The Modified Negative Decision Number in Graphs

A mapping x : V → {−1, 1} is called negative if∑u∈N v x u ≤ 1 for every v ∈ V. The maximum of the values of ∑ v∈V x v taken over all negative mappings x, is called the modified negative decision number and is denoted by β′ D G . In this paper, several sharp upper bounds of this number for a general graph are presented. Exact values of these numbers for cycles, paths, cliques and bicliques are found.


Introduction
Let G V, E be a graph with vertex set V and edge set E. For a vertex v ∈ V , the open neighbourhood of v in G is N v {u ∈ V | uv ∈ E}, and N v N v ∪ {v} is its closed neighbourhood.For a graph G, and a subset S ⊆ V , we let deg S v denote the number of vertices in S joined to v.In particular, deg V v deg v , the degree of v in G.For disjoint subsets S and T of vertices, we use E S, T for the set of edges between S and T, and let e S, T |E S, T |.Let G S denote the subgraph of G induced by S. For an integer k ≥ 3, the bipartite graph K 1,k is called a star, and the vertex of degree k is called the central vertex.Let x : V → Ê be a real-valued function.For convenience, we write x S for v∈S x v for S ⊆ V .
In 1 , we initiated the study of the negative decision number in a graph.A function x : V G → {−1, 1} is called a bad function of G if x N v ≤ 1 for every v ∈ V G .The maximum of the values of x V G , taken over all bad functions f, is called the negative decision number and is denoted by β D G .
The motivation for studying this parameter may be explained from a modelling perspective.For instance, by assigning the values −1 or 1 to the vertices of a graph, one can model networks of people in which global decisions must be made e.g., positive or negative responses .In certain circumstances, a positive decision can be made only if there are significantly more people voting for than those voting against.We assume that each International Journal of Mathematics and Mathematical Sciences individual has one vote, and each has an initial opinion.We assign 1 to vertices individuals which have a positive opinion and −1 to vertices which have a negative opinion.A voter votes "good" if there are at least two more vertices in its open neighbourhood with positive opinion than with negative opinion, otherwise the vote is "bad".We seek an assignment of opinions that guarantee a unanimous decision; namely, for which every vertex votes "bad".Such an assignment of opinions is called a uniformly negative assignment.Among all uniformly negative assignments of opinions, we are particularly interested in the minimum number of vertices individuals which have a negative opinion.The negative decision number is the maximum possible sum of all opinions, 1 for a positive opinion and −1 for a negative opinion, in a uniformly negative assignment of opinions.The negative decision number corresponds to the minimum number of individuals who can have negative opinions and in doing so force every individual to vote bad.
In the present paper, we study a variation of the negative decision number in a graph.A mapping x : V → {−1, 1} is called negative if u∈N v x u ≤ 1 for every v ∈ V .The maximum of the values of v∈V x v , taken over all negative mappings x, is called the modified negative decision number and is denoted by β D G .
The parameter β D differs significantly from β D .For instance, for a cycle C n of order n, Throughout this paper, if x is a negative mapping of G, then we let P and Q denote the sets of those vertices of G which are assigned under x the values 1 and −1, respectively, and we let p |P | and q |Q|.Hence, x V p − q.All graphs considered in this paper are simple, finite, and undirected.For a general reference on graph theory, the reader is referred to 2, 3 .Notice that β D G ∪ H β D G β D H for two disjoint graphs G and H. Hence, we assume that all graphs are connected in this paper.
In the next section, we present several sharp upper bounds on the modified negative decision number for a general graph.We also establish sharp upper bounds on this number for bipartite graphs and regular graphs.Exact values of these numbers for cycles, paths, cliques, and bicliques are found.

Main Results
In this section, we investigate the modified negative decision number of a graph.We first present an upper bound on this number for a general graph in terms of its order.
For this purpose, we define a family F of graphs as follows.Let F 3 K 1,2 , and for k ≥ 4, let F k be the graph obtained from the disjoint union of k stars K 1,k 1 by adding all possible edges between the central vertices of the k stars.See Figure 1 for an example of and this bound is sharp.
Proof.Let x be a negative mapping such that Note that every vertex in P must be joined to at least one vertex in Q.By the pigeonhole principle, at least one vertex v in Q is joined to at least |P |/|Q| n − q /q vertices in P .Hence, That is, Solving the above inequality, we obtain that To see this bound is sharp, let G ∈ F. Thus, G F k for some k ≥ 4. Hence, G has order n k k 2 , and so k √ n 1 − 1. Assigning the value −1 to each of the k central vertices of the stars, and 1 to all other vertices, we define a negative mapping Next we give an upper bound of the modified negative decision number for a general graph in terms of its order and size.Theorem 2.2.If G is a graph of order n ≥ 2 and size m, then and this bound is sharp.
Proof.Let x be a negative mapping such that As every vertex in P must be joinedto at least one vertex in Q, e P, Q ≥ p n − q.For each International Journal of Mathematics and Mathematical Sciences Hence, Thus, the total number of edges in G is 2.9 So, q ≥ 3n − 2m /5.Hence,

2.10
To see this bound is sharp, let G F k ∈ F for some k ≥ 4 and let x be the negative mapping of G defined in the proof of Theorem 2.1.As x V k 2 , β D G ≥ k 2 .On the other hand, as G has order n k k 2 and size m k k In the following theorem, we establish an upper bound of the modified negative decision number for a bipartite graph in terms of its order and we characterize the graphs attaining this bound.We define a family H of bipartite graphs as follows.
For k ≥ 1, let H k be the bipartite graph obtained from the disjoint union of 2k stars K 1,k 2 with centres {x 1 , x 2 , . . ., x k , y 1 , y 2 , . . ., y k } by adding all edges of the type x i y j , 1 ≤ i, j ≤ k.See Figure 2 for an example of

2.11
The equality holds if and only if G ∈ H.
Proof.Let x be a negative mapping of G such that β D G x V .Let X and Y be the partite sets of G. Further, let X and X − be the sets of vertices in X that are assigned the value 1 and −1 under x , respectively.Let Y and Y − be defined analogously.Then P As every vertex in X must be joined to at least one vertex in Y − , by the pigeonhole principle, at least one vertex v in Y − is joined to at least

2.16
Solving the above inequality for s t, we obtain that

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If β D G n 6 − 2 √ 2n 9, then by the above analysis, we have s t and k l s s 2 .Moreover, each vertex of X resp., Y has degree 1 and is joined to one vertex of Y − resp., X − , while each vertex of Y − is joined to all vertices of X − and s 2 vertices of X and each vertex of X − is joined to all vertices of Y − and s 2 vertices of Y .Thus, Hence, G has order n 2k k 3 , and so k 1/2 √ 2n 9 − 3 .Assigning to the k central vertices of the stars the value −1, and to all other vertices 1, we define a negative mapping Finally, we present an upper bound of the modified negative decision number for a regular graph in terms of its order in Theorem 2.5.We encourage the reader to verify our next theorem first.
Theorem 2.4.For any integer n ≥ 2, one has 1 for n odd.

2.18
For a general k-regular graph, we have the following.
for k even; 0 for k odd.

2.19
This bound is sharp.
Proof.Let x be any negative mapping of G.As G is a k-regular graph,

2.20
We discuss the following two cases.
Case 1. k is even.Since x is a negative mapping, for every vertex v ∈ V , x N v ≤ 1.By 2.20 , it follows that The sharpness follows from Theorem 2.4.

2.23
Proof.As C n is a 2-regular graph, by Theorem 2.5, we have . ., v n be the n vertices of C n in a clockwise order.We discuss the following three cases.
Case 1. n 3k for some integer k.To show that β D C n k, it suffices to show that there is a negative mapping x of C n such that x V C n k.In fact, assigning 1, 1, −1 starting with v 1 clockwise, and repeating, we produce a negative mapping x of C n satisfying x V C n k.
Case 2. n 3k 2 for some integer k.To show that β D C n k, it suffices to show that there is a negative mapping x of C n such that x V C n k.In fact, if we assign 1, 1, −1, 1, −1 to the vertices in a clockwise order when n 5, or assign 1, 1, −1, . .., 1, 1, −1, 1, −1 starting with v 1 clockwise when n > 5, then we produce a negative mapping x of C n satisfying x V C n k.
Case 3. n 3k 1 for some integer k.To show that β D C n ≥ k − 1, it suffices to show that there is a negative mapping x of C n such that x V C n k − 1.In fact, if we assign 1, −1, 1, −1 to the vertices in a clockwise order when n 4, or assign 1, 1, −1, . . ., 1, 1, −1, −1 starting with v 1 clockwise when n > 4, then we produce a negative mapping Corollary 2.7.For any integer n ≥ 2, one has

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Proof.It is easy to see that Corollary 2.7 is true for 2 ≤ n ≤ 4. So we may assume that n ≥ 5. Let v 1 , v 2 , . . ., v n be the n vertices of P n with end vertices v 1 and v n .We only show that β D P n ≤ n/3.The rest of the proof is similar to that of Corollary 2.6, so is omitted.Take any negative mapping x of P n .For i 1 or n, as deg v i 1, we have that For the biclique K n,n , we have the following result.

2.30
To prove the case when n is even, it suffices to show that there exists a negative mapping x of K n,n such that x V K n,n 0. In fact, we can produce such x by assigning x v i x u i −1 i , 1 ≤ i ≤ n.Now we prove the case when n is odd.Let x be a negative mapping of K n,n such that β D K n,n x V K n,n .Let X , X − , Y , and Y − be defined the same as in the proof of Theorem 2.

2.31
Notice that we may assume k ≤ s and l ≤ t.If k l s t, then k s and l t contradicting to the fact k s l t n, where n is odd.Hence, β D K n,n ≤ −2.
To show that β D K n,n −2, it suffices to show that there is a negative mapping x of K n,n such that x V K n,n −2.In fact, we can produce such x by assigning x u n −1, x u i −1 i−1 for 1 ≤ i ≤ n − 1 and x v j −1 j , 1 ≤ j ≤ n.

Figure 1 :
Figure 1: A copy of the graph F 4 .

Figure 2 :
Figure 2: A copy of the graph H 2 .

3 .
Then P X ∪ Y and Q X − ∪ Y − .Let |X | k, |X − | s, |Y | l and |Y − | t.Thus, β D K n,n k l − s − t 2n − 2 s t .As β D K n,n ≤ 0,by 2.30 , we have that k l ≤ s t.