Classification of Planar Graphs Associated to the Ideal of the Numerical Semigroup
Muhammad Ahsan Binyamin, Wajid Ali, Adnan Aslam, Hasan Mahmood
aa r X i v : . [ m a t h . A C ] D ec CLASSIFICATION OF PLANAR GRAPHS ASSOCIATED TO THEIDEAL OF THE NUMERICAL SEMIGROUP
MUHAMMAD AHSAN BINYAMIN, WAJID ALI, ADNAN ASLAM, HASAN MAHMOOD
Abstract.
Let Λ be a numerical semigroup and I ⊂ Λ be an ideal of Λ. Thegraph G I (Λ) assigned to an ideal I of Λ is a graph with elements of (Λ \ I ) ∗ as vertices and any two vertices x, y are adjacent if and only if x + y ∈ I .In this paper we give a complete characterization (up to isomorphism ) of thegraph G I (Λ) to be planar, where I is an irreducible ideal of Λ. This will finallycharacterize non planar graphs G I (Λ) corresponding to irreducible ideal I . Introduction and Preliminaries
In the recent years, the study of algebraic structures through the properties ofgraphs has become an exciting topic of research. This lead to many interestingresults and questions. There are many papers on assigning graphs to rings, groupsand semigroups see [1, 2, 3, 4, 5, 6]. Several authors [7, 8, 9, 10, 11] studied differentproperties of these graphs including diameter, girth, domination, central sets andplanarity. Let Λ be a numerical semigroup. A subset I ⊂ Λ is an ideal (integralideal) of Λ if I + Λ ⊂ Λ. The ideal I is irreducible ideal if it cannot be written asintersection of two or more proper ideals which contained it properly. Throughoutthe paper we consider I to be an irreducible ideal of Λ. Barucci [12] proved thatevery irreducible ideal I of numerical semigroup Λ can be written in the formΛ \ B ( x ), where B ( x ) = { y ∈ Λ : x − y ∈ Λ } , for some x ∈ Λ. Binyamin et al.[13] assigned a graph to numerical semigroup and studied some properties of thisclass of graphs. Recently Peng Xu et al. [14] assigned a graph to the ideal of anumerical semigroup with vertex set { v i : i ∈ (Λ \ I ) ∗ = (Λ \ I ) − { }} and edgeset { v i v j ⇐⇒ i + j ∈ I } . It is easy to observe that if I is an irreducible idealof Λ, then V ( G I (Λ)) = { v i : i ∈ B ∗ ( x ) } for some x ∈ Λ. For every ideal I of Λ,the graph G I (Λ) is always connected [14]. Therefore it is natural to ask when thegraph G I (Λ) is complete? It has been proved in [14], if I is an irreducible ideal ofnumerical semigroup Λ, then the graph G I (Λ) is not a complete graph whenever | V ( G I (Λ)) | ≥ cl ( G I (Λ)) of Λ is given by the formula cl ( G I (Λ)) = (cid:26) n + 1 , if n is even; n +12 , if n is odd,where n is the order of the graph G I (Λ). This shows that whenever n ≥ G I (Λ)has a subgraph isomorphic to complete graph K and hence G I (Λ) is not a planargraph. The motivation of this paper is to find all the graphs G I (Λ) that are planar.To answer this question, It is required to give a complete characterization of thegraph G I (Λ) such that the order of G I (Λ) is either 6 or 7. Key words and phrases.
Numerical Semigroup, Complete Graph, Planarity. graph G = ( V ( G ) , E ( G )) is an ordered pair with the vertex set V ( G ) and theedge set E ( G ). The cardinality of the vertex set and edge set is called the order andsize of G respectively. A graph G is connected if every pair of vertices x, y ∈ V ( G ) isconnected by a path. A graph G of order n is complete if every pair of vertices of G are adjacent and is denoted by K n . The graph G is bipartite if its vertex set V ( G )can be partitioned in to two sets V ( G ) and V ( G ) in such a way that xy ∈ E ( G )if and only if x ∈ V ( G ) and y ∈ V ( G ) or vice versa. If | V ( G ) | = | V ( G ) | = m ,then G is called a complete bipartite graph and is denoted by K m,m . A planargraph is a graph that can be drawn in the plane without crossings that is, no twoedges intersect geometrically except at a vertex to which both are incident. Twographs G and H are said to be homeomorphic if both G and H can be obtainedby a same graph by inserting vertices of degree 2 into its edges. It is well knownthat K , and K are non-planar. In order to show that a graph is planar one canuse the famous Kuratowski’s theorem which states that: A graph is planar if andonly if it contains no subgraph homeomorphic to K or K , . For more undefinedterminologies related to graph theory see [15, 16, 17].The main aim of this paper is to classify all the graphs G I (Λ) of order 6 andorder 7. This, certainly helps us to give a complete answer about the planarity ofthe graph G I (Λ). 2. Planar and Non-planar Graphs G I (Λ)Let Λ = < A > , where A = { a , a , . . . , a n } is the minimal system of generators.Let x ∈ Λ, then x = u a + u a + · · · + u n a n , where u , u , . . . , u n are non negative integers. Note that Λ can be written asΛ = n [ p =1 L ( p ) , where L ( p ) denote the collection of all those elements of Λ, which can be writtenas a linear combination of exactly p elements of A that is if x ∈ L ( p ) then x = u a i + u a i + . . . + u p a i p , where a i , a i , . . . , a i p ∈ A and u , u , . . . , u p are positiveintegers. By using the above notation, for each x ∈ Λ, we define L ( p ) x = { Σ ∈ L ( p ) : Σ = x } . Lemma 2.1.
With the notations defined above, we have B ( x ) ⊇ { v a i + v a i + · · · + v p a i p : 0 ≤ v i ≤ u i , a i , a i , . . . , a i p ∈ A } and | B ( x ) | ≥ p X i =1 u i + X ≤ i
The proof of this Lemma follows from the definition of B ( x ). (cid:3) The following Propositions provide us the bounds in term of L ( p ) x to computethe graph G I (Λ) of order 6 and 7. Proposition 2.2.
Let
Λ = < A > be a numerical semigroup of embedding dimen-sion n ≥ . Then | G I (Λ) |6 = 6 , if one of the following holds: L ( p ) x = ∅ for p ≥ | L (1) x |≥ . (3) L (2) x = ∅ and L (1) x = ∅ . (4) | L (1) x |≥ and | L (2) x |≥ . Proposition 2.3.
Let
Λ = < A > be a numerical semigroup of embedding dimen-sion n ≥ . Then | G I (Λ) |6 = 7 , if one of the following hold: (1) L ( p ) x = ∅ for p ≥ . (2) | L (3) x |≥ . (3) | L (3) x | = 1 and L (1) x = ∅ or L (2) x = ∅ . (4) | L (2) x |≥ . (5) | L (2) x | = 3 and L (1) x = ∅ . (6) | L (2) x | = 2 and | L (1) x |≥ . (7) | L (2) x | = 1 and | L (1) x |≥ . (8) | L (1) x |≥ . In the following, we give the proof of Proposition 2.2, Proposition 2.3 can beproved in a similar way.
Proof. (1) : If L ( p ) x = ∅ for p ≥ a i , a j , a k ∈ A such that x = u a i + u a j + u a k , where u , u , u ≥
1. By Lemma 2.1, a i , a j , a k , a i + a j , a i + a k , a j + a k , x belongs to B ∗ ( x ). Clearly all these elements are distinct and hence | G I (Λ) |≥ | L (1) x |≥ x = u a i , x = u a j and x = u a k , where u , u , u ≥ a i < a j < a k , then u > u > u . It follows that u ≥
5. By Lemma2.1, B ∗ ( x ) contains a i , a i , a i , a i , a i , a j , a k and therefore | G I (Λ) |≥ | L (2) x | = 1 then x = u a i + v a j , where u , v ≥
1. If u = v = 1 then x = a i + a j and Lemma 2.1 gives B ∗ ( x ) = { a i , a j , a i + a j } . For u = 1 and v = 2,we have x = a i + 2 a j and B ∗ ( x ) = { a i , a j , a j , a i + a j , a i + 2 a j } . Similarly, for u = 2 and v = 1 we get B ∗ ( x ) = { a i , a j , a i , a i + a j , a i + a j } . Now if u = 1 and v ≥ a i , a j , a j , a j , a i + a j , a i + 2 a j , a i + 3 a j are distinct elements of B ∗ ( x ).Similarly, for u ≥ v = 1 we get a j , a i , a i , a i , a i + a j , a i + a j , a i + a j ∈ B ∗ ( x ). If u , v ≥ a i , a j , a i , a j , a i +2 a j , a i + a j , a i + a j , a i +2 a j are all distinct elements of B ∗ ( x ). Hence, in all cases, we obtain | G I (Λ) |6 = 6.Now if | L (2) x | = 2 then x = u a i + v a j and x = u a k + v a l , where u , v , u , v ≥ u = 1 and v ≥ u ≥ v = 1, u , v ≥ u = 1 and v ≥ u ≥ v = 1, u , v ≥ | G I (Λ) |≥
7. If a i , a j , a k and a l are not distinct then one can easily checkthat the remaining possibilities does not holds. Now consider a i , a j , a k and a l aredistinct. If u = v = 1 and u = v = 1 then x = a i + a j = a k + a l and B ∗ ( x ) = { a i , a j , a k , a l , a i + a j } , hence | G I (Λ) | <
6. Now if u = v = 1 and u = 2, v = 1 then x = a i + a j = 2 a k + a l . This gives a i , a j , a k , a l , a k + a l , a k , a i + a j distinctelements of B ∗ ( x ) and therefore | G I (Λ) |≥
7. Similarly we get | G I (Λ) |≥
7, forall the remaining possibilities.Finally, if | L (2) x |≥ x = u a i + v a j , x = u a k + v a l and x = u a m + v a q ,where u , v , u , v , u , v ≥
1. This give a i , a j , a k , a l , a m , a q and a i + a j belongs to B ∗ ( x ) and therefore | G I (Λ) |≥
3) : If | L (1) x |≥ | L (2) x |≥
2, then we can assume x = u a i + v a j , x = u a k + v a l , x = ua m and x = va q , where u , v , u , v ≥ u, v ≥
2. Fromcase (2), we have x = a i + a j and x = a k + a l is the only possibility for which | G I (Λ) | <
6. Now if a m < a q with u > v , then x = a i + a j and x = a k + a l gives a i , a j , a k , a l , a m , a q , a m , a i + a j distinct elements of B ∗ ( x ). Hence | G I (Λ) |≥ (cid:3) In the proof of Proposition 2.2, one can observe that there are some cases, wherewe have | G I (Λ) | <
6. Therefore it is possible to get | G I (Λ) | = 6 by addingsome suitable conditions. The similar is the case for Proposition 2.3. In the nexttheorems, we classify all irreducible ideals I such that the graph G I (Λ) is of order6 or order 7. Theorem 2.4.
Let
Λ = < A > be a numerical semigroup of embedding dimension n ≥ . If | G I (Λ) | = 6 then x ∈ Λ is one of the following: (1) x = 6 a i . (2) x = 4 a i = 3 a j . (3) x = 5 a i = 2 a j . (4) x = 4 a i and x = a j + a k . (5) x = 4 a i and a i = 2 a k . (6) x = 2 a i and x = 2 a k + a l . (7) x = 3 a i , x = 2 a j and x = a k + a l . (8) x = 2 a i , x = a k + a l and x = a m + a n .Proof. Given that | G I (Λ) | = 6, then from Proposition 2.2, it follows that x ∈ Λsatisfy one of the following condition: • | L (1) x | = 1 and L ( p ) x = ∅ , ∀ p ≥ • | L (1) x | = 2 and L ( p ) x = ∅ , ∀ p ≥ • | L (1) x | = 1, | L (2) x | = 1 and L ( p ) x = ∅ , ∀ p ≥ • | L (1) x | = 2, | L (2) x | = 1 and L ( p ) x = ∅ , ∀ p ≥ • | L (1) x | = 1, | L (2) x | = 2 and L ( p ) x = ∅ , ∀ p ≥ Case − : If | L (1) x | = 1 and L ( p ) x = ∅ , ∀ p ≥ x = la i , where l ≥ B ∗ ( x ) = { a i , a i , . . . , la i } . As | G I (Λ) | = 6, it follows that l = 6. Case − : if | L (1) x | = 2 and L ( p ) x = ∅ , ∀ p ≥ x = ua i and x = va j ,where a i < a j , 1 < v < u and u is not a multiple of v . Then it follows fromLemma 2.1, we have B ∗ ( x ) = { a i , a i , . . . , ua i , a j , a j . . . , ( v − a j } . Let pa i = qa j for some q < p < u with q = 2 , , . . . , v −
1. Then pa i + ( u − p ) a i = ua i and we get qa j + ( u − p ) a i = x. This give L (2) x = ∅ , a contradiction. Therefore a i , a i , . . . , ua i , a j , a j . . . , ( v − a j are distinct elements of B ∗ ( x ). As | G I (Λ) | = 6,therefore either u = 4 and v = 3 or u = 5 and v = 2. This gives the case (2) andcase (3). Case − : Let | L (1) x | = 1 = | L (2) x | , then x = ra i and x = ua k + va l , where r ≥ u, v ≥
1. By Proposition 2.2(2), we have x = a k + a l or x = 2 a k + a l or x = a k + 2 a l . If x = a k + a l and x = ra i , it is easy to observe that a i , a k , a l are distinctelements of A and B ∗ ( x ) = { a k , a l , a i , a i , . . . , ( r − a i , x } . then | B ∗ ( x ) | = 6 gives r = 4 and we get the case (4).Now if x = 2 a k + a l and x = ra i then B ∗ ( x ) = { a k , a l , a k , a k + a l , a i , a i , . . . ,x = ra i } . Note that a i = a k is not possible, otherwise we get a l is a multiple of a i . f a i = a l then a i < a k , otherwise we get either a i = 2 a k or 2 a i = 2 a k . These bothconditions are not possible. Now a i < a k gives r > | B ∗ ( x ) | = 6 gives r = 4.Hence we get x = 4 a i and 3 a i = 2 a k which is the case (5). Now if a i , a k , a l aredistinct then | B ∗ ( x ) | = r + 4. As | B ∗ ( x ) | = 6 therefore r = 2 and we get the case(6). Also if x = a k + 2 a l and x = ra i then again we get the cases (5) and (6). Case − : If | L (1) x | = 2 and | L (2) x | = 1, then x = ra i , x = sa j and x = ua k + va l , where r, s ≥ u, v ≥
1. Assume that a i < a j then 1 < s < r and r is not a multiple of s . By Proposition 2.2(2), we have x = a k + a l or x = 2 a k + a l or x = a k + 2 a l . For u = 2 and v = 1, we get B ∗ ( x ) = { a k , a l , a k , a k + a l , a j , a j , . . . , ( s − a j , a i , a i , . . . , ra i } . Similarly, for u = 1 and v = 2 we get B ∗ ( x ) = { a k , a l , a l , a k + a l , a j , a j , . . . , ( s − a j , a i , a i , . . . , ra i } . Note that a i or a j can not be equal to a k , otherwise we get a l is a multiple of a i or a j . Now if a i = a l or a j = a l then | B ∗ ( x ) | = r + s + 2 and if a i , a j , a k , a l are different then | B ∗ ( x ) | = r + s + 3, as 1 < s < r therefore | B ∗ ( x ) | >
6. Now if u = 1 , v = 1 then a i , a j , a k , a l are different and B ∗ ( x ) = { a k , a l , a j , a j , . . . , ( s − a j , a i , a i , . . . , ra i } .This gives | B ∗ ( x ) | = r + s + 1. | B ∗ ( x ) | = 6 gives r = 3 and s = 2 which is thecase (7). Case − : If | L (1) x | = 1 and | L (2) x | = 2, then x = u a k + v a l , x = u a m + v a n and x = ra i , where r ≥ u , u , v , v ≥
1. By Proposition 2.2(2), we have x = a k + a l and x = a m + a n . In this case we have B ∗ ( x ) = { a k , a l , a m , a n , a i , a i , . . . , ra i } .Since a k , a l , a m , a n , a i are distinct, it follows that | B ∗ ( x ) | = r + 4. | B ∗ ( x ) | = 6gives r = 2 which is the case (8). (cid:3) Theorem 2.5.
Let
Λ = < A > be a numerical semigroup of embedding dimension n ≥ . If | G I (Λ) | = 7 then x ∈ Λ satisfy one of the following: (1) x = 7 a i . (2) x = 5 a i = 3 a j . (3) x = 3 a i + a j or x = a i + 3 a j . (4) x = 5 a i and x = a j + a k . (5) x = 3 a i and x = 2 a j + a k or x = a j + 2 a k . (6) x = a i + a j and x = 2 a k + a l . (7) x = 3 a i and x = a k + a l and x = a m + a n . (8) x = a i + a j and x = a k + a l and x = a m + a n . (9) x = a i + a j + a k .Proof. Given that | G I (Λ) | = 7, then from Proposition 2.3, it follows that x satisfyone of the following condition: • | L (1) x | = 1 and L ( p ) x = ∅ , ∀ p ≥ • | L (1) x | = 2 and L ( p ) x = ∅ , ∀ p ≥ • L (1) x = ∅ , , | L (2) x | = 1, L ( p ) x = ∅ , ∀ p ≥ • | L (1) x | = 1, | L (2) x | = 1, L ( p ) x = ∅ , ∀ p ≥ • | L (1) x | = 1, | L (2) x | = 1, L ( p ) x = ∅ , ∀ p ≥ • L (1) x = ∅ , , | L (2) x | = 2, L ( p ) x = ∅ , ∀ p ≥ • | L (1) x | = 1, | L (2) x | = 2, L ( p ) x = ∅ , ∀ p ≥ • L (1) x = ∅ , | L (2) x | = 3, L ( p ) x = ∅ , ∀ p ≥ • | L (3) x | = 1, L ( p ) x = ∅ , ∀ p = 3 and x = a i + a j + a k . hese possibilities can be checked in a similar way as we did in Theorem 2.4 to getthe required result. (cid:3) Theorem 2.4 and Theorem 2.5 give us all irreducible ideals I such that G I (Λ) isof order 6 or order 7. This is easy to see that there are some cases where we get thegraphs G I (Λ) which are isomorphic to each other. Our next propositions classifyall graphs G I (Λ) upto isomorphism of order 6 and order 7. Corollary 2.6.
A graph G I (Λ) of order is isomorphic to one of the graphs givenin the Table : Table 1
Type Degree sequence Graph , , , , , q qq qq q ✑✑✑✑✑✑✑✑✁✁✁✁✁✁✁✁❆❆❆ , , , , , q qq qq q ✁✁✁❆❆❆✑✑✑✑✑✑✑✑✁✁✁✁✁✁✁✁❆❆❆◗◗◗◗ , , , , , q qq qq q ✑✑✑✑✑✑✑✑✁✁✁✁✁✁✁✁❆❆❆◗◗◗◗❆❆❆✁✁✁ , , , , , q qq qq q ✑✑✑✑✑✑✑✑✁✁✁✁✁✁✁✁❆❆❆◗◗◗◗◗◗◗◗✁✁✁ ❆❆❆❆❆ , , , , , q qq qq q ✁✁✁❆❆❆✑✑✑✑✑✑✑✑✁✁✁✁✁✁✁✁❆❆❆❆❆❆❆❆ , , , , , q qq qq q ✑✑✑✑✑✑✑✑✁✁✁✁✁ ❆❆❆❆❆❆ ✁✁✁✁✁✁ ❆❆❆❆❆◗◗◗◗ roof. This is an easy consequence of Theorem 2.4. (cid:3)
Corollary 2.7.
A graph G I (Λ) of order is isomorphic to one of the graphs givenin the Table : Table 2
Type Degree sequence Graph , , , , , , q qq q qq q ✟✟✟ (cid:0)(cid:0)(cid:0) ✂✂✂✂ ❇❇❇❇❅❅❅❍❍❍✘✘✘✘✘✏✏✏✏✑✑✑✑(cid:0)(cid:0)✁✁✁ , , , , , , q qq q q qq ❅❅❅❍❍❍❍❍❤❤❤❤❤❤❤✭✭✭✭✭✭✭✟✟✟✟✟(cid:0)(cid:0)(cid:0)✘✘✘✘✘✟✟✟❆❆❆❆❆ ❏❏❏❏ ❩❩❩❩❩ ❍❍❍ ❳❳❳❳❳ , , , , , , q qq q q qq ❅❅❅❍❍❍❍❍❤❤❤❤❤❤❤✭✭✭✭✭✭✭✟✟✟✟✟(cid:0)(cid:0)(cid:0)✚✚✚✚✚✁✁✁✁✁✡✡✡✡❆❆❆❆❆ ❏❏❏❏ ❩❩❩❩❩ ❍❍❍ ❳❳❳❳❳ , , , , , , q qq q q qq ❅❅❅❍❍❍❍❍❤❤❤❤❤❤❤✭✭✭✭✭✭✭✟✟✟✟✟(cid:0)(cid:0)(cid:0)✁✁✁✁✁✡✡✡✡❆❆❆❆❆ ❏❏❏❏ ❩❩❩❩❩ ❍❍❍ ❳❳❳❳❳ , , , , , , q qq q q qq ❅❅❅❍❍❍❍❍❤❤❤❤❤❤❤✭✭✭✭✭✭✭✟✟✟✟✟(cid:0)(cid:0)(cid:0)✘✘✘✘✘✚✚✚✚✚✟✟✟✡✡✡✡❆❆❆❆❆ ❏❏❏❏ ❩❩❩❩❩ ❍❍❍ ❳❳❳❳❳ , , , , , , q qq q q qq ❅❅❅❍❍❍❍❍❤❤❤❤❤❤❤✭✭✭✭✭✭✭✟✟✟✟✟(cid:0)(cid:0)(cid:0)✘✘✘✘✘✚✚✚✚✚✁✁✁✁✁✟✟✟✡✡✡✡❆❆❆❆❆ ❏❏❏❏ ❩❩❩❩❩ ❍❍❍ ❳❳❳❳❳ roof. This is an easy consequence of Theorem 2.5. (cid:3)
Theorem 2.8.
Let Λ be a numerical semigroup of embedding dimension n ≥ .Then G I (Λ) is planar if one of the following hold: (1) | G I (Λ) |≤ . (2) | G I (Λ) | = 6 and G I (Λ) is of type , , or . (3) | G I (Λ) | = 7 and G I (Λ) is of type , or .Proof. (1) is trivial.(2) If | G I (Λ) | = 6 and G I (Λ) is of type 1 , G I (Λ) is trivially planar.Now if G I (Λ) is of type 4 then B ∗ ( x ) = { a i , a j , a k , a j + a k , a j , a i + a j , x } withdeg( v a i ) = 5, deg( v a j ) = 3, deg( v a k ) = 3, deg( v a j + a k ) = 4, deg( v a j ) = 4, anddeg( v x ) = 5.Since | G I (Λ) | = 6, therefore cl ( G I (Λ)) = 4. This shows that G I (Λ) cannot havea subgraph which is isomorphic to complete graph K . Now consider a subgraph H of G I (Λ) such that | H | = 6 and the degree sequence of H is (3 , , , , , V ( H ) = V ( G I (Λ)) and E ( H ) = E ( G I (Λ)) − { v a i v x , v a i v a j , v a j + a k v x } or V ( H ) = V ( G I (Λ)) and E ( H ) = E ( G I (Λ)) − { v a i v x , v a i v a j + a k , v a j v x } . In bothcases, one can easily see that H ≇ K , .(3) can be proved in a similar way as we proved (2). (cid:3) Theorem 2.9.
Let Λ be a numerical semigroup of embedding dimension n ≥ .Then G I (Λ) is non-planar if one of the following hold: (1) | G I (Λ) |≥ , (2) | G I (Λ) | = 6 and G I (Λ) is of type or . (3) | G I (Λ) | = 7 and G I (Λ) is of type , or .Proof. ( ) : If | G I (Λ) |≥ cl ( G I (Λ)) ≥ G I (Λ) must has asubgraph isomorphic to complete graph K .( ) : Assume that | G I (Λ) | = 6 and G I (Λ) is of type 5 then B ∗ ( x ) = { a i , a j , a i , a j , a i , a i } with deg( v a i ) = 3, deg( v a j ) = 4, deg( v a i ) = 4, deg( v a j ) = 4, deg( v a i ) =4 and deg( v a i ) = 5. Since | G I (Λ) | = 6, therefore cl ( G I (Λ)) = 4. This showsthat G I (Λ) cannot have a subgraph which is isomorphic to complete graph K .Consider a subgraph H of G I (Λ) such that V ( H ) = V ( G I (Λ)) and E ( H ) = E ( G I (Λ)) − { v a j v a i , v a i v a i , v a j v a i } . Note that we can partition the set ofvertices of H into V = { a i , a i , a i } and V = { a j , a j , a i } such that no edgehas both endpoints in the same subset and every possible edge that could connectvertices in different subsets is part of the graph. This implies H is isomorphic tocomplete bipartite graph K , and therefore G I (Λ) is non-planar.Remaining cases of ( ) and ( ) can be proved in a similar way. v a i ✉ v a i ✉ v a j ✉ v a i ✉ v a j ✉ v a i (cid:0)(cid:0)(cid:0)✏✏✏✏✏✏✏✏✏◗◗◗◗◗◗◗◗◗❅❅❅PPPPPPPPP✑✑✑✑✑✑✑✑✑✡✡✡✡✡✡✡✡✡ Fig-1 ✉ v a i ✉ v a i ✉ v a i ✉ v a j ✉ v a j ✉ v a i ❅❅❅❅❅❅❆❆❆❆❆❆✁✁✁✁✁✁ ❆❆❆❆❆❆(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) ✁✁✁✁✁✁ Fig-2 (cid:3) Conclusion
In this article, we have given a complete answer about the planarity of the graph G I (Λ) associated with the irreducible ideal of a numerical semigroup. However, forany integral ideal I , this is an open question. Conflict of Interests:
The authors hereby declare that there is no conflict ofinterests regarding the publication of this paper.
Data Availability Statement:
No data is required for this study.
Funding Statement:
This research is carried out as a part of the employmentof the authors.
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Muhammad Ahsan Binyamin, Department of Mathematics, GC University, Faisalabad,Pakistan
Email address : [email protected] Wajid Ali, Department of Mathematics, Department of Mathematics, GC University,Faisalabad, Pakistan
Email address : [email protected] Adnan Aslam, Department of Natural Sciences and Humanities, University of Engi-neering and Technology, Lahore (RCET) 54000, Pakistan.
Email address : [email protected] Hasan Mahmood, Department of Mathematics, GC University, Lahore, Pakistan
Email address : [email protected]@gcu.edu.pk