G in G +G and so,
1 2 2 1 2
∈ ∈
all of vertices of N(G +G ) that are the corresponding set to the neighborhoods of V(G ).
1 2 2
Similarity,y V(G )dominates n verticesofN(G +G ). It isshownthatS is a dominating
2 1 1 2
∈
set of N(G +G ). Therefore, the result holds.
1 2
2
Theorem 4.3 For graphs G and G ,
1 2
γ (N(G +G ))=4.
t 1 2
Proof Assume S = x,N (x),y,N (y) where x V(G ) and y V(G ). The
{ G1+G2 G1+G2 } ∈ 1 ∈ 2
vertex of N (x) in N(G +G ) is the corresponding vertex to the neighborhood of x in
G1+G2 1 2
G . So, x dominates all ofthe vertices of G and y dominates all of verticesof G . It is clearly
1 1 2
to see that x is adjacent to N (y) and y is adjacent to N (x). Therefore,S is a total
G1+G2 G1+G2
dominating set of N(G +G ) and we have γ (N(G +G )) S =4.
1 2 t 1 2
≤| |
Let D be a total dominating set of N(G + G ) that D 3. We can assume that
1 2
| | ≤
D = x,y,z . Thus, we have the following cases.
{ }
Case 1. If x,y,z V(G +G ), then since V(N(G +G )) = V(G +G ) S so, all of the
1 2 1 2 1 2
∈ ∪
vertices S are dominated by D where S is the set of all open neighborhood sets of G +G .
1 2
But, each of vertices of V(G +G ) in V(N(G +G )) is not dominated by D. Thus, it is a
1 2 1 2
contradiction.
Case2. LetoneofverticesofDbeinV(G +G )andremainedverticesbeinS ofN(G +G ).
1 2 1 2
Without loss of generality suppose that x V(G ). So, x V(G +G ) and y,z S. since x
1 1 2
∈ ∈ ∈