GRAPHOIDAL PATH COVER OF SOME SPECIAL GRAPHS · © 2019 JETIR February 2019, Volume 6, Issue 2...

© 2019 JETIR February 2019, Volume 6, Issue 2 www.jetir.org (ISSN-2349-5162)

JETIRAB06209 Journal of Emerging Technologies and Innovative Research (JETIR) www.jetir.org 1115

2 – GRAPHOIDAL PATH COVER OF SOME

SPECIAL GRAPHS

1 B.Saranya, 2G.Susi Vinnarasi, 3V.Maheswari 1,2PG Student,PG&Research Department of Mathematics,

A.P.C.Mahalaxmi College For Women,Thoothukudi,TN,India.

[email protected]

[email protected] 3PG&Reseach Department of Mathematics,

A.P.C.Mahalaxmi College For Women,Thoothukudi,TN,India.

[email protected]

Abstract

B.D.Acharya and E.Sampath kumar introduced concept of graphoidal cover . The concept of 2-

graphoidal path cover was introduced by K. Nagarajan et.al. A 2- graphoidal path cover of a graph G is a

partition of edge set of G into paths in which every vertex is an internal vertex of atmost two paths and the

minimum cardinality of it is a 2- graphoidal path covering number 2(G). In this paper we find the 2 (G)

for some special graphs.

Key words Graphoidal cover ; acyclic graphoidal cover ; 2- graphoidal path cover ; 2 – graphoidal path

covering number .

1. Introduction

A graph is a pair G=(V,E), where V is the set of vertices and E is the set of edges. Now, we

consider only nontrivial, finite, connected and simple graphs. The order and size of G are denoted by p & q

respectively. SupposeP=(v0.v1,v2,…vn-1,vn) is a path in G. Then the vertices v1,v2,……vn are called internal

vertices of P and the vertices v0,vn are called external vertices of P. If P=(v0.v1,v2,….vn-1,vn=v0) is a closed

path, then v0 may be taken as a external vertex. Let be a collection of internally edge disjoint paths in G.

A Vertex of G is said to be an internal vertex of if it is an internal vertex of some paths in ,otherwise it

is called an external vertex of .

1.1 Definition :[1]

A graphoidal cover of a graph G is a partition of E(G) into non-trivial (not necessarily open) paths

in G such that every vertex of G is an internal vertex of atmost one path in .

The minimum cardinality of a graphoidal cover of G is called the graphoidal covering number of G.

1.2 Definition:[2]

An acyclic graphoidal cover of G is a graphoidal cover of G such that every element of is a path

in G. The minimum cardinality of an acyclic graphoidal cover of G is called the acyclic graphoidal covering

number of G and is denoted by a(G).

1.3 Definition:[3]

A 2-graphoidal path cover is a collection of non-trivial paths in G such that

1) Every edge is in exactly one path in .

2) Every vertex is an internal vertex of atmost two paths in .

http://www.jetir.org/

mailto:[email protected]





Let g2 denote the collection of all 2-graphoidal path covers of G. Since E(G) is a 2-graphoidal path

cover, we have g2 ≠ . The minimum cardinality of a 2-graphoidal path cover of G is called the 2-graphoidal

path covering number of G and is denoted by 2(G)(i.e.) 2(G) = min { : g2(G)}.

1.4 Definition

A Fan Graph 𝐹𝑚.𝑛 is defined as the graph join𝑘𝑚 +𝑝𝑛 where 𝑘𝑚

is the empty graph on m

vertices and Pn is the path graphs on n vertices.

1.5Definition

An Udukkai graph Un, n≥ 2 is a graph constructed by joining two fan graghFn, n≥ 2 with two paths

Pn, n≥ 2 by sharing a common vertex at the centre.

1.6 Definition

An Octopus graph On, n≥ 2, can be constructed by joining a fan graph Fn, n≥ 2 with a star graph K1,n,

by sharing a common vertex which is the centre of the star.

1.7Notation

Let be a 2-graphoidal path cover of G. We use the following notations in proving theorems.

i. i(P) – number of internal vertices of path P in .

ii. t1() - number of internal vertices which appear exactly once in a path of .

iii. t2() - number of internal vertices which appear exactly once in a path of .

iv. t - number of vertices which are not internal in any path of .

If a 2-graphoidal path cover of G is minimum, then clearly t1() and t2() should be maximum and t

should be minimum. So we define the following:ti = max ti() where the maximum is taken from all 2-

graphoidal path covers of G(i=1,2) and t = min t where the minimum is taken from all 2-graphoidal path

covers of G.

1.8Definition

A corona G1ʘ G2 of two graphs G1 (with n1 vertices and m1 edges) and G2 (with n2 vertices and m1

edges) is defined as a graph obtained by taking one copy of G1 and n1 copies of G2, and then joining the ith

vertex of G1 with an edge to every vertex with an ith copy of G2.

1.9 Theorem[3]

For any graph G,2(G)=q-p-t2+t

2 Main Results

2.1 Proposition

Let F1,n = K1+ Pn. Then

2(F,n) = {2 𝑓𝑜𝑟 𝑛 = 3

𝑛 − 2 𝑓𝑜𝑟 𝑛 ≥ 4

Proof

Consider F1,3,




Figure 1

Clearly, 2 F1,3 = 2 where the 2- graphoidal path covers are

P1 = <v1, u2, v3, v2>

P2 = <u, v2, v1>

Consider F1,4

Let (v1, u1; v2, v3, v4) be the vertices of F1,4 and u as its centre.

Since d(u) = 4

d(vi) = 2 for i= 1,4

d(vj) = 3 for j= 2,3

Among the 5 vertices, atleast one of the vertex must appear as an external vertex, t≥1

2(G)=q-p-t2+t

2(F1,4) ≥ 7-5-1+1 = 2.

2(F1,4) ≥ 2.

Now, let P1 = <v1, u, v4, v3, v2>

P2 = <v3, u, v2, v1>

Thus, { P1, P2} is a 2- graphoidal path cover of F1,4 and so 2(F1,4) ≤ 2

Hence 2(F1,4) = 2

Now, Consider F1,n for n > 4.

Let u,v1,…vn be the vertices of F1,n with u as centre.

Since d(vi) = 2 for i= 1,n

d(vj) = 3 for j=2,3….n-1 and d(u) = n. we have, t2 ≤ 1.

Among n vertices, atleast one of the vertex must appear as an external vertex,t≥1

2(G) = q-p-t2+t

2(F1,n) ≥ 2n-1-(n+1)-1+1

= n-3+1

= n-2

2(F1,n) ≥ n-2




Now let,

P1 = <v1, u, vn,,,,,,,,,, v3, v2>

P2 = <vn-1,,,, u, v2, v1>

Then the paths P1, P2 together with remaining edges form a 2-graphoidal cover of F1,n.

= 2 + E(F1,n) – { E(P1) ∪ E(P2)}

= 2 + 2n-1 – (n +3)

= 2 + n -1-3

= 2 + n – 4

= n – 2

2(F1,n) ≤ n-2.

2.2 Proposition

For a graph 𝑇𝑚,𝑛,2 (𝑇𝑚,𝑛)= n-1.

Proof

Consider 𝑇𝑚,𝑛

Let v1,v2,…….vn,u2,u3,……..um be the vertices of Tm,n

Let q = n+m-1 p = n+m ; t2≤1 , t≥n+1

2(G) ≥ q-p-t2+t

2(Tm,n) ≥ n+m-1-(n+m)-1+n+1

2(Tm,n) ≥ n-1

Let p1=<v1,v,u2,u3,u4,……um>

P2 = <v2,v,vn>

Then the paths{ p1,p2} together with remaining edges form a 2- graphoidal cover of Tm,n




Figure 2

= 2 + E(Tm,n) – { E(P1) ∪ E(P2)}

= 2 + n+m-1-(m+2)

= 2 + n+m-1-m-2

= 2 + n-3

= n-1

2(G) = n-1

2.3 Illustration

For a graph T4,5 then 2(T4,5)=3

Proof

Figure 3

Consider T4,5

P1 = <v1,v,u2,u3,u4,u5>

P2 = <v2,v,v3>

P3 = <v,v4>

2(T4,5) = 3

2.4 Proposition

For a Udukkai graph Un, (n≥ 2) ,

2(Un )=2n

Proof

Here ∆≥ 5, q=6n-4, p = 4n-1, t1=0 t2≤1, since the graph contains cycle, at least one vertex must appear as an

external vertex,t≥4.

Then, 2

(𝐺)≥ q – p – t2+t

≥6n-4-4n+1- 1+4

2(𝐺)≥2n.




Now,P1=<v1,v2,…vn,u,u1,u2,….un> P2=<un,u,v1>

The paths P1, P2 together with the remaining edges form a 2-graphoidal path cover of Un.

Figure 4

| |=2+|E(Un)-{E(P1)+E(P2)}|

=2+|6n-4-4n+2|

=2+|2n-2| =2n.

2

(Un) ≤ 2n.

Hence, 2(Un)= 2n

2.5 Proposition

For a Octopus graph On,( n≥ 2) ,

2(On)= 2n-2

Proof

Here, ∆≥ 5, q=3n-1, p = 2n+1, t1=0 t2≤1, t≥n+1.

Then, 2

(𝐺)≥ q – p – t2+t .

≥3n-1-(2n+1)- 1+n+1

2

(𝑂𝑛)≥2n-2.

Figure 5




Now, P1= <un,un-1,…u1,u,v3>

P2=<un,u,v1>

The paths P1, P2, together with the remaining edges form a 2-graphoidal path cover of On.

| |=2+|E(On)-{E(P1)+E(P2)}|

=2+|(3n-1)-{(n+1+2)}

=2+|2n-4| =2n-2.

2

(On) ≤ 2n-2.

Hence, 2(On)= 2n-2.

2.6 Illustration

For a Octopus graph O4,( n≥ 2) ,2(O4)= 4

Proof

Figure 6

Here, P1=<v3,u,u1,u2,u3>

P2=<v1,u,u3>

P3=<u2,u>

P4=<v2,u>

Hence,2(O4)=4.

2.7 Definition

A Butterfly graph Cn:Cm@WK denote the two cycles of order m and n sharing a common vertex with k

pendant edges attached at the common vertex.




2.8 Proposition

For a butterfly graph Cm:Cn@Wk, 2(Cm:Cn@Wk)= k+2.

Proof

Figure 7

Here ∆≥5, q= m+n+k, p = m+n+k-1, t2≤ 1 , since the graph contains two cycles, at least one vertex must

appear as an external vertex in each path, t≥ k + 2.

2= q – p – t2+ t.

≥ m+n+k-(m+n+k-1)-1+k+2.

2≥k+2.

P1= (w1,u,sn-1,…,s1) P2= (s1,u,t1,…,tn-1)

The paths P1, P2 together with the remaining edges form a 2-graphoidal path cover of Cm:Cn@Wk.

||=2+|E(G)-{E(P1)+E(P2)}|

=2+|(m+n+k)-{(m+n)}|

=k+2.

2(Cm:Cn@Wk) ≤ k+2.

Hence, 2(Cm:Cn@Wk)=k+2.




References :

[1] B.D. Acharya, Further results on the Graphoidal Covering number of a graph, Graph Theory News

Letter, 17(4) (1988),1.

[2] S. Arumugam and J. Suresh Suseela, Acyclic Graphoidal Covers and path Partitions in a graph, Discrete

Mathematics, 190(1998), 67-77.

[3] P.K. Das and K. Ratan Singh, On 2-graphoidal Covering Number of A Graph, International Journal of

Pure and Applied Mathematics.

[4] R. Kaviyapriya and V. Maheswari , Ascending tree cover of some special graphs, International Journal

of Science, Engineering and Management, Vol.2, Issue 12, Dec 2017, 101-104.

[5] V. Maheswari and A. Nagarajan, Ascending Graphoidal Tree Cover, International Journal of

Mathematical Archieve, Vol.7(8),2016, 71-75.

[6] V. Maheswari and A. Nagarajan Graphoidal Path Double Covers, International Journal of Discrete

Mathematical Sciences and Cryptography, Vol.6(2013), No.425, pp.221-233.

[7] E. Sampathkumar and B.D. Acharya, Graphoidal Covers and Graphoidal Covering number of a graph,

Indian Journal of Pure and applied Maths 18(10)(19876), 882-890.


GRAPHOIDAL PATH COVER OF SOME SPECIAL GRAPHS · © 2019 JETIR February 2019, Volume 6, Issue 2...

Documents

Transcript of GRAPHOIDAL PATH COVER OF SOME SPECIAL GRAPHS · © 2019 JETIR February 2019, Volume 6, Issue 2...