On the Signed $2$-independence Number of Graphs

In this paper, we study the signed 2-independence number in graphs and give new sharp upper and lower bounds on the signed 2-independence number of a graph by a simple uniform approach. In this way, we can improve and generalize some known results in this area.


Introduction
Throughout this paper, let G be a finite connected graph with vertex set V = V (G) and edge set E = E(G). We use [13] as a reference for terminology and notation which are not defined here. The open neighborhood of a vertex v is denoted by N (v), and the closed neighborhood of v is N [v] = N (v) ∪ {v}. The minimum and maximum degree of G are respectively denoted by ∆(G) = ∆ and δ(G) = δ.
Let S ⊆ V . For a real-valued function f : V → R we define f (S) = v∈S f (v). Also, f (V ) is the weight of f . A signed 2-independence function, abbreviated S2IF, of G is defined in [14] as a function f : V → {−1, 1} such that f (N [v]) ≤ 1, for every v ∈ V . The signed 2-independence number, abbreviated S2IN, of G is α 2 s (G) = max{f (V )|f is a S2IF of G}. This concept was defined in [14] as a certain dual of the signed domination number of a graph [3] and has been studied by several authors including [8,10,11,12]. A set S ⊆ V is a dominating set if each vertex in V \S has at least one neighbor in S. The domination number γ(G) is the minimum cardinality of a dominating set [7]. A subset B ⊆ V is a 2-packing in G if for every pair of vertices u, v ∈ B, d(u, v) ≥ 3. The 2-packing number (or packing number) ρ(G) is the maximum cardinality of a 2-packing in G.
Gallant et al. [5] introduced the concept of limited packing in graphs. They exhibited some real-world applications of it to network security, NIMBY, market saturation and codes. In this paper we exhibit an application of it to signed 2-independence number in graphs. In fact as it is defined in [5], a set of vertices B ⊆ V is called a k-limited packing in G provided that for all v ∈ V , we have |N [v] ∩ B| ≤ k. The limited packing number, denoted L k (G), is the largest number of vertices in a k-limited packing set. It is easy to see that L 1 (G) = ρ(G). In [6], Harary and Haynes introduced the concept of tuple domination in graphs.
The k-tuple domination number, denoted γ ×k (G), is the smallest number of vertices in a k-tuple dominating set. When k = 2, D is called a double dominating set and the 2-tuple domination number is called the double domination number and is denoted by dd(G). In fact the authors showed that every graph G with δ ≥ k − 1 has a k-tuple dominating set and hence a k-tuple domination number.
By a simple uniform approach, we derive many new sharp bounds on α 2 s (G) in terms of several different graph parameters. Some of our results improve some known bounds on the S2IN of graphs in [8,11,12].
The authors noted that most of the existing bounds on α 2 s (G) are lower bounds. In section 2, we prove that α 2 s (G) ≥ 2⌊ δ+2ρ(G) 2 ⌋ − n, for a graph G of order n. Also in section 3, by a simple connection between the concepts of limited packing and tuple domination, we obtain the exact value of the signed 2-independence numbers of regular graphs. In particular, we bound the signed 2-independence numbers of cubic graphs from below and above just in terms of order as, − n 3 ≤ α 2 s (G) ≤ 0.

Main results
At this point we are going to present some sharp upper bounds on α 2 s (G). First, let us introduce some notation. Let f : Theorem 2.1. Let G be a graph of order n. Then and this bound is sharp. Using , we obtain the desired upper bound. For sharpness it is sufficient to consider the complete graph K n .
In [8] the author established a relationship between the signed 2-independence number and the domination number of a graph as follows.

and this bound is sharp.
Now we are going to improve Theorem 2.2. We shall need the following result, which can be found implicit in [4] and explicit in [2] as Corollary 81.
Proof. Let f be a maximum S2IF of G. We have shown that For sharpness it is sufficient to consider the complete graph K n .
By the concept of limited packing we can present a sharp lower bound on α 2 s (G) that involves the packing number.
Theorem 2.5. Let G be a connected graph of order n. Then and this bound is sharp.
Repeating these inequalities, we have Now let B be a maximum (⌊ δ 2 ⌋ + 1)-limited packing set in G. We define f : We deduce that as desired. Considering the graph K n we can see that this bound is sharp.
Volkmann in [11] proved that if G is a graph of order n, then 2 − n ≤ α 2 s (G). Moreover if n ≥ 3, then 4 − n ≤ α 2 s (G). Obviously, the lower bound in Theorem 2.5 is an improvement of the first inequality and when δ ≥ 2 this improves the second, as well. At the end of this section we exhibit a short comment about signed 2-independence number of bipartite graphs. The following upper bound on α 2 s (G) of a bipartite graph was obtained by Wang [12]. Theorem 2.6. ( [12]) If G is a bipartite graph of order n ≥ 2, then Furthermore, the bound is sharp.
We now improve the bound in the previous theorem.
Theorem 2.7. Let G be a bipartite graph of order n. Then and this bound is sharp.
Proof. Let f be a maximum S2IF of G. Let X and Y be the partite sets of G. For convenience we define X + = X ∩ V + , X − = X ∩ V − and let Y + and Y − be defined, analogously. Obviously, Since every vertex in X + has at least ⌈ δ 2 ⌉ neighbors in Y − , by the pigeonhole principle, there exists a vertex v in Y − that is joined to at least A similar argument shows that Using inequalities (2) and (3) we have Using |V + | = n − |V − |, we obtain . Now, by using the value of |V − | we derive the desired bound.

Remarks on signed 2-independence in regular graphs
Zelinka [14] obtained the following sharp upper bound on α 2 s (G) for regular graphs G.
Theorem 3.1. ( [14]) If G is an r-regular graph of order n, then α 2 s (G) ≤ n r+1 when r is even and α 2 s (G) ≤ 0 when r is odd.
We note that the bound in Theorem 2.1 implies the previous result. The authors in [9] proved the following result. Lemma 3.1. ([9]) Let G be a graph. Then the following statements hold.
Now, by the above lemma we are able to obtain the exact value of the signed 2-independence number of regular graphs, first in terms of order and limited packing number, second in terms of order and tuple domination number. At the end we bound α 2 s (G) of a cubic graph G from above and below, just in terms of the order. First we need the following lemma.
Lemma 3.2. Let G be a graph of order n, then . Now let f be a maximum S2IF of G. In the proof of Theorem 2.1 we have shown that   In [1], the authors showed that if G is a cubic graph of order n, then n 3 ≤ L 2 (G). Moreover, the upper bound L 2 (G) ≤ n 2 was presented in [5] for a cubic graph G. Therefore Corollary 3.2 leads to − n 3 ≤ α 2 s (G) ≤ 0 for cubic graphs.