Uploaded by mail

Basic Maximal Total Strong Dominating Functions

International Journal of Trend in Scientific Research and Development(IJTSRD Volume 4 Issue 4 June 2020 Available
Online www.ijtsrd.com e-ISSN 2456 - 6470
Basic Maximal Total Strong Dominating Functions
Dr. M. Kavitha1,∗ ,
1
Department of Mathematics, KPR Institute of Engineering and Technology, Coimbatore - 641407, India.
Let G = (V, E) be a simple graph. A subset D of V (G) is called a total strong dominating set of G, if for every
u ∈ V (G), there exists a v ∈ D such that u and v are adjacent and deg(v) ≥ deg(u). The minimum cardinality
of a total strong dominating set of G is called total strong domination number of G and is denoted by γts (G).
Corresponding to total strong dominating set of G, total strong dominating function can be defined. The minimum
weight of a total strong dominating function is called the fractional total strong domination number of G and is
t (G). A study of total strong dominating functions is carried out in this paper.
denoted by γsf
Keywords: Total Strong Dominating Function, Maximal Total Strong Dominating Function, Basic Maximal Total
Strong Dominating Function
Introduction: Corresponding to total strong dominating sets in a graph, total strong dominating functions may be
defined. The minimality of a total strong dominating function can be characterised. Convex combination of minimal
total strong dominating function is defined and studied. A total strong dominating function is basic, if it cannot
be expressed as a covex combination of minimal total strong dominating functions. Basic total strong dominating
functions are characterised.
Definition 0.1:
Let G = (V, E) be a simple graph without strong isolates. A function
f : V (G) → [0, 1] is called a Total Strong Dominating Function (TSDF), if f (Ns (u)) =
X
f (v) ≥ 1, ∀ u ∈
v∈Ns (u)
V (G), where
Ns (u) = {x ∈ N (u) : deg (x) ≥ deg (u)}.
Definition 0.2:
A TSDF is called a Minimal Total Strong Dominating Function (MTSDF), if whenever g : V (G) → [0, 1] and
g < f , g is not a TSDF.
Definition 0.3: Let G be a graph without isolated vertex. A TSDF (MTSDF) is called a basic TSDF (basic
MTSDF) denoted by BTSDF (BMTSDF) if it cannot be expressed as a proper convex combination of two distinct
TSDFs (MTSDFs ).
Remark 0.4: A BTSDF need not be a BMTSDF.
71
*Corresponding author: M. Kavitha, E-mail: [email protected]
International Journal of Trend in Scientific Research and Development(IJTSRD Volume 4 Issue 4 June 2020
Available Online www.ijtsrd.com e-ISSN 2456 - 6470
Lemma 0.5: Let f and g be two distinct MTSDFs of a graph G with Bfs = Bgs and Pf = Pg . Let δ(v) = f (v)−g(v),
for every v ∈ V . Then
(i) If f (v) = 0 or f (v) = 1, then δ(v) = 0.
X
(ii)
δ(v) = 0, ∀ v ∈ Bfs .
u∈Ns (v)
(iii)
X
δ(v) = 0, ∀ v ∈ Bgs .
u∈Ns (v)
Proof:
(i) We have, f (v) = 1 if and only if g(v) = 1 and
f (v) = 0 if and only if g(v) = 0.
Therefore, (i) follows.
(ii) Let v ∈ Bfs .
Then v ∈ Bgs .
X
X
δ(v)
=
(f (u) − g(u))
v∈Ns (v)
u∈Ns (v)
X
=
X
f (u) −
u∈Ns (v)
g(u)
u∈Ns (v)
= 1 − 1 (since v ∈ Bfs and v ∈ Bgs )
= 0.
Therefore, (ii) follows.
(iii) is similar to (ii).
Lemma 0.6: Let f and g be convex linear combination of MTSDFs g1 , g2 ,...,gn such that f is minimal. Then
n
n
\
[
Bfs = Bgs =
Bgsi , Pf = Pg =
Pgi and g is minimal.
i=1
i=1
Proof:
Let v ∈ Pf .
Then f (v) > 0.
n
n
X
X
Let f =
λi gi , 0 < λi < 1,
λi = 1.
i=1
i=1
Suppose gi (v) = 0, ∀ i. Then f (v) = 0, a contradiction.
n
[
Therefore, gi (v) > 0, for at least one i. Therefore, v ∈
Pgi .
Therefore, Pf ⊆
n
[
Pgi . Suppose v ∈
i=1
i=1
n
[
Pgi .
i=1
Then g(vi ) > 0, for some i. Therefore, f (v) > 0.
n
[
Therefore, v ∈ Pf . Therefore,
Pgi ⊆ Pf .
Therefore, Pf =
n
[
i=1
Pgi . Similarly, Pg =
i=1
Therefore, Pf = Pg =
n
[
n
[
Pgi .
i=1
Pgi . Let v ∈ Bfs .
i=1
Then f (Ns (v)) = 1. Therefore,
n
X
λi gi (Ns (v)) = 1.
i=1
Suppose v ∈
/ Bgsi , for some i, 1 ≤ i ≤ n. Then gi (Ns (v)) > 1.
Since gi (Ns (v)) ≥ 1, for all j , 1 ≤ j ≤ n,
72
International Journal of Trend in Scientific Research and Development(IJTSRD Volume 4 Issue 4 June 2020
Available Online www.ijtsrd.com e-ISSN 2456 - 6470
n
X
λi gi (Ns (v)) >
i=1
λi .
i=1
= 1, a contradiction.
Therefore, v ∈
Bgsi ,
Therefore, Bfs ⊆
Let v ∈
n
X
∀ i. Therefore, v ∈
n
\
Bgsi .
i=1
Bgsi .
i=1
n
\
Bgsi . Then gi (Ns (v)) = 1, for all i, 1 ≤ i ≤ n.
i=1
Therefore,
n
\
n
X
λi gi (Ns (v)) =
i=1
n
X
λi = 1.
i=1
Therefore, f (Ns (v)) = 1. Therefore, v ∈ Bfs .
n
n
\
\
Therefore,
Bgsi ⊆ Bfs . Therefore, Bfs =
Bgsi .
i=1
Similarly,
Bgs
=
n
\
Bgsi .
Hence
Bfs
=
i=1
Bgs
i=1
n
\
=
Bgsi .
i=1
Since f is minimal, Bfs weakly dominates Pf .
n
n
\
[
s
That is,
Bgi weakly dominates
Pgi .
i=1
i=1
That is, Bgs weakly dominates Pg .
Hence g is a MTSDF.
Theorem 0.7: Let f be a MTSDF. Then f is a BMTSDF if and only if there does not exist an MTSDF g such
that Bfs = Bgs and Pf = Pg .
Proof:
Suppose f is a BMTSDF.
Suppose there exists a MTSDF g such that Bfs = Bgs and Pf = Pg .
Let S = {a ∈ < : ha = (1 + a)g − af } be a TSDF and Bhs a = Bfs and Pha = Pf .
Then S is a bounded open interval.
Let S = (k1 , k2 ).
Then k1 < −1 < 0 < k2 .
Also hk1 and hk2 are MTSDFs .
hk1 = (1 + k1 )g − k1 f .
(1 + k1 )g hk1
−
.
Therefore, f =
k1
k1
1 + k1
−1
and λ2 =
.
Let λ1 =
k1
k1
Therefore, λ1 and λ2 are positive and λ1 + λ2 = 1.
Therefore, f is a convex combination of g and hk1 .
Therefore, f is not a BMTSDF, a contradiction.
Therefore, there does not exist a MTSDF g such that Bfs = Bgs and Pf = Pg .
Conversely, suppose f is not a BMTSDF.
n
X
Then there exists MTSDFs g1 , g2 ,...,gn such that f =
λi gi ,
where 0 < λi < 1 and
n
X
i=1
λi = 1.
i=1
73
International Journal of Trend in Scientific Research and Development(IJTSRD Volume 4 Issue 4 June 2020
Available Online www.ijtsrd.com e-ISSN 2456 - 6470
Let g =
n
X
µi gi , 0 < µi < 1 and
i=1
Then by lemma 0.6, Bfs = Bgs =
n
X
µi = 1.
i=1
n
\
n
[
i=1
i=1
Bgsi and Pf = Pg =
Pgi and since f is a MTSDF, g is a MTSDF.
Thus there exists a MTSDF g such that Bfs = Bgs and Pf = Pg .
Hence the theorem.
Theorem 0.8: Let f be a MTSDF of a graph G = (V, E) with Bfs = {v1 , v2 , ..., vm } and Pf0 = {u ∈ V : 0 <
f (u) < 1} = {u1 , u2 , ..., un }.
Let A 
= [aij ] be a m × n matrix defined by
 1, if v weakly dominates u
i
j
aij =
 0, otherwise.
n
X
Consider the system of linear equations given by
aij xj = 0, 1 ≤ i ≤ m.
j=1
Then f is a BMTSDF if and only if the above system does not have a non-trivial solution.
Proof:
Suppose f is not a BMTSDF.
Then there exists a MTSDF g such that Bfs = Bgs and Pf = Pg .
Let xj = f (uj ) − g(uj ), 1 ≤ j ≤ n.
Suppose xj = 0, ∀ j , 1 ≤ j ≤ n.
If f (v) = 0, then v ∈
/ Pf = Pg .
Therefore, g(v) = 0.
Therefore, f (v) − g(v) = 0, ∀ v ∈
/ Pf .
That is, f (v) = g(v), ∀ v ∈
/ Pf .
If f (v) = 1, then by Theorem ??, g(v) = 1 and hence f (v) − g(v) = 0.
Therefore, f = g , a contradiction.
Therefore, there exists, some j , 1 ≤ j ≤ n such that xj 6= 0.
Let f (uj1 ) − g(uj1 ) 6= 0.
n
n
X
X
aij xj =
aij (f (uj ) − g(uj ))
j=1
j=1X
=
(f (u) − g(u))
u∈Ns (vi )
(since aij = 1 because vi weakly dominates u)
= 0, by lemma0.5.
Since xj 6= 0, the left hand side has a non-trivial solution.
Conversely, let {x1 , x2 , ..., xn } be a non-trivial solution for the system of
linear equations.
Define g: V (G) → [0, 1] as follows:
 f (v),
if v ∈
/ Pf0
g(v) =
f (v) + xj , if v = uj , 1 ≤ j ≤ n, where M is to be suitably chosen.
M
Since {x1 , x2 , ..., xn } is a non-trivial solution, g 6= f .
xj
Since 0 < f (uj ) < 1, choose Mj > 0 such that 0 < (f (uj ) + M
) < 1, for each j, 1 ≤ j ≤ n.
j
0
Let M = max{M1 , M2 , ..., Mn }.
Choose M to be equal to M’.
74
International Journal of Trend in Scientific Research and Development(IJTSRD Volume 4 Issue 4 June 2020
Available Online www.ijtsrd.com e-ISSN 2456 - 6470
X
For any v ∈ V , g(Ns (v)) =
g(u)
u∈Ns (v)
=
X
g(u) +
u∈Ns (v)∩P
=
X
0
f
g(u)
0
f
u∈Ns (v)−P
X
xi
(f (vi ) + 0 ) +
M
u∈Ns (v)−Pf0
ui ∈Ns (v)∩Pf0
X
f (u)
1 X
xi
M0
i
u∈Ns (u)
X
= f (Ns (v)) + M1 0
xi
=
If v ∈ Bfs , then
X
xi =
n
X
i
X
f (u) +
i
aij xi = 0.
j=1
(since v weakly dominates Pf and hence aij = 1).
Therefore, g(Ns (v)) = f (Ns (v)) = 1.
Suppose v ∈
/ Bfs .
Then f (Ns (v)) > 1.
Choose M 00 > 0 such that g(Ns (v)) > 1, ∀ v ∈
/ Bfs .
Let M = max{M 0 , M 00 }.
For this choice of M , we have
X
0 ≤ g(v) ≤ 1 and
g(v) ≥ 1, ∀ v ∈ V
u∈Ns (v)
Therefore, g is a TSDF.
From what we have seen above,
Bfs = Bgs and Pf = Pg .
Since f is a MTSDF, Bfs weakly dominates Pf .
Therefore, Bgs weakly dominates Pg . Therefore, g is a MTSDF.
Hence f is not a BMTSDF.
Corollary 0.9: Let G = (V, E) be a graph without isolated vertices. Let S be a minimal total strong dominating
set of G. Then χs is a BMTSDF.
Proof:
Clearly, χs is a MTSDF.
Let f = χs .
Pf0 = φ.
Therefore from the above theorem, χs is a BMTSDF.
Example 0.10:
Consider Hajo’s Graph H3 :
vs 1
sv3
v2 s
v4 s
s
v5
Define f1 and f2 as follows:
f1 (v1 ) = f1 (v4 ) = f1 (v2 ) = f1 (v6 ) = 0.
f1 (v3 ) = f1 (v5 ) = 1.
75
sv6
International Journal of Trend in Scientific Research and Development(IJTSRD Volume 4 Issue 4 June 2020
Available Online www.ijtsrd.com e-ISSN 2456 - 6470
0 vs 1
f1 :
v2 0s
0
v4 s
f2 (v1 ) = f2 (v4 ) = f2 (v6 ) = 0.
1
f2 (v2 ) = f2 (v3 ) = f2 (v5 ) = .
2
sv13
s
sv6
0
1 v5
0vs 1
1
1
2v
v22s
f2 :
0
v4 s
s3
s
v5
1
2
s0v6
f1 is a TSDF.
Bfs1 = {v1 , v3 , v4 , v5 }.
Pf1 = {v3 , v5 }.
Bfs1 weakly dominates Pf1 .
Therefore, f1 is a MTSDF.
Therefore, f2 is a TSDF.
Bfs2 = V .
Bfs2 weakly dominates Pf2 .
f2 is a MTSDF.
Pf01 = φ.
Let A = [aij ]m×n be a m × n matrix defined by



1, if vi ∈ Bfs weakly dominates uj in Pf0


aij =
.



 0, otherwise.
Consider the system of linear equations given by
a11 x1 + a12 x2 + . . . + a1n xn = 0
.
.
.
am1 x1 + am2 x2 + . . . + a1n xn = 0.
Since Pf01 = φ, the system of equations does not occur and hence does not have a non-trivial solution.
Therefore, f1 is a BMTSDF.
f2 is a TSDF.
Bfs2 = V, Pf2 = {v2 , v3 , v4 }.
Pf02 = {v2 , v3 , v4 }.
76
International Journal of Trend in Scientific Research and Development(IJTSRD Volume 4 Issue 4 June 2020
Available Online www.ijtsrd.com e-ISSN 2456 - 6470
A = [aij ]6×3

1

0

0
=
1


0
0
1
1
0
0
0
1

0

0

0

1


0
1
6×3


x1 + x2 = 0




x =0
2
imply x1 = 0 , x2 = 0


x
=
0
1




x2 = 0
Therefore, f2 is a BMTSDF.
Example 0.11:
0
1
u
P5 :
1
u
v1
v2
1
0
u
u
u
v3
v4
v5
f1 is a TSDF.
Bfs1 = {v1 , v2 , v4 , v5 }.
Pf1 = {v2 , v3 , v4 }.
Bfs1 weakly dominates Pf1 .
v2 v3 v4
A = [aij ]4×3 =
v1
v2
v4
v5
1
0
0
0
0
1
1
0
0
0
0
x1
x2
x3
1
x1 = 0, x2 = 0, x3 = 0.
Therefore, f1 is a BMTSDF.
Example 0.12:
0
t
P9 :
Consider
v1
1
1
0
0
1
1
1
0
v2
v3
v4
v5
v6
v7
v8
v9
t
t
t
t
t
t
Define f1 on V (P6 ) as follows:
f1 (v1 ) = f1 (v4 ) = f1 (v5 ) = f1 (v9 ) = 0.
f1 (v2 ) = f1 (v3 ) = f1 (v6 ) = f1 (v7 ) = f1 (v8 ) = 1.
f1 is a TSDF.
Bfs1 = {v1 , v2 , v3 , v4 , v5 , v6 , v8 , v9 }.
Pf1 = {v2 , v3 , v6 , v7 , v8 }.
Bfs1 weakly dominates Pf1 .
Therefore, f1 is a MTSDF.
77
t
t
International Journal of Trend in Scientific Research and Development(IJTSRD Volume 4 Issue 4 June 2020
Available Online www.ijtsrd.com e-ISSN 2456 - 6470
A =
v2 v3 v6 v7
v8
v1
1
0
0
0
0
v2
v3
0
0
0
0
1
1
0
0
0
0
v4
0
1
0
0
0
v5
0
0
1
0
0
v6
0
0
0
1 0
v8
0
0
0
1
0
v9
0
0
0
0
1
xi = 0, ∀ i, 1 ≤ i ≤ 5.
Therefore, f1 is a BMTSDF.
Example 0.13:
0
t
P9 :
Consider
v1
1
1
0
0
v2
v3
v4
v5
t
t
t
t
x1
x2
x3
x4
x5
1
1
1
0
v6
v7
v8
v9
t
0
0
0
0
0
0
0
0
x1
x2
x1
x2
x3
x4
x4
x5
t
t
t
Define f1 on V (P9 ) as follows:
f1 (v1 ) = f1 (v4 ) = f1 (v5 ) = f1 (v9 ) = 0.
f1 (v2 ) = f1 (v3 ) = f1 (v6 ) = f1 (v7 ) = f1 (v8 ) = 1.
f1 is a TSDF.
Bfs1 = {v1 , v2 , v3 , v4 , v5 , v6 , v7 , v8 , v9 }.
Pf1 = {v2 , v3 , v6 , v7 , v8 }.
Bfs1 weakly dominates Pf1 .
Therefore, f1 is a MTSDF.
 
x1
 
x2   


 
1 0 0 0 0
0
 

   x1   
0 1 0 0 0
  0

 x
x   
1 0 0 0 0  1   2  0

 x2  x3   
0 1 0 0 0     0

     
A=
 x3  = x4  =  
0 0 1 0 0 
   

     0
0 0 0 1 0 x4  x5  0


   

 x5
x6   
  0
0 0 0 1 0
x7 
 
0 0 0 0 1
0
x 
 8
x9
xi = 0, ∀ i, 1 ≤ i ≤ 5.
Therefore, f1 is a BMTSDF.
Example 0.14:
1
1
0
0
t
t
t
t
Let P11 :
v1 v2 v3 v4
0
1
1
0
1
1
0
v5
v6
v7
v8
v9
v10
v11
t
t
Define f1 on V (P11 ) as follows:
f1 (v1 ) = f1 (v4 ) = f1 (v5 ) = f1 (v8 ) = f1 (v11 ) = 0.
78
t
t
t
t
t
International Journal of Trend in Scientific Research and Development(IJTSRD Volume 4 Issue 4 June 2020
Available Online www.ijtsrd.com e-ISSN 2456 - 6470
f1 (v2 ) = f1 (v3 ) = f1 (v6 ) = f1 (v7 ) = f1 (v9 ) = f1 (v10 ) = 1.
f1 is a TSDF.
Bfs1 = {v1 , v2 , v3 , v4 , v5 , v6 , v7 , v9 , v10 , v11 }.
Pf1 = {v2 , v3 , v6 , v7 , v9 , v10 }.
Bfs1 weakly dominates Pf1 .
Therefore, f1 is a MTSDF.
A =
v2 v23 v6 v7
v8
v1
1
0
0
0
0
v2
v3
0
0
0
0
1
1
0
0
0
0
v4
0
1
0
0
0
v5
0
0
1
0
0
v6
0
0
0
1 0
v8
0
0
0
1
0
v9
0
0
0
0
1
1
1
1
0
0
1
1
1
1
0
v2
v3
v4
v5
v6
v7
v8
v9
v10
v11
xi = 0, ∀ i, 1 ≤ i ≤ 6.
Therefore, f1 is a BMTSDF.
Again consider
0
t
P11 :
v1
t
t
t
t
x1
x2
x3
x4
x5
t
t
t
Define f2 on V (P11 ) as follows:
f2 (v1 ) = f2 (v5 ) = f2 (v6 ) = f2 (v11 ) = 0.
f2 (v2 ) = f2 (v3 ) = f2 (v4 ) = f2 (v7 ) = f2 (v8 ) = f2 (v9 ) = f2 (v10 ) = 1.
f2 is a TSDF.
Bfs2 = {v1 , v2 , v4 , v5 , v6 , v7 , v10 , v11 }.
Pf2 = {v2 , v3 , v4 , v7 , v8 , v9 , v10 }.
Bfs2 weakly dominates Pf2 .
79
0
0
0
0
0
0
0
0
x1
x2
x1
x2
x3
x4
x4
x5
t
t
t
International Journal of Trend in Scientific Research and Development(IJTSRD Volume 4 Issue 4 June 2020
Available Online www.ijtsrd.com e-ISSN 2456 - 6470
v2
v3
v4
v7 v8 v9 v10
v1
1
0
0
0
0 0
0
v2
0
1
0
0
0 0
0
v4
v5
0
0
1
0
0
0
1
0
0 0
0 0 0
v6
0
0
0
1
0
v7
0
0
0
v10
0
0
0
0
0
1 0 0
0 1 0
v11
0
0
0
0
0 0 1
B =
x1
x2
x3
x4
x5
x6
x7
0
0 0
0
0
0
0
0
0
0
xi = 0, ∀ i, 1 ≤ i ≤ 7.
Therefore, f2 is a BMTSDF.
Example 0.15:
Let G = C2n+1 .
Let V (G) = {v1 , v2 , ..., v2n+1 }.
Case (i)
Let n ≡ 0 (mod 2).
Let n = 2k .
Then 2n + 1
= 4k + 1.
 1, if i ≡ 1, 2 (mod 4)
Let f (vi ) =
 0, if i ≡ 3, 0 (mod 4).
Then f is a TSDF.
Bfs = {v2 , v3 , v4 , v5 , ..., v4k+1 }
Pf = {v1 , v5 , ..., v4k+1 , v2 , v6 , ..., v4k−2 }
Clearly, Bfs weakly dominates Pf .
Therefore, f is a MTSDF.
v1 v2 v5
A =
v2
1 0
0 0
v3
0 1
0 0
v4
0 0
0 0
.
1 0
0 1
v5
.
v6 . . .
...
...
...
...
v4k−1v4k−2
v4k+1
0
0
0
x1
0
0
0
0
x2
0
0
0
0
x3
0
0
0
0
x4
.
0
.
.
.
.
.
.
.
.
.
v4k+1
1
x2k+1
0
0
0 0
...
0
0
0
80
International Journal of Trend in Scientific Research and Development(IJTSRD Volume 4 Issue 4 June 2020
Available Online www.ijtsrd.com e-ISSN 2456 - 6470
Therefore, xi = 0, ∀ i, 1 ≤ i ≤ 2k + 1.
Therefore, f is a BTMSDF.
Case (ii)
Let n ≡ 1 (mod 2).
Let n = 2k + 1.
Then 2n + 1
= 4k + 3.
 1, if i ≡ 1, 2 (mod 4)
Let f (vi ) =
 0, if i ≡ 3, 0 (mod 4).
Then f is a TSDF.
Bfs = {v1 , v2 , v5 , v6 , ..., v4k+2 }.
Pf = {v1 , v2 , v5 , v6 , ..., v4k+1 , v4k+2 }.
Clearly, Bfs weakly dominates Pf .
Therefore, f is a MTSDF.
v2
v1 v2 v5 v6 . . .
0 1 0 0 ...
1 0 0 0 ...
v5
0
0
v6
0.
0
v1
A =
v4k+1 v4k+2
0
0
x1
0
x2
0
0
x3
0
0
x4
.
0
.
.
.
.
x2k+1
0
0
0
0
1
...
0
1
0 ...
0
.
.
.
.
.
v4k+2
0
0
1
0 0 ...
Therefore, xi = 0, ∀ i, 1 ≤ i ≤ 2k + 2.
Therefore, f is a BTMSDF.
Case (iii)
Let G = C2n .
Let V (G) = 
{v1 , v2 , ..., v2n }.
 1, if i ≡ 1, 2 (mod 4)
Let f (vi ) =
 0, otherwise.
Clearly,
f is a TSDF.
 V,
if n ≡ 2 (mod 4)
Bfs =
 V − {v1 , v2n }, if n ≡ 1 (mod 4).

{v , v , ..., v
1 5
2n−3 , v2 , v6 , ..., v2n−2 }, if n ≡ 2 (mod 4)
Pf =
{v1 , v5 , ..., v2n−1 , v2 , v6 , ..., v2n },
if n ≡ 1 (mod 4).
81
0
.
International Journal of Trend in Scientific Research and Development(IJTSRD Volume 4 Issue 4 June 2020
Available Online www.ijtsrd.com e-ISSN 2456 - 6470
Clearly, Bfs weakly dominates Pf .
Therefore, f is a MTSDF.
Let n ≡ 2 (mod 4).
v1 v2 v5 v6 . . . v2n−3 v
2n−2
v1
0
1
0 0 ...
0
0
v2
1
0
0
...
0
0
0
0 0 ...
0
0
0
.
A =
.
.
.
.
.
v2n
1
AX = 0 implies X = 0.
Therefore, f is a BMTSDF.
Let n ≡ 1 (mod 4).
v1 v2 v5 v6 . . . v2n−1
v2n
v2
1
0
0
...
0
0
v3
0
1
0 0 ...
0
0
0
0 0 ...
0
0
.
A =
.
.
.
v2n−1
AX = 0 implies X = 0.
Therefore, f is a BMTSDF.
.
.
0
1
R EFERENCES
[1]
[2]
[3]
[4]
Acharya, B.D., The Strong Domination Number of a Graph and Related Concepts, J.Math. Phys. Sci., 14,471-475(1980).
Allan R.B., and Laskar.R.C., On Domination and Independent Domination Number of a Graph, Dicrete Math., 23, 73-76(1978).
Arumugam.S. and Regikumar.K., Basic Minimal Dominating Functions, Utilitas Mathematica, 77, pp. 235-247(2008).
Arumugam.S. and Regikumar.K., Fractional Independence and Fractional Domination Chain in Graphs, AKCE J.Graphs Combin., 4,
No.2, pp.161-169(2007).
[5] Arumugam.S. and Sithara Jerry.A., A Note On Independent Domination in Graphs, Bulletin of the Allahabad Mathematical Society,
Volume 23, Part 1.(2008).
[6] Arumugam.S. and Sithara Jerry.A., Fractional Edge Domination in Graphs, Appl. Anal. Discrete Math.,(2009).
82
International Journal of Trend in Scientific Research and Development(IJTSRD Volume 4 Issue 4 June 2020
Available Online www.ijtsrd.com e-ISSN 2456 - 6470
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[20]
[21]
[22]
[23]
[24]
[25]
[26]
[27]
[28]
[29]
Berge.C., Theory of Graphs and its Applications, Dunod, Paris(1958).
Berge.C., Graphs and Hypergraphs, North-Holland, Amsterdam(1973).
Cockayne E.J., Dawes R.M. and Hedetniemi.S.T., Total domination in graphs, Networks, 10, 211-219(1980).
Cockayne E.J., Favaron O., Payan C. and Thomasaon A.G., Contributions to the theory of domination, independence and irredundance
in graphs, Discrete Math., 33, 249-258(1981).
Cockayne E.J., MacGillivray G. and Mynhardt C.M., Convexity of minimal dominating functions and universal in graphs, Bull. Inst.
Combin. Appl., 5, 37-38(1992).
Cockayne E.J. and Mynhardt C.M., Convexity of minimal dominating functions of trees: A survey, Quaestiones Math., 16, 301-317(1993).
Cockayne E.J., MacGillivray G. and Mynhardt C.M., Convexity of minimal dominating functions of trees-II, Discrete Math., 125,
137-146(1994).
Cockayne E.J. and Mynhardt C.M., A characterization of universal minimal total dominating functions in trees, Discrete Math., 141,
75-84(1995).
Cockayne.E.J., Fricke.G., Hedetniemi.S.T. and Mynhardt.C.M., Properties of Minimal Dominating Functions of Graphs, Ars Combin.,
41,107-115(1995).
Cockayne E.J., MacGillivray G. and Mynhardt C.M., Convexity of minimal dominating functions of trees, Utilitas Mathematica, 48,
129-144(1995).
Cockayne.E.J. and Hedetniemi.S.T., Towards a Theory of Domination in Graphs, Networks, 7,247-261(1977).
Cockayne.E.J. and Mynhardt.C.M., Minimality and Convexity of dominating and related functions in Graphs, A unifying theory, Utilitas
Mathematica, 51,145-163(1997).
Cockayne.E.J. and Mynhardt.C.M. and Yu.B., Universal Minimal Total Dominating Functions in Graphs, Networks, 24, 83-90(1994).
Grinstead D. and Slater P.J., Fractional Domination and Fractional Packing in Graphs, Congr., Numer., 71, 153-172(1990).
Harary.F., Graph Theory, Addison-Wesley, Reading, Mass(1969).
Hedetniemi.S.T. and Laskar.R.L., Bibliography on Domination in Graphs and Some Basic Definitions of Domination Parameters,
Discrete Math., 86,pp 257-277(1990).
Ore.O, Theory of Graphs, AMS(1962).
Regikumar K., Dominating Functions - Ph.D thesis, Manonmaniam Sundaranar University(2004).
Sampathkumar.E and Pushpa Latha.L., Strong Weak Domination and Domination Balance in a Graph, Discrete Math., 161,235242(1996).
Scheinerman E.R. and Ullman D.H., Fractional Graph Theory: A rational approach to the theory of graphs, John Wiley and Sons
Inc.,(1997).
Terasa W. Haynes, Stephen T. Hedetneimi, Peter J. Slater, Fundamentals of Domination in Graphs, Marcel Dekker Inc.,(1998).
Terasa W. Haynes, Stephen T. Hedetneimi, Peter J. Slater, Domination in Graphs: Advanced Topics Marcel Dekker Inc., New York(1998).
Yu.B., Convexity of Minimal Total Dominating Functions in Graphs, J.Graph Theory, 24,313-321(1997).
83