Accurate Independent Domination in Graphs

2018·0.15 MB·

by  B. Basavanagoud

International J.Math. Combin. Vol.2(2018), 87-96
Accurate Independent Domination in Graphs
B.Basavanagoud
(DepartmentofMathematics,KarnatakUniversity,Dharwad-580003,India)
Sujata Timmanaikar
(DepartmentofMathematics,GovernmentEngineeringCollege,Haveri-581110,India)
E-mail: [email protected],[email protected]
Abstract: Adominating set D of agraph G=(V,E)is an independent dominating set, if
theinducedsubgraphhDihasnoedges. AnindependentdominatingsetDofGisanaccurate
independent dominating set if V −D has noindependent dominating set of cardinality |D|.
The accurate independent domination number i (G) of G is the minimum cardinality of an
a
accurate independent dominating set of G. In this paper, we initiate a study of this new
parameter and obtain some results concerning this parameter.
Key Words: Domination, independent domination number, accurate independent domi-
nation number,Smarandache H-dominatingset.
AMS(2010): 05C69.
§1. Introduction
All graphs considered here are finite, nontrivial, undirected with no loops and multiple edges.
For graph theoretic terminology we refer to Harary [1].
LetG=(V,E)beagraphwith V =pand E =q. Let∆(G)(δ(G))denotethemaximum
| | | |
(minimum)degreeand x ( x )theleast(greatest)integergreater(less)thanorequaltox. The
⌈ ⌉ ⌊ ⌋
neighborhood of a vertex u is the set N(u) consisting of all vertices v which are adjacent with
u. The closed neighborhood is N[u] = N(u) u . A set of vertices in G is independent if
∪{ }
no two of them are adjacent. The largest number of vertices in such a set is called the vertex
independence number ofGandis denotedbyβ (G). ForanysetS ofverticesofG,the induced
o
subgraph S is maximal subgraph of G with vertex set S.
h i
The corona of two graphs G and G is the graph G = G G formed from one copy
1 2 1 2
◦
of G and V(G ) copies of G where the ith vertex of G is adjacent to every vertex in the
1 1 2 1
| |
ith copy of G . A wounded spider is the graph formed by subdividing at most n 1 of the
2
−
edges of a star K for n 0. Let Ω(G) be the set of all pendant vertices of G, that is the
1,n
≥
set of vertices of degree 1. A vertex v is called a support vertex if v is neighbor of a pendant
vertex and d (v) > 1. Denote by X(G) the set of all support vertices in G, M(G) be the set
G
1Supported by University Grant Commission(UGC), New Delhi, India through UGC-SAP- DRS-III, 2016-
2021: F.510/3/DRS-III/2016 (SAP-I).
2ReceivedJanuary11,2018,Accepted May25,2018.
88 B.BasavanagoudandSujataTimmanaikar
of vertices which are adjacent to support vertex and J(G) be the set of vertices which are not
adjacent to a support vertex. The diameter diam(G) of a connected graph G is the maximum
distance between two vertices of G, that is diam(G) = max d (u,v). A set B V is
u,v∈V(G) G
⊆
a 2-packing if for each pair of vertices u,v B,N [u] N [v]=φ
G G
∈ ∩
Aproper coloring ofagraphG=(V(G),E(G))isafunctionfromtheverticesofthegraph
to a set of colors such that any two adjacent vertices have different colors. The chromatic
number χ(G) is the minimum number of colors needed in a proper coloring of a graph. A
dominatorcoloring ofagraphGisapropercoloringinwhicheachvertexofthegraphdominates
every vertex of some color class. The dominator chromatic number χ (G) is the minimum
d
number of color classes in a dominator coloring of a graph G. This concept was introduced by
R. Gera at.al [3].
AsetD ofverticesinagraphG=(V,E)isadominating set ofG,ifeveryvertexinV D
−
isadjacenttosomevertexinD. Thedomination number γ(G)ofGistheminimumcardinality
of a dominating set. For a comprehensive survey of domination in graphs, see [4, 5, 7].
Generally, if D H, such a dominating set D is called a Smarandache H-dominating
h i ≃
set. A dominating set D of a graph G = (V,E) is an independent dominating set, if the
inducedsubgraph D hasnoedges,i.e.,aSmarandacheH-dominatingsetwithE(H)= . The
h i ∅
independent domination number i(G)istheminimumcardinalityofanindependentdominating
set.
AdominatingsetDofG=(V,E)isanaccuratedominatingset ifV Dhasnodominating
−
setofcardinality D . Theaccurate domination number γ (G) ofGisthe minimumcardinality
a
| |
of an accurate dominating set. This concept was introduced by Kulli and Kattimani [6, 9].
AnindependentdominatingsetD ofGisanaccurate independent dominating set ifV D
−
has no independent dominating set of cardinality D . The accurate independent domination
| |
number i (G) of G is the minimum cardinality of an accurate independent dominating set of
a
G. This concept was introduced by Kulli [8].
For example, we consider the graph G in Figure 1. The accurate independent dominating
sets are 1,2,6,7 and 1,3,6,7 . Therefore i (G)=4.
a
{ } { }
2 1 6 7
b b bb b
G:
b b b
3 4 5
Figure 1
§2. Results
Observation 2.1
1. Everyaccurateindependent dominating setis independent anddominating. Hence itis
a minimal dominating set.
AccurateIndependentDominationinGraphs 89
2. Every minimal accurate independent dominating set is a maximal independent domi-
nating set.
Proposition 2.1 For any nontrivial connected graph G, γ(G) i (G).
a
≤
Proof Clearly, every accurate independent dominating set of G is a dominating set of G.
Thus result holds.
2
Proposition 2.2 If G contains an isolated vertex, then every accurate dominating set is an
accurate independent dominating set.
Now we obtain the exact values of i (G) for some standard class of graphs.
a
Proposition 2.3 For graphs P ,W and K , there are
p p m,n
(1) i (P )= p/3 if p 3;
a p
⌈ ⌉ ≥
(2) i (W )=1 if p 5;
a p
≥
(3) i (K )=m for 1 m<n.
a m,n
≤
Theorem 2.1 For any graph G, i (G) p γ(G).
a
≤ −
Proof Let D be a minimal dominating set of G. Then there exist at least one accurate
independent dominating set in (V D) and by proposition 2.1,
−
i (G) V D p γ(G).
a
≤| |−| |≤ −
Notice that the path P achieves this bound.
4
2
Theorem 2.2 For any graph G,
p/△+1 i (G) p△/△+1
a
⌈ ⌉≤ ≤⌊ ⌋
and these bounds are sharp.
Proof It is known that p/ △ +1 γ(G) and by proposition 2.1, we see that the lower
≤
bound holds. By Theorem 2.1,
i (G) p γ(G),
a
≤ −
p p/△+1
≤ −
p△/△+1.
≤
Notice that the path P ,p 3 achieves the lower bound. This completes the proof.
p
≥ 2
Proposition 2.4 If G=K ,r 3, then
m1,m2,m3,···,mr ≥
i (G)=m if m <m <m <m .
a 1 1 2 3 r
···
90 B.BasavanagoudandSujataTimmanaikar
Theorem2.3 ForanygraphGwithoutisolatedverticesγ (G) i (G)ifG=K ,r
a ≤ a 6 m1,m2,m3,···,mr ≥
3. Furthermore, the equality holds if G=P (p=4,p 3),W (p 5) or K for 1 m<n.
p p m,n
6 ≥ ≥ ≤
Proof Since we have γ(G) γ (G) and by Proposition2.1,γ (G) i (G).
a a a
≤ ≤
Let γ (G) i (G). If G = K , r 3 then by Proposition 2.4, i (G) =
a ≤ a m1,m2,m3,···,mr ≥ a
m if m < m < m < m and also accurate domination number is p/2 + 1 i.e.,
1 1 2 3 r
··· ⌊ ⌋
γ (G)= p/2 +1>m =i (G), a contradiction.
a 1 a
⌊ ⌋ 2
Corollary 2.1 For any graph G, i (G)=γ(G) if diam(G)=2.
a
Proposition 2.5 For any graph G without isolated vertices i(G) i (G). Furthermore, the
a
≤
equality holds if G=P (p 3), W (p 5)or K for 1 m<n.
p p m,n
≥ ≥ ≤
Proof Every accurate independent dominating set is a independent dominating set. Thus
result holds.
2
Definition 2.1 The double star S is the graph obtained by joining the centers of two stars
n,m
K and K with an edge.
1,n 1,m
Proposition2.6 ForanygraphG,i (G) β (G).Furthermore,theequalityholdsifG=S .
a o n,m
≤
Proof Sinceeveryminimalaccurateindependentdominatingsetisanmaximalindependent
dominating set. Thus result holds.
2
Theorem 2.4 For any graph G, i (G) p α (G).
a 0
≤ −
Proof Let S be a vertex cover of G. Then V S is an accurate independent dominating
−
set. Then i (G) V S p α (G).
a 0
≤| − |≤ − 2
Corollary 2.2 Fr any graph G, i (G) p β (G)+2.
a 0
≤ −
Theorem 2.5 If G is any nontrivial connected graph containing exactly one vertex of degree
△(G)=p 1, then γ(G)=i (G)=1.
a
−
Proof Let G be any nontrivial connected graph containing exactly one vertex v of degree
deg(v)=p 1. Let D be a minimal dominating set of G containing vertex of degree deg(v)=
−
p=1. Then D is a minimum dominating set of G i.e.,
D =γ(G)=1. (1)
| |
Also V D has no dominating set of same cardinality D . Therefore,
− | |
D =i (G). (2)
a
| |
Hence, by (1) and (2) γ(G)=i (G)=1.
a
2
AccurateIndependentDominationinGraphs 91
Theorem 2.6 If G is a connected graph with p vertices then i (G) = p/2 if and only if
a
G=H K ,where H is any nontrivial connected graph.
1
◦
Proof Let D be any minimal accurate independent dominating set with D = p/2. If
| |
G = H K then there exist at least one vertex v V(G) which is neither a pendant vertex
1 i
6 ◦ ∈
nor a support vertex. Then there exist a minimal accurate independent dominating set D′
containing v such that
i
D′ D v p/2 v p/2 1,
i i
| |≤| |−{ }≤ −{ }≤ −
which is a contradiction to minimality of D.
Conversely, let l be the set of all pendant vertices in G = H K such that l = p/2.
1
◦ | |
If G = H K , then there exist a minimal accurate independent dominating set D V(G)
1
◦ ⊆
containing all pendant vertices of G. Hence D = l =p/2.
| | | | 2
Now we characterize the trees for which i (T)=p ∆(T).
a
−
Theorem 2.7 For any tree T, i (T) = p ∆(T) if and only if T is a wounded spider and
a
−
T =K , K .
1 1,1
6
Proof Suppose T is wounded spider. Then it is easy to verify that
i (T)=p ∆(T).
a
−
Conversely, suppose T is a tree with i (T) = p ∆(T). Let v be a vertex of maximum
a
−
degree ∆(T) and u be a vertex in N(v) which has degree 1. If T N[v]=φ then T is the star
−
K ,n 2. Thus T is a double wounded spider. Assume now there is at least one vertex in
1,n
≥
T N[v]. Let S be a maximal independent set of T N[v] . Then either S v or S u
− h − i ∪{ } ∪{ }
is an accurate independent dominating set of T. Thus p=i (T)+∆(T) S +1+∆(T) p.
a
≤| | ≤
This implies that V N(v) is an accurate independent dominating set. Furthermore, N(v) is
−
also an accurate independent dominating set.
The connectivity of T implies that each vertex in V N[v] must be adjacent to at least
−
one vertex in N(v). Moreover if any vertex in V N[v] is adjacent to two or more vertices in
−
N(v), then a cycle is formed. Hence each vertex in V N[v] is adjacent to exactly one vertex
−
in N(v). To show that ∆(T)+1 vertices are necessary to dominate T, there must be at least
one vertex in N(v) which are not adjacent to any vertex in V N[v] and each vertex in N(v)
−
has either 0or1 neighbors in V N[v]. Thus T is a wounded spider.
− 2
Proposition 2.7 If G is a path P , p 3 then γ(P )=i (P ).
p p a p
≥
We characterize the class of trees with equal domination and accurate independent domi-
nation number in the next section.
§3. Characterization of (γ,i )-Trees
a
For any graph theoretical parameter λ and µ, we define G to be (λ,µ)-graph if λ(G) =
92 B.BasavanagoudandSujataTimmanaikar
µ(G). Here we provide a constructive characterizationof (γ,i )-trees.
a
To characterize (γ,i )-trees we introduce family τ of trees T = T that can be obtained
a 1 k
as follows. If k is a positive integer, then T can be obtained recursively from T by the
k+1 k
following operation.
Operation O Attach a path P (x,y,z) and an edge mx, where m is a support vertex of a tree
3
T.
τ = T/obtainedfromP byfinitesequenceof operationsof O
5
{ }
b b b b b
b
b b b b
b b
b
b
Tree T belonging to family τ
1
Observation 3.1 If T τ, then
∈
1. i (T)= p+1/3 ;
a
⌈ ⌉
2. X(T)isaminimaldominatingsetaswellasaminimalaccurateindependentdominating
set of T;
3. V D is totally disconnected.
h − i
Corollary 3.1 If tree T with p 5 belongs to the family τ then γ(T) = X(T) and i (T) =
a
≥ | |
X(T).
| |
Lemma 3.1 If a tree T belongs to the family τ then T is a (γ,i ) tree.
a
−
Proof IfT =P , p 3thenfromproposition2.7T isa(γ,i ) tree.NowifT =P , p 3
p a p
≥ − ≥
then we proceed by induction on the number of operations n(T) required to construct the tree
T. If n(T)=0 then T P by proposition 2.7 T is a (γ,i )-tree.
5 a
∈
Assume now that T is a tree belonging to the family τ with n(T) = k, for some positive
integer k and each tree T′ τ with n(T′) < k and with V(T′) 5 is a (γ,i )-tree in which
a
∈ ≥
X(T′) is a minimal accurate independent dominating set of T′. Then T can be obtained from
a tree T′ belonging to τ by operation O where m V(T′) (M(T′) Ω(T′)) and we add
∈ − −
AccurateIndependentDominationinGraphs 93
path (x,y,z) and the edge mx. Then z is a pendant vertex in T and y is a support vertex
and x M(T). Thus S(T) = X(T′) y is a minimal accurate independent dominating set
∈ ∪{ }
of T. Therefore i (T) X(T) = X(T′) +1. Hence we conclude that i (T) = i (T′)+1.
a a a
≥ | | | |
By the induction hypothesis and by observation 3.1(2) i (T′) = γ(T′) = X(T′). In this way
a
| |
i (T)= X(T) and in particular i (T)=γ(T).
a a
| | 2
Lemma 3.2 If T is a (γ,i ) tree, then T belongs to the family τ.
a
−
Proof If T is a path P , p 3 then by proposition 2.7 T is a (γ,i ) tree. It is easy to
p a
≥ −
verifythatthe statementistrue for alltreesT with diameterless thanorequalto 4. Hence we
mayassume thatdiam(T) 4.LetT be rootedata supportvertexm ofa longestpathP. Let
≥
P be a m z path and let y be the neighbor of z. Further, let x be a vertex belongs to M(T).
−
Let T be a (γ,i )-tree. Now we proceed by induction onnumber of vertices V(T) of a (γ,i )-
a a
| |
tree. Let T be a (γ,i )-tree and assume that the result holds good for all trees on V(T) 1
a
−
vertices. By observation3.1(2) since T is (γ,i )-tree it contains minimal accurate independent
a
dominatingsetD thatcontainsallsupportverticesofa tree. Inparticular m,y D andthe
{ }⊂
vertices x and z are independent in V D .
h − i
Let T′ = T (x,y,z). Then D y is dominating set of T′ and so γ(T′) γ(T) 1.
− −{ } ≤ −
Any dominating set can be extended to a minimal accurate independent dominating set of
T by adding to it the vertices (x,y,z) and so i (T) i (T′)+1. Hence, i (T′) γ(T′)
a a a
≤ ≤ ≤
γ(T)+1 i (T) 1 i (T′). Consequently,we musthave equalitythroughoutthis inequality
a a
≤ − ≤
chain. In particular i (T′) = γ(T′) and i (T) = i (T′) + 1. By inductive hypothesis any
a a a
minimalaccurateindependentdominatingsetofatreeT′ canbeextendedtominimalaccurate
independent dominating set of a tree T by operation O. Thus T τ.
∈ 2
Asanimmediateconsequenceoflemmas3.1and3.2,wehavethefollowingcharacterization
of trees with equal domination and accurate independent domination number.
Theorem 3.1 Let T be a tree. Then i (T)=γ(T) if and only if T τ.
a
∈
§4. Accurate Independent Domination of Some Graph Families
In this sectionaccurateindependent dominationof fan graph,double fan graph, helm graph and
geargraph areconsidered. Wealsoobtainthecorrespondingrelationbetweenotherdominating
parameters and dominator coloring of the above graph families.
Definition 4.1 A fan graph, denoted by F can be constructed by joining n copies of the cycle
n
graph C with a common vertex.
3
Observation 4.1 Let F be a fan. Then,
n
1. F is a planar undirected graph with 2n+1 vertices and 3n edges;
n
2. F has exactly one vertex with ∆(F )=p 1;
n n
−
3. Diam(F )=2.
n
94 B.BasavanagoudandSujataTimmanaikar
Theorem 4.1([2]) For a fan graph F ,n 2, χ (F )=3.
n d n
≥
Proposition 4.1 For a fan graph F ,n 2, i (F )=1.
n a n
≥
Proof By Observation 4.1(2) and Theorem 2.5 result holds.
2
Proposition 4.2 For a fan graph F ,n 2, i (F )<χ (F ).
n a n d n
≥
Proof By Proposition 4.1 and Theorem 4.1, we know that χ (F ) = 3. This implies that
d n
i (F )<χ (F ).
a n d n
2
Definition 4.2 A double fan graph, denoted by F isomorphic to P +2K .
2,n n 1
Observation 4.2
1. F is a planar undirected graph with (n+2) vertices and (3n 1) edges;
2,n
−
2. Diam(G)=2.
Theorem 4.2([2]) For a double fan graph F ,n 2, χ (F )=3.
2,n d 2,n
≥
Theorem 4.3 For a double fan graph F , n 2, i (F ) = 2, i (F ) = 1, i (F ) = 3
2,n a 2,2 a 2,3 a 2,5
≥
and i (F )=2 if n 7.
a 2,n
≥
Proof Our proof is divided into cases following.
Case 1. Ifn=2 and n 7,thenF , n 2hasonlyoneaccurateindependent dominating
2,n
≥ ≥
set D of D =2. Hence, i (F )=2.
a 2,n
| |
Case 2. If n = 3, thenF has exactly one vertex of ∆(G) = p 1. Then by Theorem 2.5,
2,3
−
i (F )=1.
a 2,n
Case 3. If n=5and D be a independent dominating setof G with D =2,then (V D) also
| | −
has an independent dominating set of cardinality 2. Hence D is not accurate.
Let D be a independent dominating set with D =3, then V D has no independent
1 1 1
| | −
dominating set of cardinality 3. Then D is accurate. Hence, i (F )=3.
1 a 2,n
Case 4. If n=4 and 6, there does not exist accurate independent dominating set.
2
Proposition 4.3 For a double fan graph F , n 7,
2,n
≥
γ(F )=i(F )=γ (F )=i (F )=2
2,n 2,n a 2,n a 2,n
.
Proof LetF , n 7beaDoublefangraph. Then2k formsaminimaldominatingsetof
2,n 1
≥
F suchthatγ(F )=2.Sincethis dominatingsetis independentandin(V D)thereis no
2,n 2,n
−
independent dominating set of cardinality 2 it is both independent and accurate independent
dominating set. Also it is accurate dominating set. Hence,
γ(F )=i(F )=γ (F )=i (F )=2.
2,n 2,n a 2,n a 2,n
2
AccurateIndependentDominationinGraphs 95
Proposition 4.4 For Double fan graph F , n 7
2,n
≥
i (F ) χ (F ).
a 2,n d 2,n
≤
Proof The proof follows by Theorems 4.2 and 4.3.
2
Definition 4.3([1]) For n 4,the wheel W is defined to be the graph W =C +K . Also
n n n−1 1
≥
it is defined as W =C +K .
1,n n 1
Definition 4.4 A helm H is the graph obtained from W by attaching a pendant edge at
n 1,n
each vertex of the n-cycle.
Observation 4.3 A helm H is a planar undirected graphwith (2n+1) verticesand 3nedges.
n
Theorem 4.4([2]) For Helm graph H ,n 3,χ (H )=n+1.
n d n
≥
Proposition 4.5 For a helm graph H , n 3, i (H )=n.
n a n
≥
Proof LetH ,n 3beahelmgraph. Thenthereexistaminimalindependentdominating
n
≥
set D with D =n and (V D) has no independent dominating set of cardinality n. Hence D
| | −
is accurate. Therefore i (H )=n.
a n
2
Proposition 4.6 For a helm graph H ,n 3
n
≥
γ(H )=i(H )=γ (H )=i (H )=n.
n n a n a n
Proposition 4.7 For a helm graph H ,n 3
n
≥
i (H )=χ (H ) 1.
a n d n
−
Proof Applying Proposition 4.5 , i (H )=n=n+1 1=χ (H ) 1 by Theorem 4.4,
a n d n
− −
χ (H )=n+1. Hence the proof.
d n
2
Definition 4.5 A gear graph G also known as a bipartite wheel graph, is a wheel graph W
n 1,n
with a vertex added between each pair of adjacent vertices of the outer cycle.
Observation 4.4 A gear graph G is a planar undirected graph with 2n+1 vertices and 3n
n
edges.
Theorem 4.5([2]) For a gear graph G ,n 3,
n
≥
χ (G )= 2n/3 +2.
d n
⌈ ⌉
96 B.BasavanagoudandSujataTimmanaikar
Theorem 4.6 For a gear graph G ,n 3 ,i (G )=n.
n a n
≥
Proof It is clear from the definition of gear graph G is obtained from wheel graph W
n 1,n
with a vertex added between each pair of adjacent vertices of the outer cycle of wheel graph
W . These n vertices forms an independent dominating set in G such that (V D) has no
1,n n
−
independentdominatingsetofcardinalityn. Therefore,thesetD withcardinalitynisaccurate
independent dominating set of G . Therefore i (G )=n.
n a n
2
Corollary 4.1 For any gear graph G , n 3,γ(G )=i(G )=n 1.
n n n
≥ −
Proposition 4.8 For a gear graph G ,n 3,
n
≥
i (G )=γ (G ).
a n a n
Proposition 4.9 For a graph G ,n 3
n
≥
i (G )=γ(G )+1=i(G )+1.
a n n n
Proof Applying Theorem 4.6 and Corollary 4.1, we know that i (G )=n =n 1+1 =
a n
−
γ(G )+1=i(G )+1.
n n
2
References
[1] F. Harary, Graph Theory, Addison-Wesley, Reading, Mass, (1969).
[2] K.Kavitha and N.G.David, Dominator coloring of some classes of graphs, International
Journal of Mathematical Archive, 3(11), 2012, 3954-3957.
[3] R.Gera, S. Horton, C. Rasmussen, Dominator Colorings and Safe Clique Partitions, Con-
gressus Numerantium , (2006).
[4] T.W.Haynes,S.T.Hedetniemi andP.J.Slater,Fundamentals of Domination in Graphs,
Marcel Dekker, Inc., New York, (1998).
[5] T.W.Haynes,S.T.HedetniemiandP.J.Slater,Domination in Graphs- Advanced Topics,
Marcel Dekker, Inc., New York, (1998).
[6] V. R. Kulli and M.B.Kattimani, The Accurate Domination Number of a Graph, Technical
Report 2000:01,Dept.Mathematics, Gulbarga University, Gulbarga, India (2000).
[7] V.R.Kulli,TheoryofDominationinGraphs,VishwaInternationalPublications,Gulbarga,
India (2010).
[8] V. R. Kulli, Advances in Domination Theory-I, Vishwa International Publications, Gul-
barga, India (2012).
[9] V. R. Kulli and M.B.Kattimani, Accurate Domination in Graphs, In V.R.Kulli, ed., Ad-
vancesin DominationTheory-I,VishwaInternationalPublications,Gulbarga,India(2012)
1-8.