eu SIMPLE GRAPHOIDAL COVERING NUMBER OF PRODUCT OF GRAPHS

A graphoidal cover of G is a set ψ of (not necessarily open) paths in G, such that every path in ψ has at least two vertices, every vertex of G is an internal vertex of at most one path in ψ and every edge of G is in exactly one path in ψ. The minimum cardinality of a graphoidal cover of G is called the graphoidal covering number of G and is denoted by η. If every two paths in ψ have at most one common vertex, then it is called simple graphoidal cover of G. The minimum cardinality of a simple graphoidal cover of G is called simple graphoidal covering number of G and is denoted by ηs. Here we determine the simple graphoidal covering number of Product of Graphs. AMS Subject Classification: 05C70, 05C76


Introduction
By a graph G = (V, E), we mean a finite undirected graph without loop or multiple edges.The order and size of the G are denoted by p and q respectively.For theoretical terminology of graph we refer Harary [10].All the graphs considered in this paper are assumed to be connected and nontrivial.If P = (v 1 , v 2 , v 3 , ....v n ) be a path or cycle in a graph G, the vertices v 2 , v 3 , v 4 , ....v n−1 are called internal vertices of P and v 1 , v n are called external vertices of P. Two paths P and Q are said to be internally disjoint if no vertex of G is an internal vertex of both P and Q.The concept of graphoidal cover was introduced by Acharya and Sampath Kumar [1] [2].Arumugam and Pakkiam [5], [6], [7], [8], [9] determined the graphoidal covering number of several families of curve.Further Arumugam and Suresh Suseela [3] introduced acyclic graphoidal cover.The concept of simple graphoidal cover was introduced by Arumugam and Shahul Hamid [4].Definition 1. (see [1]) A graphoidal cover of G is a set ψ of (not necessarily open) paths in G satisfying the following conditions.
(i) Every path in ψ has at least two vertices.
(ii) Every vertex of G is an internal vertex of at most one path in ψ.
(iii) Every edge of G is in exactly one path in ψ.
The minimum cardinality of a graphoidal cover of G is called the graphoidal covering number of G and is denoted by η.Definition 2. (see [4]) A simple graphoidal cover of a graph G is a graphoidal cover ψ of G such that any two paths in ψ have at most one vertex in common.The minimum cardinality of a simple graphoidal cover of G is called simple graphoidal covering number of G and is denoted by η s .Definition 3. (see [4]) Let ψ be a collection of internally disjoint paths in G.A vertex of G is said to be an interior vertex of if it is an internal vertex of some path in ψ.Any vertex which is not an interior vertex of is said to be an exterior vertex of ψ.Theorem 4. For any simple graphoidal cover ψ of a (p,q) of graph G, let t ψ denote the number of exterior vertices of ψ and let t= min t ψ , where the minimum is taken overall simple graphoidal covers ψ of G, Then η s (G)=q-p+t.
(ii) There exists a simple graphoidal cover of G without exterior vertices (iii) There exists a set P of internally disjoint and edge disjoint paths without exterior vertices such that any two paths in P have at most one vertex in common.
Theorem 6.For the complete graph K n (n ≥ 3), we have

and n is even
Given two graphs G and H, their product is a graph whose vertex set is V(G) X V(H)= {(g, h) : g ∈ V (G) and h ∈ V (H)}, while the edge set varies according to the nature of the product.Various kinds of graph products are defined in [12].We consider only Cartesian product, Strong product and Lexicographic product of two paths or cycles.
The Cartesian Product of two graphs G and H denoted by G • H has the edge set E(G • H )= {((g, h), (g The Strong Product of graphs G and H is denoted by G ⊗ H where E(G ⊗ H )={((g, h), (g The Lexicographic Product or Composition of two graphs G and H is de- For the positive integer m and n, we use the following notations For all the product graphs G, we have considered the vertex set V(G)= {w ij : 1 ≤ i ≤ m; 1 ≤ j ≤ n} where w i,j = (u i , v j ).

Main Results
Theorem 7.
When m = 2 and n ≥ 3, The collection of paths of G are is a simple graphoidal cover of G in which w 11 , w 2n are not internal.Therefore η s (G) ≤ q − p + 2. Now, let ψ be any simple graphoidal cover of G.If ψ contains only paths, then atleast three vertices can not be made internal.If ψ contains atleast one cycle, then atleast two vertices can not be made internal.
When m, n ≥ 3, then we have the following four cases.
Case (i) when m = n = odd, The collection of paths of G are P = (w 11 , w 12 , w 22 , w 21 , w 11 ) is a simple graphoidal cover of G in which w 11 is not internal.Therefore η s (G) ≤ q − p + 1.Now, let ψ be any simple graphoidal cover of G.If ψ contains only paths, then atleast three vertices can not be made internal.If ψ contains atleast one cycle, then atleast one vertex can not be made internal.T heref oret ψ ≥ 1.
Case(ii) When m = odd and n= even, The collection of paths of G are P = (w 11 , w 12 , w 22 , w 21 , w 11 ) is a simple graphoidal cover of G in which w 11 is not internal.Therefore η s (G) ≤ q − p + 1.Now, let ψ be any simple graphoidal cover of G.If ψ contains only paths, then atleast three vertices can not be made internal.If ψ contains atleast one cycle, then atleast one vertex can not be made internal.T heref oret ψ ≥ 1. Hence η s (G) ≥ q − p + 1.Thus η s (G) = q − p + 1.
Case (iii) When m=even and n = odd, The collection of paths of G are P = (w 11 , w 12 , w 22 , w 21 , w 11 ) is a simple graphoidal cover of G in which w 11 is not internal.Therefore η s (G) ≤ q − p + 1.Now, let ψ be any simple graphoidal cover of G.If ψ contains only paths, then atleast three vertices can not be made internal.If ψ contains atleast one cycle, then atleast one vertex can not be made internal.T heref oret ψ ≥ 1.
Case(iv) When m = n=even or m = n = 2.The collection of paths of G are P = (w 11 , w 12 , w 22 , w 21 , w 11 ) is a simple graphoidal cover of G in which w 11 is the only vertex which is not internal.Therefore η s (G) ≤ q − p + 1.Now, let ψ be any simple graphoidal cover of G.If ψ contains only paths, then atleast three vertices can not be made internal.If ψ contains atleast one cycle, then atleast one vertex can not be made internal.
The collection of paths of G are is a simple graphoidal cover of G in which w 11 is not internal.Therefore η s (G) ≤ q − p + 1.Now, let ψ be any simple graphoidal cover of G.Here we have two cases.
Case(i) when m=n, If ψ contains only paths, then atleast mn-(m+n)> 1 vertices can not be made internal, since every path is of length 2. If ψ contains atleast one cycle, then atleast one vertex can not be made internal.
Case(ii) when m = n, If ψ contains only paths, then atleast mn-(m+n)-1 vertices can not be made internal, since every path is of length 2. If ψ contains atleast one cycle,then atleast one vertex can not be made internal.
The collection of paths of G are is a simple graphoidal cover of G in which w 11 and w n1 are not internal.Therefore η s (G) ≤ q − p + 2. Now, let ψ be any simple graphoidal cover of G.If ψ can contains only paths, then atleast four vertices can not be made internal.If ψ can contains atleast one cycle, then atleast two vertices can not be made internal.T heref oret ψ ≥ 2. Hence η s (G) ≥ q−p+2.Thus η s (G) = q−p+2.
Theorem 10.For the graph G= P m ⊙ P n , where m, n ≥ 2 then When m = n= 2, Then G= K 4 , by Theorem 6, η s (G) = 4 When m=2 and n=3, the collection of paths of G are ψ contains only paths, then no vertex can be made internal, since every path is an edge.If ψ contains atleast one cycle, then atleast one vertex can not be made internal.Therefore t ψ ≥ 1. Hence η s (G) ≥ q − p + 1.Thus η s (G) = q − p + 1. when m ≥ 3 and n ≥ 4, The collection of paths of G are is a simple graphoidal cover of G in which all the vertices made internal.By corollary (5), η s (G) = q − p.
Theorem 15.Therefore η s (G) ≤ q − p + 2. Now let ψ be any graphoidal cover of G.Here we have two sub cases.
Sub case(i) When m=2 and n=3, If ψ contains only paths, no vertices can not be made internal, since each path is an edge.If ψ contains atleast one cycle, two vertices can not be made internal, since each path is an edge.T heref oret ψ ≥ 2. Hence η s (G) ≥ q − p + 2. Thus η s (G) = q − p + 2.