ON 2 ACYCLIC SIMPLE GRAPHOIDAL COVERING OF BICYCLIC GRAPHS

A 2−simple graphoidal cover of G is a set ψ of (not necessarily open) paths in G such that every edge is in exactly one path in ψ and every vertex is an internal vertex of at most two paths in ψ and any two paths in ψ has at most one vertex in common. The minimum cardinality of the 2−simple graphoidal cover of G is called the 2−simple graphoidal covering number of G and is denoted by η2s. A 2−simple graphoidal cover ψ of a graph G is called 2−acyclic simple graphoidal cover if every member of ψ is a path. The minimum cardinality of a 2−acyclic simple graphoidal cover of G is called the 2−acyclic graphoidal covering number of G and is denoted by η2as. This paper discusses 2−acyclic simple graphoidal cover on bicyclic graphs.


INTRODUCTION
In graph theory, Graph Decomposition is the one of the fastest-growing research topics and plays a major role in Road Network, Block design and so on. A decomposition of a graph G is a collection of edge disjoint subgraphs H 1 , H 2 , . . . , H n of G such that every edge of G is in exactly one H i . Several authors [1] [3][7] [9] [12] impose different conditions and parameters to

PRELIMINARIES
All the graph G = (V, E) in this paper is a nontrivial, simple-connected, and undirected graphs. The number of elements of V is said to be the order of G is expressed by p and the number of elements in the E are said to be the size of G is expressed by q. For graph theoretic terminology, Harary [8] is referred. The vertices u 0 and u l are called external vertices of P and u 1 , u 2 , . . . , u l−1 are internal vertices of P, where P = (u 0 , u 1 , u 2 , . . . , u l−1 , u l ) is a path or cycle in G. Two paths P 1 and P 2 are said to be internally disjoint if no vertex of G is an internal vertex of both P 1 and P 2 . The graphoidal cover introduced and discussed by Acharya and Sampath Kumar [1] [2]. 2−graphoidal path cover introduced by Nagarajan et.al [9]. 2−graphoidal cover extensively studied and discussed by Das and Singh [7]. The authors [12] discuss about 2−acyclic simple graphoidal cover and discusses 2−acyclic simple graphoidal covering on standard graph.
In this paper, the authors determine the 2−acyclic simple graphoidal cover on bicyclic graphs. Definition 2.1 (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(G). Definition 2.2 (9). An 2−graphoidal cover of a graph G is a collection ψ of paths (not necessarily open) in G such that (i) Every path in ψ has atleast two vertices.
(ii) Every edge is exactly in one path ψ.
(ii) Every vertex is an internal vertex of at most two paths in ψ. The minimum cardinality of a 2−graphoidal cover of G is called the 2− graphoidal covering number of G and is denoted by η 2 (G). Definition 2.3 (12). A 2−simple graphoidal cover of a graph G is a 2−graphoidal cover ψ of G such that any two paths in ψ have at most one vertex in common. The minimum cardinality of a 2−simple graphoidal cover of G is called the 2−simple graphoidal covering number of G and is denoted by η 2s (G).
Definition 2.4 (12). A 2−acyclic simple graphoidal cover of G is said to be 2− simple graphoidal cover ψ of G such that every member ψ of G is a path. The minimum cardinality of a 2−acyclic simple graphoidal cover of G is called the 2−acylic simple graphoidal covering number of G and is denoted by η 2as (G).
Definition 2.5. Let ψ be a collection of internally disjoint paths in G. A vertex v of G is said to be an interior vertex of ψ if it is an internal vertex of some path in ψ. Otherwise, it is said to be an exterior vertex.
Notations 2.6 (9). Let ψ be a 2−acyclic simple graphoidal cover of G. The following notations are used in the theorems. Here i ψ (P),t 1 (ψ),t 2 (ψ),t ψ denotes the number of internal vertices of the path P, the number of vertices appear as internal vertex exactly in one path ψ , the number of vertices appears as internal vertex exactly in two paths of ψ and the number of vertices are not internal in ψ respectively.
If 2−acyclic simple graphoidal cover ψ of G is minimum, then it is clear that t 1 (ψ),t 2 (ψ) should be maximum and t(ψ) should be minimum. We define t i = max t i (ψ) (i = 1, 2) where the maximum is taken over all 2−acyclic simple graphoidal covers of ψ of G and t = min t ψ where the minimum is taken over all 2−acyclic simple graphoidal cover ψ of G.
Corollary 2.8 (12). There exists a 2−acyclic simple graphoidal cover ψ of G in which every vertex is internal vertex in exactly 2 paths in ψ of G if and only if η 2as (G) = q − 2p. ON (12). Let G be a unicycle graph with n pendent vertices. Let C be the unique cycle on G. Let l be the number of vertices of degree greater than 2 on C. Then where m is the total number of vertices of degree ≥ 4 on G.
Definition 2.11 (6). A one-point union of two cycles is a simple graph obtained from two cycles, say C l and C m where l, m ≥ 3, by identifying one and the same vertex from both cycles. Without loss of generality, we assume C l = (u 0 , u 1 , . . . , u l−1 , u 0 ) and C m = (u 0 , u l , u l+1 , . . . , u m+l−2 , u 0 ).
This graph is denoted by U(l : m).
Definition 2.12 (6). A long dumbbell graph is a simple graph obtained by joining two cycles C l and C m where l, m ≥ 3, with a path of length i, i ≥ 1. Without loss of generality, we may assume C l = (u 0 , u 1 , . . . , u l−1 , u 0 ), P i = (u l−1 , u l , . . . , u l+i−1 ) and C m = (u l+i−1 , u l+i , .., u l+m+i−2 , u l+i−1 ). This graph is denoted by D(l : m : i).

MAIN RESULTS
Theorem 3.1. Let G be a bicyclic graph with n pendant vertices. Also let U(l : m) be the unique bicycle in G and let l be the number of vertices of degree greater than 2 on C . Then Proof. Let C l = (u 0 , u 1 , . . . , u l−1 , u 0 ) and C m = (u 0 , u l , . . . , u l+m−2 , u 0 ) be two cycles sharing a common vertex say u 0 with q = p + 1.
is a minimum 2−acyclic simple graphoidal cover of G so that η 2as (G) = 4.
Case 2. When l = 1 and deg(u 0 ) ≥ 5 Let P = (w, x), where P be a path on U(l : m) and w, x ∈ C m . Take G 1 = G − P is a unicyclic graph with (n + 2) pendent vertices and m vertices is of degree ≥ 4 with l = 1. Hence by theorem 2.9, η 2as (G 1 ) = (n + 2) + 2 − m = (n + 4) − m. Let ψ 1 be the minimum 2−acyclic simple graphoidal cover of G 1 . Then ψ = ψ 1 ∪ {P} is a minimum 2−acyclic simple graphoidal cover of G, hence η 2as (G) ≤ (n + 5) − m. For any 2−acyclic simple graphoidal cover of G, n pendent vertices and atleast four vertices in U(l : m) are external and atmost m vertices are internal twice. Therefore t ψ ≥ (n + 4),t 2 (ψ) ≤ m. Hence t ≥ (n + 4),t 2 ≤ m so that η 2as (G) = Case 3. When l = 2 and let u 0 , v be the two vertices is of degree greater than two on U(l : m) be a path on U(l : m) and w, x ∈ C m . It is clear that Case 4. When l = 3 and let u 0 , u, v be the vertices is of degree greater than two on U(l : m) and Then there are two subcases. Therefore   Therefore t ψ ≥ (n + 2),t 2 (ψ) ≤ m. Hence t ≥ (n + 2),t 2 ≤ m so that η 2as (G) = q − p − t 2 + t = Therefore t ψ ≥ (n + 2),t 2 (ψ) ≤ m. Hence t ≥ (n + 2),t 2 ≤ m so that η 2as Subcase 5.2. Suppose u, v ∈ C l and w ∈ C m , then there are two subcases. Take G 1 = G − P, where P = (w, y) be a path on U(l : m) and y ∈ C m . It is clear that G 1 is a unicyclic graph with (n + 1) pendent vertices and m vertices is of degree ≥ 4 with l = 3. By theorem 2.9, η 2as (G 1 ) = (n + 1) − m. Let ψ 1 be the minimum 2−acyclic simple graphoidal cover of G 1 . Then ψ = ψ 1 ∪ {P} is a 2−acyclic simple graphoidal cover of G, hence η 2as (G) ≤ (n + 2) − m. For any 2−acyclic simple graphoidal cover of G, n pendent vertices and atleast one vertex in U(l : m) are external and atmost m vertices are internal twice. Therefore t ψ ≥ Take G 1 = G − P, where P = (w, y) be a path on U(l : m) and y ∈ C m . It is clear that G 1 is a unicyclic graph with (n + 1) pendent vertices and (m − 1) vertices is of degree ≥ 4 with l = 3. By theorem 2.9, η 2as (G 1 ) = (n + 1) − (m − 1) = (n + 2) − m. Let ψ 1 be the minimum 2−acyclic simple graphoidal cover of G 1 . Let P 1 be a path in ψ 1 in which x is an external vertex. Therefore Case 6. When l ≥ 5 and let u 0 , u, v, w, x be the vertices is of degree greater than two on U(l : m) subcases.
where P = (y, z) be a path on U(l : m) and y, z ∈ C m . It is clear that Therefore Subcase 6.2. When u, v, w ∈ C l and x ∈ C m and let P = (x, y) be a path in U(l : m) and y ∈ C m .
Then there are two subcases. Therefore be the minimum 2−acyclic simple graphoidal cover of G 1 . Let P 1 be a path in ψ 1 in which x is an external vertex. Then ψ = (ψ 1 − P 1 ) ∪ {P 1 ∪ P} is a 2−acyclic simple graphoidal cover of For any 2−acyclic simple graphoidal cover of G, n pendent vertices and atleast one vertex in U(l : m) are external and atmost m vertices are internal twice. Therefore t ψ ≥ (n + 1),t 2 (ψ) ≤ m. Hence t ≥ (n + 1),t 2 ≤ m so that Subcase 6.3. Suppose u, v ∈ C l and w, x ∈ C m and let P = (w, x) be a path in U(l : m). Then there are three subcases.
For any 2−acyclic simple graphoidal cover of G, n pendent vertices are external and atmost m vertices are internal twice. Therefore Take G 1 = G − P is a unicyclic graph with n pendent vertices and (m − 2) vertices is of degree Let ψ 1 be the minimum 2−acyclic simple graphoidal cover of G 1 . Let P 1 be a path in ψ 1 in which x is an external vertex and P 2 be a path in ψ 1 in which w is an external vertex. Then ψ = ( For any 2−acyclic simple graphoidal cover of G, n pendent vertices are external and atmost m vertices are internal twice. Therefore t ψ ≥ n, Here p = 33, q = 34, n = 14, l = 6 and m = 4, then Case 2. Suppose l = 2 and let P denote (u l+i−1 , w) section of C m such that it has atleast one internal vertex say u i and w ∈ C m . Let P 1 and P 2 denote the (u l+i−1 , u i ) and (u i , w) section of P respectively. Then there are two subcases.
Case 3. When l = 3 and let u be the only vertex is of degree greater than 2 other than u l−1 , u l+i−1 . Let P denote (u l+i−1 , w) section of C m such that it has atleast one internal vertex say u i and w ∈ C m . Let P 1 and P 2 denote the (u l+i−1 , u i ) and (u i , w) section of P respectively.
where Let P = (y, z) be a path in C m and y, z ∈ C m . It is clear that G 1 is a unicyclic graph with (n + 2) pendnent vertices and m vertices is of degree ≥ 4 with l > 3.
Hence by therorem 2.9, η 2as (G 1 ) = (n + 2) − m. Let ψ 1 be the minimum 2−acyclic simple graphoidal cover of G 1 . Take ψ = ψ 1 ∪ {P} is a 2−acyclic simple graphoidal cover of G so and let P denotes (u l+i−1 , w) section of C m such that it has atleast one internal vertex say u i .
and let P denotes (w, x) section of C m . Then there are three subcases.
Take G 1 = G − P where P = (y, z) be a path in C m and y, z ∈ C m . It is clear that G 1 is a unicyclic graph with (n + 2) pendnent vertices and m vertices is of degree ≥ 4 with l > 3.
Hence by theorem 2.9, η 2as (G 1 ) = (n + 2) − m. Let ψ 1 be the minimum 2−acyclic simple graphoidal cover of G 1 . Take ψ = ψ 1 ∪ {P} is a 2−acyclic simple graphoidal cover of G, hence 3, u, v, w ∈ C l , x ∈ C m and let P denotes (u l+i−1 , x) section of C m such that it has atleast one internal vertex say u i . Let P 1 and P 2 denote the (u l+i−1 , u i ) and (u i , x) section of P respectively.

CONFLICT OF INTERESTS
The author(s) declare that there is no conflict of interests.