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[Volume 2 issue 8 August 2014]

Page No.565-573 ISSN :2320-7167

INTERNATIONAL JOURNAL OF MATHEMATICS AND COMPUTER RESEARCH

IJMCR www.ijmcr.in| 2:8|August|2014|565-573 | 565

Path Double Covering Number of a Graph

Gayathri Thiyagarajan1, Meena Saravanan2

1Department of Mathematics, Sri Manakula Vinayagar Engineering College, Puducherry-605 107, India, E-

mail: gayathiyagu@gmail.com

Department of Mathematics, Government Arts and Science College, Chidambaram-608 102, India. Email:

meenasaravanan14@gmail.com

ABSTRACT

A path double cover of a graph G is a collection of paths in G such that every edge of G belongs to exactly two paths in . The minimum cardinality of a path double cover is called the path double covering number of G and is denoted by PD

 

G .In this paper we determine the exact value of this parameter for several classes of graphs.

Key words: Graphoidal covers; path double covers; path double covering number of a graph; bicyclic graphs.

INTRODUCTION

A graph is a pairG

V E,

, where V is the set of vertices and E is the set of edges. Here we consider only nontrivial,finite, connected undirected graph without loops or multiple edges. The order and size of G are denoted by p and q respectively. For graph theoretic terminology we refer to Harary [7].The concept of graphoidal cover was introduced by B.D Acharya and E. Sampathkumar [1] and the concept of acyclic graphoidal cover was introduced by Arumugam and Suresh Suseela [4].The reader may refer [7] and [2] for the terms not defined here.

Let pv v v1, 2, 3, ,vr be a path or a cycle in a graph G

V E,

. Then vertices v v2, 3, ,vr1 are called internal vertices of P and v1 and vrare called external vertices of P. Two paths P and Q of a graph G are said to be internally disjoint if no vertex of G is an internal vertex of both P and Q .If Pv v v0, 1, 2, vr and

r 0, 1, 2, , r

Q v w w w w are two paths in G then the walk obtained by concatenating P and Q at vr is denoted by P Q and the path v vr, r1, ,v0 is denoted by P1.[3]. Bondy [5] introduced the concept of path double cover of a graph. This was further studied by Hao Li [8].

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IJMCR www.ijmcr.in| 2:8|August|2014|565-573 | 566 Definition 1.1[3]: A path double cover (PDC) of a graph G is a collection of paths in G such that every edge of G belongs to exactly two paths in .

The collection may not necessarily consist of distinct paths in G and hence it cannot be treated as a set in the standard sense. For any graph G

V E,

, let denote the collection of all paths of length one each path appearing twice in the collection. Clearly is a path double of G and hence the set of all path double covers of G is non-empty.

Arumugam and Meena [3] introduced the concept of path double covering number of a graph G.

Definition 1.2[3]: The minimum cardinality of a path double cover of a graph G is called path double covering number of G and is denoted by PD

 

G

In [9] it has been observed that for any graph G PD

 

G2qand equality holds if and only if G is isomorphic to qK and the following results have been proved . 2

Theorem1.3[3]: Let be any path double cover of a graph G. Then P 2q ip where

 

p P

ip i p

where i(p) is the number of internal vertices of .

Theorem1.4[3]: PD 2q i where i= maxiP the maximum being taken over all path double covers of G Theorem 1.5 [3]: Let G be a graph with

1,if there exists a path double cover such that every non pendant vertex of G is an internal vertex of d(v)paths in then is minimum path double cover and

PD P

 

Theorem 1.6 [3]: For any tree T,

PD

 

Tn where n is the number of pendent vertices of T.

Theorem 1.7 [3]: For any graph G,

PD

 

T  .Further for any tree T,

PD

 

T  if and only if T is homeomorphic to a star.

Definition 1.8 [6]: A triangular cactus is a connected graph all of whose blocks are triangles. A triangular snake is a triangular cactus whose block-cutpoint-graph is a path (a triangular snake is obtained from a path

1, 2,..., n

v v v by joining vi and vi1 to a new vertex wi for i1, 2,...,n1).

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IJMCR www.ijmcr.in| 2:8|August|2014|565-573 | 567 Definition 1.9 [6]: A double triangular snake consists of two triangular snakes that have a common path.

That is a double triangular snake is obtained from a path v v1, 2,...,vnby joining vi and vi1to a new vertex wi for i1, 2,...,n1and to a new vertex ui for i1, 2,...,n1.

Definition 1.20 [6]: The book Bm is the graph SmP2 where Sm is the star with m + 1 vertex

Definition 1.21 [6]: A gear graph denoted Gn is a graph obtained by inserting an extra vertex between each pair of adjacent vertices on the perimeter of a wheel graph Wn. Thus, Gn has 2n+1 vertices and 3n edges.

Gear graphs are examples of square graphs, and play a key role in the forbidden graph characterization of square graphs. Gear graphs are also known as cogwheels and bipartite wheels.

Definition 1.22 [6]: A helm graph, denoted Hn is a graph obtained by attaching a single edge and node to each node of the outer circuit of a wheel graph Wn.

Definition 1.23 [6]: A graph G is called the flower graph with n petals if it has 3n+1 vertices which form an n- cycle.

Definition [6]: A shell Sn is the graph obtained by taking n-3 concurrent chords in a cycle Cn on n vertices.

The vertex at which all the chords are concurrent is called the apex vertex. The shell is also called fan Fn-1.

i.e.. Sn F -1n P -1 K .n1

Definition 1.24 [6]: The cartesian product of two paths is known as grid graph which is denoted by PmPn. In particular the graph LnPnP2 is known as ladder graph.

Definition 1.25 [6]: A web graph is the graph obtained by joining the pendant vertices of a helm to form a cycle and then adding a single pendant edge to each vertex of this outer cycle.

Theorem 2.1: Let G be a triangular snake graph, then PD

 

G 4

Proof: Let V G

  

v v v1, , ,2 3 , ,v w wn 1, 2,...,wn1

,n is odd.

The path double covering of G is as follows.

 

1 1, 2, 3, , n P v v v v

 

2 1, w ,1 2, w ,2 , n 1, wn 1, n

P v v v v

1 2

2 2

P P Pis a path double cover of G .

 

4

PD G

  

Since PD

 

G   4

 

4

PD G

   is a minimum path double covering number of G.

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IJMCR www.ijmcr.in| 2:8|August|2014|565-573 | 568 Theorem 2.2: Let G be a triangular cactus graph, then PD

 

G 2n, where n is the number of triangles in the graph.

Proof: Let V G

  

v v v v0, 11, 12, 21,v22, ,v vn1, n2

Case1: n is even and n > 4

The path double covering of G is as follows.

1, 2, 0, 12, 11

, 1,3,5,..., 1

i i i i i

P v v v v v i n

12, 11, 0, 1, 2

1,3,5, 7,..., 1

j j j j j

P v v v v v j n

1, 0, 2

, 1to

k k k

P v v v k n

 

i

 

j

 

k

P P P P is a path double cover of G .

2 2 2 n n

P    n n

 

2

PD G n

 

Since PD

 

G   2n

 

2

PD G n

   is a minimum path double covering number of G.

Case 2: n is odd and n >5

The path double covering of G is as follows.

 

1 11, 12, 0, n1, n2

P v v v v v

 

2 21, 22, 0, n2, n1

P v v v v v

 

3 12, 11, 0, 21, 22

P v v v v v

1, 2, 0, 1,2, 1,1 ,

3,5, 7, , 2

i i i i i

P v v v v v i n

1,2, 1,1, 0, 1, 2

, j 3,5, 7, , 2

j j j j j

P v v v v v n

1, 0, 2

, k 1, 2,...,

k k k

P v v v n

1, 2, 3, ,i j, k

P P P P P P P is a path double cover of G .

1 1

3 1 1 2

2 2

n n

P n n

      

 

2

PD G n

 

Since PD

 

G   2n

 

2

PD G n

   is a minimum path double covering number of G.

Theorem 2.3: Let G be a flower graph with n petals then PD

 

G 2n

Proof: Let V G

  

v v v v0, 11, 12, 21,v22, ,v vn1, n2

Case1: n is even.

The path double covering of G is as follows.

1, 2, 3, 0, 1 3, 12, 11

1,3,5, , 1

i i i i i i i

P v v v v v v v i n

13, 12, 11, 0, 1, 2, 3

1,3,5, , 1

j i i i i i i

P v v v v v v v i n

1, 0, 3

, 1to

k i i

P v v v i n

i, j, k

P P P P is a path double cover of G .

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IJMCR www.ijmcr.in| 2:8|August|2014|565-573 | 569

1 1 2

P      n n n n

 

2

PD G n

  is a minimum path double covering of G.

Case 2: n is odd.

The path double covering of G is as follows.

 

1 11, 12, 13, 0, 23, 22, 21

P v v v v v v v

 

2 n3, n2, n1, 0, 21, 22, 23

P v v v v v v v

 

3 13, 12, 11, n3, n2, n1

P v v v v v v

1, 2, 3, 0, 3, 2, 1

, 3,5, , 2

i i i i i i i

P v v v v v v v i n

13 12, 11, 0, 1, 2, 3

, 3,5, , 2

j i i i i i i

P v v v v v v v i n

1, 0, 3

, 1to

k i i

P v v v i n

1, 2, 3, ,i j, k

P P P P P P P is a path double cover of G .

1 1

3 1 1 2

2 2

n n

P       n n

 

2

PD G n

  is a minimum path double covering of G.

Theorem 2.4: Let G be a Pm(QSn) graph. The path double covering of G is PD

 

G 4n2

Proof: Let V G

  

v1, ,v lm, i1 l rin, i1 r win, i1 ,win

i1ton Case1: n is even.

The path double covering of G is as follows.

, , 1, 1, , 1, 1, , 1, 11, 11, 12, 12, , 1 , 1

1 to 1

i in in in in i i i i i i i i i n i n

P w r w r w l v v l w l w l w i n

, , 1, 1, , 1, 1, , 1, 1,1, 1,1, 1,2, 1,2, , 1, , 1,

1 to 1

i in in in in i i i i i i i i i n i n

Q w r w r w r v v r w r w r w i n

, 1, 1, 2, 2, , ,

1, 2,...,

i i i i i i in in

R v l w l w l w i n

, 1, 1, 2, 2, , ,

1, 2,...,

i i i i i i in in

S v r w r w r w i n

 

1 1 2 4 2

PD G n n n n

      Case 2: n is odd.

The path double covering of G is as follows.

, , 1, 1, , 1, 1, , 1, 11, 11, 12, 12, , 1 , 1

1 to 2

i in in in in i i i i i i i i i n i n

P w r w r w l v v l w l w l w i n

, , 1, 1, , 1, 1, , 1, 1,1, 1,1, 1,2, 1,2, , 1, , 1,

1 to 2

i in in in in i i i i i i i i i n i n

Q w r w r w r v v r w r w r w i n

, 1, 1, 2, 2, , ,

1, 2,...,

i i i i i i in in

R v l w l w l w i n

, 1, 1, 2, 2, , ,

1, 2,...,

i i i i i i in in

S v r w r w r w i n

 

1 , , , 1, 1, , 1

n nn nn n n n n

P w l w l v v

 

1 , , , 1, 1, , 1

n nn nn n n n n

Q w r w r v v

PD

 

G     n 1 n 1 2n4n2

Theorem 2.5: Let G be a Cm(QSn) graph. The path double covering of G is PD

 

G 4n

Proof: Let V G

  

v1, ,v lm, i1 l rin, i1 r win, i1 ,win

i1ton Case1: n is even.

The path double covering of G is as follows.

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IJMCR www.ijmcr.in| 2:8|August|2014|565-573 | 570

, , 1, 1, , 1, 1, , 1, 11, 11, 12, 12, , 1 , 1

1 to 1

i in in in in i i i i i i i i i n i n

P w r w r w l v v l w l w l w i n

, , 1, 1, , 1, 1, , 1, 1,1, 1,1, 1,2, 1,2, , 1, , 1,

1 to 1

i in in in in i i i i i i i i i n i n

Q w r w r w r v v r w r w r w i n

, 1, 1, 2, 2, , ,

1, 2,...,

i i i i i i in in

R v l w l w l w i n

, 1, 1, 2, 2, , ,

1, 2,...,

i i i i i i in in

S v r w r w r w i n

1,

n n

P v v

P Q R Si, i, i, i, 2Pn

 

 

1 1 2 2 4

PD G n n n n

        Case 2: n is odd.

The path double covering of G is as follows.

, , 1, 1, , 1, 1, , 1, 11, 11, 12, 12, , 1 , 1

1 to 2

i in in in in i i i i i i i i i n i n

P w r w r w l v v l w l w l w i n

, , 1, 1, , 1, 1, , 1, 1,1, 1,1, 1,2, 1,2, , 1, , 1,

1 to 2

i in in in in i i i i i i i i i n i n

Q w r w r w r v v r w r w r w i n

, 1, 1, 2, 2, , ,

1, 2,...,

i i i i i i in in

R v l w l w l w i n

, 1, 1, 2, 2, , ,

1, 2,...,

i i i i i i in in

S v r w r w r w i n

 

1 , , , 1, 1, , 1

n nn nn n n n n

P w l w l v v

 

1 , , , 1, 1, , 1

n nn nn n n n n

Q w r w r v v

1,

n n

P v v

P Q R Si, i, i, i, 2Pn

 

 

1 1 2 2 4

PD G n n n n

       

Theorem 2.6: Let G be a Ladder graph. The path double covering of G is PD

 

G 3

Proof: Let V G

  

u u u1, 2, 3,...,u l ln, , ,...,1 2 ln

The path double covering of G is as follows.

1 1 2 2 3 3 4 1 1

1

1 1 2 2 3 3 4 1 1

, , , , , , , , , , , , , , ,if n is even , , , , , , , , , , , , , , ,if n is odd

i i i i n n

i i i i n n

l u u l l u u l l u u u l P l u u l l u u l l u u l u

 

1 2 2 3 3 4 4 1

2

1 1 2 2 3 3 4 1

, , , , , , , , , , ,if n is even , , , , , , , , , , , if n is odd

n n n

n n n

l l u u l l u l l u P l u u l l u u u u l

 

 

3 n, n1, n 2, , 2, , , ,1 1 2 , n1, n Pu u u u u l l l l

 

3

PD G   

is a minimum path double covering of G.

Theorem 2.7: Let G be a fan graph with n vertices. The path double covering of G is PD

 

G n

Proof: Let V G

  

x x1, 2, ,xn

The path double covering of G is as follows.

 

1 1 2 3 n

Px x x x

 

2 n 1 2 3 n1

Px x x x x

 

3 n1 n 1

Px x x

Let G1 G

P P P1, 2, 3

is a tree with n-3 pendant vertices.

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IJMCR www.ijmcr.in| 2:8|August|2014|565-573 | 571

 

1 3

PD G n

  

 

3 3

PD G n n

    

PD

 

G   n

is a minimum path covering of G.

Theorem 2.8: Let G be a mobius graph. The path double covering of G is PD

 

G 4

Proof: Let V G

  

v v1, ,2 , ,v w wn 1, 2, ,wn

The path double covering of G is as follows.

 

1 1, ,2 , n1, ,n n, n1, , 2, 1

Pv v v v w w w w

 

2 n, 1, , ,1 2 2, 3, , ,3 4 , n1, n1, n Pv w v v w w v v v w w

 

3 1, 1, 2, , ,2 3 , n1, n Pv w w v v w w

 

4 1, ,n n, 1

Pw v w v

1, 2, 3, 4

PP P P P is a minimum path covering of G

 

4

PD G

    is a minimum path covering of G.

Theorem 2.9: Let G be a shell graph with n vertices. The path double covering of G is PD

 

G n

Proof: Let V G

  

v v1, ,2 ,vn

The path double covering of G is as follows.

 

1 1, ,2 , n1, n Pv v v v

 

2 1, ,n n1, , ,3 2

Pv v v v v

 

3 2, ,1 3

Pv v v

 

4 n, ,1 n1

Pv v v

 

5 3, ,1 n1

Pv v v

 

1 1, 2, 3, 4, 5

G  G P P P P P is a tree with n-5 pendant vertices.

 

1 5

PD G n

  

(By corollary 1.1)

 

5 5

PD G n n

       is a minimum path covering of G.

Since PD

 

G   n

PD

 

G

  

Theorem 2.10: Let G be a gear graph with n vertices. The path double covering of G is PD

 

G  n 1

Proof: Let V G

  

v v v0, , ,1 2 , ,v w wn 1, 2, ,wn

The path double covering of G is as follows.

 

1 0, , , , , , ,...., ,1 1 2 2 3 3 n n

Pv v t v t v t v t

 

2 0, , , , , , ,....,n n 1 1 2 2 n1, n1

Pv v t v t v t v t

 

3 n1, , , ,n 0 1 n

Pt v v v t

 

1 1, 2, 3

G  G P P P is a tree with n-2 pendant vertices.

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 

1 2

PD G n

  

(By corollary 1.1)

 

2 3 1

PD G n n

      is a minimum path covering of G.

Theorem 2.11: Let G be a web graph with n vertices. Then PD

 

G  n 1

Proof: Let V G

  

v v v0, , ,1 2 , ,v w wn 1, 2, ,wn

Her w w1, 2, ,wn are pendant vertices.

is adjacent to 0and

i i

v v w

The path double covering of G is as follows.

 

1 1, , , , ,..., ,1 0 2 3 n n

Pw v v v v v w

 

2 0, , , ,n 1 2 , n1, n 1

Pv v v v v w

 

3 0, n1, , , ,n 1 2 2

Pv v v v v w

 

4 n 1, n1, , ,0 n n

Pw v v v w

 

5 2, , , ,2 0 1 1

Pw v v v w

 

1 1, 2, 3, 4, 5

G  G P P P P P is a tree with n-4 pendant vertices.

 

1 4

PD G n

  

(By corollary 1.1)

 

4 5 1

PD G n n

      is a minimum path covering of G.

Theorem 2.12: Let G be a double triangular snake graph, then PD

 

G   6

Proof: Let V G

  

v1,, , ,v v2 3 , ,v w wn 1, 2, ,w u un, ,1 2, ,un

, n is odd.

The path double covering of G is as follows.

 

1 1, 1, 2, 2, 3, 3, , n 1, n 1, n P v w v w v w v w v

 

2 1, u ,1 2, u ,2 3, u ,3 , n 1, un 1, n

P v v v v v

 

3 1, 1, 2, 3, 3, 4, v ,5 5, , n 2, n 1, n P v w v v w v w w v v

 

4 1, u ,1 2, 3, u ,3 4, v , u ,5 5 , un 2, n 1, n

P v v v v v v

 

5 1, 2, 2, 3, 4, 4, 5, 6,..., n 1, n 1, n P v v u u v u u v v u v

 

6 1, 2, 2, v ,3 4, 4, v , v ,5 6 , n 1, n 1, n P v v w v w v w v

 

6

PD G

   is a minimum path double covering number of G.

Note: Observe that for the following graphs the path double covering number is PD

 

G  

1. t-ply

2. Multipleshell 3. Book graph

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