On Distance-Two Domination of Composition of Graphs

On Distance-Two Domination of Composition of

Graphs

Yung-Ling Lai

Department of Computer Science and Information Engineering, National Chiayi University, Taiwan

Email:[email protected]

Shou-Bo Jeng

Department of Computer Science and Information Engineering, National Chiayi University, Taiwan

Email:[email protected] Abstract―For a graph G

 

V E, , let N v₁( ) and

2( )

N v denote the set of vertices that are at distance one and two from v respectively. A subset DV G

 

is said to be a D_3,2,1-dominating set of G if every vertex vV satisfies w_D



v 3 where w_D



v 3 v  D 2 N1v  D N2v  . The minimum cardinalityD

of a D_3,2,1-dominating set of G , denoted as _3,2,1

 

G , is called the D_3,2,1-domination number of G . In this paper we obtained the D_3,2,1-domination number of the composition of two paths and a path with a cycle.

Index Terms― D_3,2,1 -domination, composition,

3,2,1

D -domination number.

I. INTRODUCTION

We consider only simple and connected graphs. A graph G

 

V E, contains a set V of vertices and a set E of edges. The distance

 

,

d x y of two vertices x and y is the length

of the shortest xy path. The

distance-k-neighborhood N_k



v of vertex v ,

defined as N_k



v  



u v d u v

 

,  , is the setk



of those vertices that are at distance k from v .

Figure 1 shows an example of a graph G with



, , , , ,



V a b c d e f where N a₁

  

b d, ,

 



2 , ,

N a c e f . For a graph G( , )V E , a

dominating set DV of G is a set of vertices such that for each u V D, N u₁



D. The distance-k-dominating set of a graph G is defined as the subset DV such that for each

u V D ,







1 i k i

N u D

   . The D3,2,1

-domination problem proposed by [12] in 2006 is

similar to distance-2-domination problem, which

may be used to solved the resource sharing problem that are modeled by graphs. For each vertex v ,

the weight of v is defined as w_D



v 3 v



D



2 N v D

 _ N2



v D for some D .V

D is called a D_3,2,1-dominating set of graph G if and only if for each v V G  , wD



v  .3 The D_3,2,1 -domination number 3,2,1

 

G of a

graph G is then the minimum cardinality among all

3,2,1

D -dominating set of G. D is an optimal

3,2,1

D -dominating set of G if D _3,2,1

 

G . Due to the short history, unlike the related

distance-k-domination has many results (see

[1,3-5,7-9,13,16] for k1, and [2,6,10-11,14-15] for k1), D_3,2,1-domination problem has only been solved for a very limited class of graphs. The D_3,2,1-domination number is known for paths,

cycles and a full binary tree B [12]._n [17]

discussed D_3,2,1 -domination problem of a

-double loop network DL n a b



; ,



according to

different value of a b, . This paper established the

3,2,1

D -domination number for the composition of

two paths and a path with a cycle.

Figure 1 N a₁

  

b d, , N₂

 

a c e f, ,



a b c

d e

f G:

II. RESULTS

The composition (also called lexicographic

product) G H

 

of two graphs G and H with vertex set V G

 





v_x1 x n



and V H

 





u_x1 x m



respectively, is the graph with vertex

set V G H



 



V H

   

V G and

 

u v₁, ₁ is

adjacent to

 

u v₂, ₂ , if either v is adjacent to₁ v₂

in G or v₁ andv₂ u is adjacent to₁ u₂ in H. Figure 3 shows an example of a P P .₃[ ₄] In this paper, we will use c_i 



(u v_j, ) 1_i  j m



for

1 i n to denote the set of vertices from i-th

column of G H .

 

Figure 3: An example of graph P P .3[ 4]

Since D_3,2,1-domination requires every vertex has weight at least 3, Fact 1 is trivial.

Fact 1: Let G be a graph of order n2. Then

3,2,1( )G 2

  .

Lemma 1: Let GP P_n[ _m] for n10,m6 and

D be an optimal D_3,2,1-dominating set of G . Then in any four consecutive columns of G, there is at least one vertex in D.

Proof: Suppose to the contrary that there is no

vertex in D in four consecutive columns

1 2 3

, , ,

i i i i

c c_ c_ c_. In order to have all vertices

1 2

i i

vc_c_ satisfy w v_D( ) , there must be at3 least three vertices of D in c_i1 and c_i4. In this case, the best possible is using six vertices to dominate ten columns (from c_i3 to c_i6).

If n=11 or n=12, then we must have D  , but7 there exist a D_3,2,1-dominating set D'



 

u v₁, ₂ ,

   

u v1, 3 , u v1, 6 , u v1, 7 , u v1, 10



, u v1, 11





such that

' 6 7

D    , contradict to the fact thatD D is optimal. If n>12, we discuss in following cases: Case 1: ci7D n n

 

0 , there exist a

3,2,1 D -dominating set D'





u v1, i-2



, u v1, i-1



,



u v1, i3



, u v2, i3



, u v1, i5













6 -3 - i_{k i k} D _ D c    . Since i_{k i}_6_-3



Dc_k



=6, we must have D'  ,D

which contradict to the fact that D is optimal.

Case 2: c_i8D n n

 

0 and c_i7D 0 ,

there exist a D_3,2,1-dominating set D'





u v₁, _i-2



,



u v1, i-1



, u v1, i3



, u v2, i3



, u v1, i5









D







7 -3 - ik i D ck     . Since 7-3





=6 i k i D ck     , we must have D'  , which contradict to the factD

that D is optimal.

Case 3:



c_i7,c_i8



D 0, in order to fulfill the condition w_D



v  for all3 v



c_i7,c_i8



, we

must have c_i9D 3 , there exist a

3,2,1 D -dominating set D'





u v₁, _i-2



, u v₁, _i-1



,



u v1, i2



, u v1, i3



, u v1, i6



, u v1, i7



, u v1, i10



,









11







1, 11 - -3 i i k i k u v_ D _ Dc . Since





11 -3 =9 i k i D ck  

  , we must have D'  , whichD

contradict to the fact that D is optimal. 

Lemma 2: Let D be an optimal D_3,2,1

-dominating set of GP P_n[ _m] for m6 and n is even. If c₁D or c_nD, then

2

D n .

Proof: Without loss of generality, assume

1

c D. Consider following cases:

Case 1: c₁D 1 . In order to fulfill the

condition w v_D( ) for all3 v , we must havec₁

either c₂D 1 or c₂D and

3 2

Subcase 1.1: c₂D 1. Assume that from c₁

through c , there are no more than two vertices in₄

D. Since the vertices in c₄ can not get enough weight from those two vertices, in order to fulfill the condition w v_D( ) for all3 v , we mustc₄

have either c5D 1 or c6D 2.

If c5D =1, the sub graph from c5 to cn

is the same as G in case 1.

If c5D 2, then the case from c5 to cn

is the same as G in case 2.

If c₆D =2, since the vertices in c₆ are distance 4 from c , hence the vertices in2 c6 can

only get weight from c6 . SinceD m6, there

must be some vertices in c6 without enough

weight from those two vertices in c .6 Hence it is

impossible for those four vertices to

3,2,1

D -dominate c through₁ c .₈

If c₆D y= 2 , assume that from c₁

through c_{2 +4y} , there are no more than y2

vertices in D. Since the vertices in c_2y3 are at least distance 2 - 3y from c , and₆ y3 , the vertices in c_2y3 can not get any weight from the vertices in c₁ through c_2y4 . Then we must have c_{2 +5y} D 3, which again will bring us to case 2.

Subcase 1.2: c₂D  x 1. Assume that from

1

c through c2x2, there are no more than x1

vertices in D. Since the vertices in c2 +1x are at

least distance 2 -1x from c , and2 x2, hence

the vertices in c2x1 can not get any weight from

the vertices in c through1 c2x2, then we must

have c_2x3D 3, which again will bring us to case 2.

Subcase 1.3: c2Dand c3D  x 2 .

Assume that from c through₁ c_2x2, there are no more than x1 vertices in D. The vertices in

2 +1x

c are at least distance 2 - 2x from c .₃ If

2

x , the vertices in c₅ are distance 2 from c ,₃

and c₃D 2, hence the weight of the vertices in c₅ can only get two from the vertices in c₁

through c .₆ Then we must have c₇D 1. If

7 =1

c D , then the case from c₇ to c_n is the same as G in case1. If c₇D 2, then the case from c₇ to c_n is the same as G in case 2. If x3, since the vertices in c_{2 +1x} are at least distance 2 - 2x from c , the vertices in₃ c_2x1 can not get any weight from the vertices in c through₁

2x 2

c _ , then we must have c_2x3D 3, which again will bring us to case 2.

Case 2: c1D 2 . Consider following

subcases:

Subcase 2.1: c₁D =2. Assume that from c₁

through c , there are no more than two vertices in₄

D. Since the vertices in c₃ are distance 2 from

1

c , the vertices in c₃ can only get two weight from the vertices in c through₁ c , then we must₄

have c₅D 1. If c₅D =1, then the case from c₅ to c_n is the same as G in case1. If

5 2

c D  , then the case from c₅ to c_n is the same as G in case2.

Subcase 2.2: c1D  x 2. Assume that from

1

c through c_{2 x} , there are no more than x

vertices in D. Since the vertices in c_{2 -1x} are at least distance 2 - 2x from c , and₁ x3, hence the vertices in c_{2 -1x} can not get any weight from the vertices in c through₁ c ._{2 x} In order to fulfill the condition w v_D( ) for all3 vc_{2 -1x} , we must have c2x1D 3 , then the case from c2x1 to

n

c is the same as G in case2.

By all the cases above, we know that if n is even,

for any optimal D_3,2,1 -dominating set D of

 

n m

Algorithm 1: A way to “partition” the graph

 

n m

P P by the vertices in the D3,2,1-dominating set,

which is used in the proof of Theorem 1.

Input: Integers n, m, and a D_3,2,1-dominating set D with D n 2 of P P .n

 

m

Task: Find max k , min k



1   ,k kn



D₁

and D₂ such that 1 1

1 1 , 1 2 k k i i D _c D D  , 2 1 , 2 2 n n k i k i

D _{ }c D D . Notice that if such k exists, then for odd n, k k; and for even n ,

1 k k. Method: 1 k← 0; x← 0; k’← 0; y← 0; z← 0; 2 D₁; D₂ ; 3 if ( n % 2 = 0 ) z← 2; 4 else z← 1; 5 for ( i = 1; i < n; i++ ){ 6 if ( c_iD ) x← x+1; 7 else x← x-1-2



c_iD 1



; 8 D₁D₁  ;



c_i D



9 if ( x = z ){ 10 k← i-z+1; 11 } 12 } 13 for( i = n; i > k; i-- ){ 14 if ( c_iD ) y← y+1;

15 else y← y-1-2



ciD 1



;

16 D₂ D₂ 



c_i D



17 if( y = z ){ 18 k’← i+z-1; 19 } 20 } 21 Return k, D ,₁ D ;₂

Theorem 1: Let GP P_n[ _m] for m6. Then





3,2,1 2 2 2, 2 3; [ ] 1, 6 10; , . n n m n n P P n or n otherwise       _   _ 

Proof. For 2 n 3, let D



  

u v1, 1 , u v1, 2



.

Since d



( , ), ( , )u v_{i 1} u v_{1 1}



 d



( ,u v_{i 3}), ( , )u v_{1 1}



2 and d



( , ), ( ,u v_{i 1} u v_{1 2})

 

d ( ,u v_{i 3}), ( ,u v_{1 2})



 for1

2 i m , we have wD



( , )u vi 1



wD



( ,u vi 3)



1 2 3

   for 2 i m . Similarly, for each

2 i m, vertex ( ,u v_{i 2}) is at distance 1 from vertex ( , )u v_{1 1} and at distance no more than 2 from

1 2

( ,u v ), so we have w_D



( ,u v_{i 2})



   for2 1 3

2 i m. Hence D is a D3,2,1-dominating set

of P P .m[ 2] By Fact 1, 3,2,1



P Pm[ 2]



 . Next2

we show the upper bound of 3,2,1



P Pn[ m]



for the

rest cases:

Case 1: n4 and n 2 ( mod 4) . Consider









1, n1 , 1, n 2



D u v _ u v_ 

 



1, | 4 2 or 4 3, for 0 4 1



n j u v j i i  i _ .

Since for all vertex

 

u v_k, _j V D- ,





1 ( ,k j) 1

N u v D and N₂



( ,u v_{k j})



D1, we

have wD



 

u vi, j



 for3 1 i m and 1 j n, which implies D is a D_3,2,1-dominating set of

[ ]

n m

P P . Let nk ( mod 4), then D 2₄n _k

 2 n _{. Hence} 3,2,1



[ ]



2 n n m P P  _.

Case 2:n2 ( mod 4). This case may be divided

into two subcases:

Subcase 2.1: n6, 10. The same D in case 1

will work in this case and D 2₄n 2

 

4 2

2n  n 1

 .

Subcase 2.2: n14 and n2 ( mod 4) .

Consider D



 

u v₁, _j j 4i 2 or 4i3 for





1 1



1 2



1 5



1 7



4 0 n 3 ( , ), , , , , , n n n n i u v _ u v _ u v_ u v_  _ 



2 7





, u v, n as shown in figure 4. Then D is a

3,2,1

D -dominating set of P P_n[ _m] and D 

4 2n 1  2 n _{, which implies} 3,2,1



P Pn[ m]



2 n _.

Figure 4: A pattern of D_3,2,1-dominating set for

[ ]

n m

P P where n2 ( mod 4) and n14.

Next we show the lower bound of





3,2,1 P Pn[ m]

 . First we consider the case of

4, and 6,10

n n . Suppose to the contrary that





3,2,1 [ ] 2 1

n n m

P P

 _{ .} Then there exists a

3,2,1

D -dominating set D such that _D   for₂n 1 even n and _D ₂n

 for odd n . Consider

following two cases:

Case 1: n is even. By using algorithm 1 to

“partition”the graph, we can get two consecutive columns c c_k, _k1 such that



c_kc_k1



D= ,

1 1 1 1 2 k k i ci D D        and n _{2 2} i k ci D D    1 2 n k _{. Let} 1

D , D₂ be two sets produced by

algorithm 1, notice that -1

1 2 k D  and - -1 2 2 n k D  .

Since D₁ can D_3,2,1-dominate the vertices from

1

c to c_k-1 , and D₂ can D_3,2,1 -dominate the vertices from c_k2 to c , to ensure that the_n

vertices in c_k and c_k1 can also be

3,2,1

D -dominated by D₁ , one of the followingD₂

must be satisfied: (1) c_k-1D = 1 and c_k+2D = 1 (2) c_k-1D = 1, c_k+2D= and c_k+3D = 2 (3) c_k-1D=, c_k-2D = 2 and c_k+2D = 1 (4)



c_k-1c_k+2



D= and -2 = +3 k k c D c D = 3

The first three conditions have either

-1 = 1

k

c D or c_k+2D = 1. Without loss of

generality, assume c_k-1D = 1. By Lemma 2,

 

1 -1 2

D k which contradict to the fact that

 

1 = -1 2

D k , so it is impossible.

For (4), since c_k-1 to c_k+2 are four consecutive columns without any vertex in D_3,2,1 -domination set, which contradict to Lemma1, hence it is also impossible.

Case 2: n is odd. By using algorithm 1 to

“partition”the graph, we can get one column c ,_k

such that c_kD=, 1 1 1 1 2 k k i ci D D        and 1 2 2 n n k i k ci D D        . Since D₁ can D_3,2,1

-dominate the vertices from c to₁ c_k-1, and D₂

can D_3,2,1-dominate the vertices from c_k1 to c ,_n

to ensure that the vertices in c_k can also be

3,2,1

D -dominated by D₁ , one of the followingD₂

must be satisfied: (1) c_k-1D =1,c_k+1D= and c_k+2D=1 (2) c_k+1D =1,c_k-1D= and c_k-2D =1 (3)



c_k-1c_k1



D=,c_k-2D=2 and +2 =1 k c D (4)



c_k-1c_k1



D=,c_k+2D =2 and -2 =1 k c D (5)



c_k-1c_k1



D= and c_k-2 D =3 (6)



c_k-1c_k1



D= and c_k+2D=3

Condition (1) and (2) have either c_k-1D = 1 or c_k+1D = 1, without loss of generality, assume

-1 = 1

k

c D . By Lemma 2, D₁ 

 

k-1 2

which contradict to the fact that D₁ =

 

k-1 2, so it is impossible.

Condition (3) to (6) have either c_k-2D 2 or ck+2D 2, without loss of generality, assume

+2 2

k

c D  . Consider following subcases.

Subcase 2.1: If c_k2D =2, in order to fulfill the

condition w v_D( ) for all3 vc_k2 , we must have



c_k3c_k4



D x= 1, assume that from

i c D n c 1 n c 2 n c 3 n c -4 n c 9 n c cn8 cn7 cn6 cn5 0 0 2 0 1 0 0 1 1 0 0 1 1 0 5 c c₆ c₇ c₈ 0 1 1 0 1 c c₂ c₃ c4 4 Repeatedn -2 times. 







column

1

k

c _ through c_{k 2x 4}, there are no more than x2

vertices in D. Since from c_k1 through c_{k 2x 4},

 



N v1 N2 v



D2 1 for all vck 2x 3, we

must have ck 2x 5D 1. By lemma 2, the

subgraph from c_{k 2x 5} to c_n must have more than n   (k 2₂x 5) 1 vertices in D_3,2,1-dominating set, hence _D2 n   (k 2₂x 5) 1  _x 2 n k₂ _which

contradict to 2 2

n k

D  , so it is impossible.

Subcase 2.2: If c_k2D  x 3 , assume that

from c_k1 through c_k2x, there are no more than

x vertices in D . Since from c_k1 through

2

k x

c _ ,



N v₁

 

N₂ v



D₂  for all vc_k2x, we must have ck 2x 1D 3. By lemma 2, the

subgraph from c_{k 2x 1} to c_n must have more than

( 2 1) 1 2

n   k x

vertices in D_3,2,1-dominating set, hence

( 2 1) 1

2 2 2

n k x n k

D     x  which contradict to the

fact that 2 2

n k

D  , hence it is also impossible.

Next we consider the case of n6 .

According to the definition of D_3,2,1-domination, if



3

D

w v  for all v , thenc₁  3_i1c_i D 2 . Similarly, if wD



v  for all3 v , thenc6

6 4 2 ici D    . Hence 3,2,1



6[ ]



4 2 1 n m P P     . For the case of n10 , according to the definition of D_3,2,1-domination, if w_D



v  for3 all v , thenc₁  3_i1c_i D 2 . Similarly, if



3

D

w v  for all v , thenc₁₀ 10_i8c_iD 2. Consider following three cases:

Case 1:



c c c₁, ₂, ₃



D 3 and



c c c₈, ₉, ₁₀



D 2  . In this case,  v c₆ 3 v



D 





1 2 2 N v D N v D 3 , therefore,





3,2,1 P P10[ m] 6   . Case 2:



c c c₁, ₂, ₃



D 2 and



c c c₈, ₉, ₁₀



D 3  . In this case,  v c₅ 3 v



D 





1 2 2 N v D N v D 3 , therefore,





3,2,1 P P10[ m] 6   . Case 3:



c c c₁, ₂, ₃



D 2 and



c c c₈, ₉, ₁₀



D 2

 . In this case, according to the definition of

3,2,1

D -domination, in order to have wD



v 3

1

v c

  , we must have c3D 1. Similarly, in

order to have wD



v 3   , we must havev c10 8 1 c D  . Therefore,  v



c₅c₆



3 v



D





1 2 2 N v D N v D 1  _  _  . Since



5 6 1 = v c c N v   , for wD



v 3  v



c5 ,c6





c c c c4, 5, 6, 7



D 2 , therefore, 3,2,1



P P10[ m]



6  .

From all the cases above, we have the lower bound of P P_n[ _m] for n4 is the same as the

upper bound. By sandwich theorem, we proved

the theorem. 

[ ]

n m

P C changes each column of P P_n[ _m]

from a path to a cycle, which does not changes the distance between columns. Hence Corollary can be obtained from Theory 1 directly.

Corollary 1 Let GP C_n[ _m] for m6. Then





3,2,1 2 2 2, 2 3; [ ] 1, 6 10; , . n n m n n P C n or n otherwise       _   _  III. Conclusion

The D_3,2,1-domination is related to distance -two-domination, which has many applications in resource allocations. This paper established the

3,2,1

D -domination number of the composition of a path with a path and a path with a cycle by giving detail proofs for each case. The author expect to study on the same problem for the composition of a cycle with a path and a cycle with a cycle in the near future.

REFERENCE

[1] R.B. Allan, R. Laskar and S.T. Hedetniemi,

“A note on total domination,” Discrete

[2] J. Cyman, M. Lemanska and J. Raczek, “Lower bound on the distance k-domination number of a tree,”Math. Slovaca, Vol.56(2), pp.235-243, 2006.

[3] P. Dankelmann, “Average distance and the

domination number,” Discrete Applied

Mathematics, Vol.80, pp.21-35, 1997.

[4] O. Favaron and D. Kratsch, “Ratios of

domination parameters, Advances in Graph Theory,”Vishwa Int. Pub., pp.173-182, 1991.

[5] M. Fischermann and L. Volkmann, “Graphs

having distance-n domination number half their order,”Discrete Applied Mathematics, Vol.120, pp.97–107, 2002.

[6] A. Hansberg, D Meierling and L Volkmann,

“Distance domination and distance

irredundance in graphs,” The Electronic

Journal of Combinatorics, Vol.14, pp.#R35, 2007.

[7] J.H. Hattingh and M.A. Henning, “The ratio of the distance irredundance and domination numbers of a graph,”Journal of Graph Theory, Vol.18, pp.1–9, 1994.

[8] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, “Domination in Graphs: Advanced Topics,” Marcel Decker Inc, New York, 1998.

[9] T. Haynes, S.T. Hedetniemi and P.J. Slater,

“Fundamentals of Domination in Graphs,”

Marcel Decker Inc, New York, 1998.

[10] M.A. Henning, O.R. Oellermann and H.C.

Swart, “Bounds on distance domination

parameters,” Journal of Combinatorics,

Information System Sciences, Vol.16,

pp.11-18, 1991.

[11] M.A. Henning, O.R. Oellermann and H.C.

Swart, “Relations between distance

domination parameters,” Mathematica

Pannonica, Vol.5, pp.69-79, 1994.

[12] F. R. Hsu, “Distance-two domination of

graph,” Master Thesis, Department of

Mathematic, National Central University, Taiwan, 2006.

[13] Shougui Li, “On connected k-domination

numbers of graphs,”Discrete Mathematics, Vol.274, pp.303–310, 2004.

[14] D. Meierling and L. Volkmann, “A lower bound for the distance k-domination number of trees,” Results in Mathematics, Vol.47, pp.335-339, 2005.

[15] Fang Tian and Jun-Ming Xu, “A note on

distance domination numbers of graphs,” Australasian Journal of Combinatorics, Vol.43, pp.181-190, 2009.

[16] D.B. West, “Introduction to Graph Theory (2nd ed.),”Prentice-Hall, NJ, 2001.

[17] K.Y. Zhen, “Distance-two domination of

double-loop networks,” Master Thesis,

Department of Mathematic, National Central University, Taiwan, 2008.