δ -Complement of a Graph

: Let G ( V , X ) be a ﬁnite and simple graph of order n and size m . The complement of G , denoted by G , is the graph obtained by removing the lines of G and adding the lines that are not in G . A graph is self-complementary if and only if it is isomorphic to its complement. In this paper, we deﬁne δ -complement and δ (cid:48) -complement of a graph as follows. For any two points u and v of G with deg u = deg v remove the lines between u and v in G and add the lines between u and v which are not in G . The graph thus obtained is called δ -complement of G . For any two points u and v of G with deg u (cid:54) = deg v remove the lines between u and v in G and add the lines between u and v that are not in G . The graph thus obtained is called δ (cid:48) -complement of G . The graph G is δ ( δ (cid:48) ) -self-complementary if G ∼ = G δ ( G ∼ = G δ (cid:48) ) . The graph G is δ ( δ (cid:48) ) -co-self-complementary if G δ ∼ = G ( G δ (cid:48) ∼ = G ) . This paper presents different properties of δ and δ (cid:48) -complement of a given graph.


Introduction
Let G be a simple, finite and undirected graph. The number of lines in G is the size of G, denoted by m. The number of lines incident on a point v is the degree of v denoted by deg v. An open neighborhood of a point v, denoted by N(v) is the set of all points that are adjacent to v. A graph G is Eulerian if it contains a closed trail that covers all the lines of G. A graph G is Hamiltonian if it contains a cycle that visits all the points exactly once. The complement of G is the graph G, obtained from G by removing all the lines of G and adding the lines between the points that are not in G. The graph G is said to be self-complementary if and only if G is isomorphic to G. For more information on self-complementary graphs, one can refer [1][2][3]. For all notations and terminologies we refer to [4,5].

Motivation
1. The concept of δ/δ -complement of a graph can be used in a scenario where the user changes the adjacency of points based on his requirements in order to optimize his end goal. For example, employees working in a company have various skills, qualifications, and strengths.
In comparison to the graph, let graph G specify a task. Each point in G represents an employee of the company. A line connecting two points symbolizes a shared competence between two employees. We explore the following two scenarios.
(i) An employee intends to work with other workers having similar skills, with whom he has not previously worked but does not wish to work with the same skilled employees with whom he has previously worked. This situation is very well described by the δ-complement of a graph.
(ii) Suppose an employee wants to work with workers of different skills with whom he has not previously worked and wishes to discontinue his work with employees of different skills with whom he has previously worked. This situation is definitely achieved via δ -complement of a graph.
2. Let v ij , 1 ≤ i ≤ n be the labeling of the points of a graph, where j represents the degree of a point. Suppose that each degree signifies a certain activity. The number of points with the label v ik represent the number of persons carrying out the kth activity. The line connecting two points indicates that the information, resources, or time required for the execution of the activity has already been shared. Since each member is equally capable, δ(δ )-complement can be used to determine the output when non-adjacent or non-linked members of the same(different) activity are made to share information, resources, or time.
For example, The graph G in Figure 1 consists of five people v 1 , v 2 , v 3 , v 4 and v 5 who are assigned to accomplish three activities, namely, 1, 2, and 3. The person v 1 is in charge of activity 1, the persons v 3 , v 4 and v 5 are in charge of activity 2 and the person v 2 is in charge of activity 3. It is observed that v 3 , v 4 and v 4 , v 5 of activity 2 have already shared information among themselves. Hence, v 3 and v 5 share the information and continue to execute the activity 2 in G δ . The paper is organized as follows. In Section 2, we define δ and δ -complement of a graph and obtain their characterization. In Sections 3 and 4 we discuss some properties of δ and δ -complement of a graph, respectively.

δ/δ -Complement of a Graph
In this section, we define a new graph complement based on the degree sequence. We begin by defining the δ-complement of a graph. Definition 1. Let G(V, X) be any graph of order n and size m. For any two points u and v in V with deg u = deg v remove the lines between u and v in G and add the lines of G between u and v. The graph thus obtained is called δ-complement of G and is denoted by G δ .
Definition 3. Let G(V, X) be any graph of order n and size m. For any two points u and v in V with deg u = deg v remove the lines between u and v in G and add the lines of G between u and v. The graph thus obtained is called δ -complement of G and is denoted by G δ .   The Table 1 displays the degree sequence and number of lines m(G δ ) in δ-complement of various graphs.

Graph
Degree Sequence of δ-Complement

Corollary 2.
For any graph G, Let G = H. Then Equation (2) implies, Similarly we can prove statement 2.
We now characterize δ-self-complement of a graph based on its degree sequence.

Proposition 2.
The graphs G and G δ are degree preserving if and only if V(G) can be partitioned is a partition of the point set of G such that V i has 4k + 1 points with regularity 2k. First, note that every point of each partite is adjacent to all the points of the remaining partites. Otherwise, G cannot be partitioned into l partites such that each V i consisting of 4k + 1 points with regularity 2k.
Thus the graphs G and G δ are degree preserving.
Suppose that the graph G cannot be partitioned into l partites such that each V i consists of 4k + 1 points with regularity 2k. Then the adjacency between the points of different degree is the same in both G and G δ . However, the adjacency between the points of the same degree varies in G δ . Thus, the degree sequence does not remain the same in both G and G δ . Hence, G and G δ are not degree preserving. Proof. Suppose each point of a given degree, say η 0 is adjacent to exactly half of the number of points of different degrees, then in both G and G δ a point of degree η 0 is adjacent to exactly half of the number of points of different degrees. The adjacency of points of degree η 0 remains the same in both G and G δ by definition of G δ . Since each point of a given degree is adjacent to exactly half of the number of points of different degrees, the structure of both G δ and G remains the same. Hence G δ ∼ = G.

Remark 1. A graph G is δ-self-complementary if and only if V(G) can be partitioned as
Suppose each point of a given degree η 0 is not adjacent to exactly half of the number of points of different degrees, then in G a point v of degree η 0 is adjacent to either less or more than the number of points of different degrees as in G. Also in G δ , point v is adjacent to the same number of points of different degrees as in G. Proof. If each point v of a given degree in G is adjacent to exactly half of the number of points of different degrees, then v is adjacent to half of the remaining points of different degrees in G δ and to the same points of given degree as in G. Since every point v of a given degree is adjacent to exactly half of the number of points of different degrees, the adjacency between every pair of points is preserved in G δ . Hence G ∼ = G δ .
Suppose each point of a given degree is not adjacent to exactly half of the number of points of different degrees in G. Then a point of a given degree is adjacent to either less or more than the number of points of different degrees and to equal number of points of the same degree as in G. Thus G G δ .

Proposition 5.
The graphs G and G δ are degree preserving if and only if V(G) can be partitioned as {V 1 , V 2 , . . . , V l }, δ(G) ≤ l ≤ ∆(G) such that V i is 2k regular graph of order 4k + 1.
Proof. Suppose {V 1 , V 2 , . . . , V l }, δ(G) ≤ l ≤ ∆(G) is a partition of the point set of G such that V i has 4k + 1 points with regularity 2k. Every point of each partite is adjacent to all the points of the remaining partites. Otherwise, G cannot be partitioned into l partites such that each V i consisting of 4k + 1 points with regularity 2k. Let order of |V i | = 4k i + 1, 0 ≤ k i ≤ n−1 4 , i = 1, 2, . . . , l. There are k 1 k 2 . . . k l lines between the points of different partites. Then both the graphs G and G δ consist of l connected components where each connected component is a 2k i regular graph. Therefore, the graphs G and G δ are degree preserving.
Suppose the graph G cannot be partitioned into l partites such that each V i consists of 4k + 1 points with regularity 2k. Then the adjacency between the points of different degree is the same in both G and G δ . The adjacency between the points of equal degree remains the same in G δ as in G but it varies in G. Thus, the degree sequence does not remain the same in both G δ and G. Hence, G and G δ are not degree preserving.

Remark 2. A graph G is δ -co-self-complementary if and only if V(G) can be partitioned as
Proposition 6. G δ and G δ of a graph G are degree preserving if and only if G is n−1 2 -regular graph.
Proof. Suppose G is n−1 2 −regular graph, then both G δ and G δ are n−1 2 −regular graph and hence both G δ and G δ are degree preserving graphs. Conversely, from Proposition 2, G and G δ are degree preserving if and only if G contains 4k + 1 points of the same degree and all the points of a given degree are adjacent to exactly 2k points of the same degree. Also from Proposition 4, G ∼ = G δ if and only if each point of a given degree is adjacent to exactly half of the number of points of different degrees. Thus Proposition 3 and 5 together imply that G is n−1 2 −regular graph of order n.

Some Properties of δ-Complement of a Graph
Theorem 1. A graph G δ is a complete graph if and only if G is a complete multipartite graph with the partition of the point set Proof. Since the lines between V i and V j , i = j of G remain in G δ and each V i is isomorphic to complete subgraph, the resultant G δ is a complete graph. Conversely, suppose that G is not a complete multipartite graph, then G may have a line between the points of the same degree or there may exist two non-adjacent points of different degrees in G. If G has a line between the points u and v of same degree, then in G δ , there is no line between u and v and hence G δ is not complete. If G has two non-adjacent points of different degrees, then those two points are non-adjacent in G δ also. Hence G δ is not a complete graph.

Remark 3.
The δ-complement of the star graph of order n is isomorphic to complete graph K n .

Theorem 2.
If G is a disconnected graph of order n with each connected component of G is r−regular, then G δ is a connected (n − r − 1)−regular graph.
Proof. Let G 1 , G 2 , . . . , G n be connected r−regular components of G. Then the degree of Theorem 3. Let G be a connected graph. The graph G δ is disconnected if 1.
G is a complete graph; 2.
G is a connected n − 2 regular graph of order n; 3.
G has a point v of degree k which is only adjacent to every point of degree k.
Proof. Let G be a connected graph.

1.
If G is a complete graph then G δ is completely disconnected by the definition of G δ ; 2.
Suppose G is a connected n − 2 regular graph of order n. Then n must be even and G δ is isomorphic to n 2 K 2 ; 3.
Suppose G has a point v of degree k that is only adjacent to every point of degree k.
Then v is not adjacent to any of the points in G δ . Thus, G δ is disconnected.

Remark 4.
The converse of Theorem 3 need not be true. For example, consider a regular graph C in Figure 4. C is a connected 3-regular graph of order 6, where 3 = n − 1, n − 2. However, the graph C δ is a disconnected 2-regular graph. Consider a non-regular graph of order 8 in Figure 5. Here, no point is adjacent to all the points of the same degree. However, D δ is disconnected. Theorem 4. Let G be an Eulerian graph of order n. Then G δ is Eulerian if G has odd number of points of the same degree and every point of a given degree is adjacent to at most half of the points of same degree and at least half of the points of each different degrees.
Proof. Let G be an Eulerian graph of order n. G has an odd number of points of the same degree. Consider point v ∈ V(G) of degree m. Either v is adjacent to points of degree m or points of degree other than m.
Suppose v is adjacent to an even number of points of degree m and an even number of points of degree other than m. As G has an odd number of points of the same degree and by definition of G δ , even number of lines incident on v are removed and an even number of lines that were non-incident on v in G are added. Hence, the degree of v is even in G δ .
Suppose v is adjacent to an odd number of points of degree m and an odd number of points of degree other than m. In G δ , v is adjacent to an odd number of points of degree m since G has an odd number of points of the same degree and to an odd number of points of a different degree. Therefore, v is of even degree in G δ .
In addition, it is given that every point of a given degree is adjacent to at most half of the points of the same degree and at least half of the points of each different degrees. In G δ , every point is adjacent to at least half of the points of same degree and at most half of the points of each different degrees. Therefore, the degree of each vertex of G δ is at least n−1 2 and hence it is connected. Therefore, the graph G δ is Eulerian. G is an r−regular graph with n ≥ 3, r ≤ n−2 2 ; 2.
G is a non-regular graph with no point v adjacent to any point with a degree, as that of v.
Proof. Let G be a Hamiltonian graph.

1.
If G is an r−regular graph with n ≥ 3, r ≤ n−2 2 , then G δ is n − r − 1 regular graph with n − r − 1 ≥ n 2 . From this, it follows that G δ is Hamiltonian; 2.
Suppose G is a non-regular graph with no point v adjacent to any point with degree as that of v. As every adjacent points of Hamiltonian cycle have different degrees in G, we see that the Hamiltonian cycle of G remains in G δ .
Remark 5. The converse of Theorem 5 need not be true. For example, consider the graph K 8 − C 8 in Figure 6. The graph K 8 − C 8 is a 5-regular graph and 5 > n−2 2 = 3. It is a Hamiltonian graph with Hamiltonian cycle v 1 v 7 v 5 v 3 v 8 v 2 v 4 v 6 v 1 . The δ-complement of the graph K 8 − C 8 is C 8 , which is also a Hamiltonian graph.
Conversely, suppose that u is adjacent to less than m−1 2 points of the equal degree, then in G δ the point u will be adjacent to more than m−1 2 points of the equal degree and the same number of points of unequal degree and hence the degree of u in G δ will be more than the degree of u in G. A similar argument holds if u is adjacent to more than m−1 2 points of the same degree. Thus deg u in G δ = deg u in G.
G is regular or; 2.
G is an even order bi-regular graph of degree r and s such that every point of a particular degree is adjacent to half of the points of different degree.
Proof. Suppose that G is r−regular. Then G δ is a (n − r − 1)−regular graph. Thus deg v in G + deg v in G δ = n − 1 for all v ∈ V(G). Suppose G is a bi-regular graph of degree r and s such that every point of degree r is adjacent to half of the points of degree s. Let n 1 and n 2 be the number of points of degree r and s respectively. Consider a point v of degree r. Suppose v is adjacent to n 2 2 points of degree s and x points of degree r. In G δ the point v is adjacent to n 2 2 points of degree s and n 2 − x − 1 points of degree r. Thus, deg v in G + deg v in G δ = x + n 2 2 + n 1 − x − 1 + n 2 2 = n 1 + n 2 − 1 = n − 1.
Proof. Let G be a Hamiltonian graph. From Table 2, it is clear that G ∼ = G δ if G is regular. Hence, G δ is Hamiltonian.
Remark 7. The converse of Remark 6 need not be true. For example, consider a non-regular graph J in Figure 8. Both the graphs J and J δ are Hamiltonian with Hamiltonian cycles v 1 v 2 v 3 v 4 v 5 v 6 v 1 and v 1 v 3 v 5 v 4 v 6 v 2 v 1 respectively.

Conclusions
In this paper, the authors defined the δ and δ -complement of a graph and explored properties such as degree preserving, δ(δ )-self-complementary, δ(δ )-co-self-complementary, and connectedness. As these complements are based on like/unlike characters, they can be easily applied in social network problems and optimization problems. There is a huge scope to study these complements in comparison with existing self-complementary graphs.