On the $2$-packing differential of a graph

Let $G$ be a graph of order $n(G)$ and vertex set $V(G)$. Given a set $S\subseteq V(G)$, we define the external neighbourhood of $S$ as the set $N_e(S)$ of all vertices in $V(G)\setminus S$ having at least one neighbour in $S$. The differential of $S$ is defined to be $\partial(S)=|N_e(S)|-|S|$. In this paper, we introduce the study of the $2$-packing differential of a graph, which we define as $\partial_{2p}(G)=\max\{\partial(S): S\subseteq V(G) \text{ is a }2\text{-packing}\}.$ We show that the $2$-packing differential is closely related to several graph parameters, including the packing number, the independent domination number, the total domination number, the perfect differential, and the unique response Roman domination number. In particular, we show that the theory of $2$-packing differentials is an appropriate framework to investigate the unique response Roman domination number of a graph without the use of functions. Among other results, we obtain a Gallai-type theorem, which states that $\partial_{2p}(G)+\mu_{_R}(G)=n(G)$, where $\mu_{_R}(G)$ denotes the unique response Roman domination number of $G$. As a consequence of the study, we derive several combinatorial results on $\mu_{_R}(G)$, and we show that the problem of finding this parameter is NP-hard. In addition, the particular case of lexicographic product graphs is discussed.


Introduction
The differential of a set S ⊆ V (G) is defined as ∂(S) = |N e (S)| − |S|, while the differential of a graph G is defined to be ∂(G) = max{∂(S) : S ⊆ V (G)}.
As described in [17], the definition of ∂(G) was given by Hedetniemi about twenty-five years ago in an unpublished paper, and was also considered by Goddard and Henning [10]. After that, the differential of a graph has been studied by several authors, including [2,5,17,22,24] Lewis et al. [17] motivated the definition of differential from the following game, what we call graph differential game. "You are allowed to buy as many tokens as you like, say k tokens, at a cost of one dollar each. You then place the tokens on some subset D of k vertices of a graph G. For each vertex of G which has no token on it, but is adjacent to a vertex with a token on it, you receive one dollar. Your objective is to maximize your profit, that is, the total value received minus the cost of the tokens bought". Obviously, ∂(D) = |N e (D)| − |D| is the profit obtained with the placement D, while the maximum profit equals ∂(G).
In order to introduce a variant of this game, we need the following terminology. The 2-packing number of G, denoted by ρ(G), is defined to be ρ(G) = max{|S| : S ∈ ℘(G)}. Now, we consider a version of the graph differential game in which we impose two additional conditions: (1) every vertex which has no token on it is adjacent to at most one vertex with a token on it; and (2) no vertex with a token on it is adjacent to a vertex with a token on it. In this case, any placement D of tokens is a 2-packing and so this version of the game can be called 2-packing graph differential game, as the maximum profit equals the 2-packing differential of G, which is defined as ∂ 2p (G) = max{∂(S) : S ∈ ℘(G)}.
We define a ∂ 2p (G)-set as a set S ∈ ℘(G) with ∂(S) = ∂ 2p (G). The same agreement will be assumed for optimal parameters associated to other characteristic sets defined in the paper.
Let G be the graph shown in Figure 1. In the graph differential game, if we place the tokens on vertices a, b and c, then we obtain the maximum profit ∂(G) = 7. In contrast, in the 2-packing graph differential game, the maximum profit ∂ 2p (G) = 6 is given by the placement of token on vertices a and b. a b c Figure 1: A graph G with ∂(G) = 7 and ∂ 2p (G) = 6.
In this paper we show that the 2-packing differential is closely related to several graph parameters, including the packing number, the independent domination number, the total domination number, the perfect differential, and the unique response Roman domination number. In particular, we show that the theory of 2-packing differentials is an appropriate framework to investigate the unique response Roman domination number of a graph without the use of functions.
The rest of the paper is organized as follows. Section 2 is devoted to provide some general bounds on the 2-packing differential, in terms of different parameters such as the maximum and minimum degrees of the graph, the number of vertices, the number of edges, among others. We show that the bounds are tight and, in some cases, we characterize the graphs achieving them. As a consequence of the study, we show that the problem of finding ∂ 2p (G) is NP-hard. In Section 3 we obtain a Nordhaus-Gaddum type theorem for both the addition and the product of the 2-packing differential of a graph and its complement. Section 4 is devoted to the study of the relationship between the 2-packing differential and the perfect differential of a graph. In Section 5 we prove a Gallai-type theorem which states that ∂ 2p (G) + µ R (G) = n(G), where µ R (G) denotes the unique response Roman domination number of G. We derive several consequences of this result, including the fact that the problem of finding µ R (G) is NP-hard. We finally show the case of the lexicographic product graph in Section 6, where we obtain general lower and upper bounds of the 2-packing differential and the unique response Roman domination number. We also compute the exact value of both parameters for the lexicographic product of a path and any other graph.

Basic results and computational complexity
In this section we present lower and upper bounds of the 2-packing differential of a graph. In some cases, we also characterize the graphs achieving such bounds. Moreover, these results will provide the computational complexity of the computation of the 2-packing differential.
We begin with the following theorem that will be a key result in the rest of the paper. The bounds presented here are in terms of the maximum and minimum degrees of the graph and the 2-packing number.
Theorem 2.1. If G is a graph with no isolated vertex, then the following statements hold.
Proof. Notice that for any S ∈ ℘(G), we have that Therefore, (i) follows. From (i) we derive (ii) and (iii). We proceed to prove (iv). If there exists a ρ(G)-set S such that deg(v) = ∆(G) for Finally, we prove (v). Suppose on the contrary that there exists a ρ(G)-set S ′ and a vertex v ∈ S ′ such that deg(v) > δ(G). Then, ∂ 2p (G) = max Figure 2 shows an example of a graph G with ∂ 2p (G) = ρ(G)(δ(G) − 1), the set of black-coloured vertices is the unique ρ(G)-set.
The first consequence of the preceding result is the determination of the complexity of the computation of the 2-packing differential. Given a graph G and a positive integer t, the 2-packing problem is to decide whether there exists a 2-packing S in G such that |S| is at least t. It is well known that the 2-packing problem is NP-complete. Hence, the optimization problem of finding ρ(G) is NP-hard. Furthermore, it was shown in [16] that the 2-packing remains NP-complete for regular bipartite graphs. Therefore, from Theorem 2.1 (ii) we derive the following result.
Corollary 2.2. The problem of finding the 2-packing differential of a graph is NP-hard, even for regular bipartite graphs.
As a second consequence of Theorem 2.1 we can derive the exact value of the 2packing differential in particular graph families. To this end, we need the following definitions. A Observe that γ(G) ≥ ρ(G) and D(G) ∩ ℘(G) = ∅ if and only if there exists a γ(G)-set which is a ρ(G)-set. A graph G with D(G) ∩ ℘(G) = ∅ is called an efficient closed domination graph and in such case γ(G) = ρ(G) . Furthermore, Meir and Moon [19] showed that γ(T ) = ρ(T ) for every tree T . Even so, there are trees in which If G is a k-regular nonempty graph and there exists S ∈ D(G) ∩ ℘(G), then n(G) = |S| + |N(S)| = |S| + |S|k = ρ(G)(1 + k). Therefore, the following result is a direct consequence of Theorem 2.1 (ii). Corollary 2.3. Let k be a positive integer. If G is a k-regular efficient closed domination graph, then The next statement is a direct consequence of Theorem 2.1 (i). We can also involve the order n(G) and the size m(G) of the graph and we obtain the following upper bound of the 2-packing differential.
Theorem 2.5. For any graph G with no isolated vertex, Furthermore, the equality holds if and only if there exists a ρ(G)-set S such that Therefore, the upper bound follows.
From the previous procedure we deduce , and by the upper bound we conclude that the equality holds.
Notice that for any k-regular graph, 2 m(G) = n(G)δ(G) = n(G)k, which implies The bound presented in Theorem 2.5 has a special behaviour in efficient closed domination graphs, as we show in the following two results.
Theorem 2.6. If G is an efficient closed domination graph with no isolated vertex, then the following statements hold.
By Theorems 2.5 and 2.6 we deduce the following result.
Theorem 2.7. For any efficient closed domination graph G with δ(G) ≥ 1, Furthermore, the equality holds if and only if there exists a ρ(G)-set S such that Proof. By combining the bounds given in Theorems 2.5 and 2.6, we deduce that 2 m( , then we have equalities in the above inequality chain, and by Theorem 2.5 we conclude that there exists a ρ(G)-set S such that deg . We now present lower and upper bounds of the 2-packing differential in terms of some dominating-type parameters. We begin with the following definition.
A set S ⊆ V (G) which is both dominating and independent is called an independent dominating set. Moreover, the independent domination number i(G) is In the following result we present both bounds and we characterize graphs attaining the upper one.
Theorem 2.8. For any graph G with no isolated vertex, Furthermore, the following statements are equivalent.
. Let S 1 be an independent dominating set of the subgraph induced by S ′′ . Notice that this subgraph does not have isolated vertices, which implies that |S 1 | < |S ′′ |. Also, observe that the set S ∪ S ′ ∪ S 1 is an independent dominating set of G. Hence, Also, notice that |N(S)| ≤ ∆(G)|S|. Hence, Hence, the upper bound follows. Now, if (i) holds, then we have equalities in the previous inequality chains. From these, we deduce that Finally, if (ii) holds, i.e., there exist an i(G)-set D and a 2- Therefore, from the upper bound we deduce that the equality holds.
The converse of the result above does not hold. For instance, let G be a graph Next, we consider a particular case of the equivalence given in Theorem 2.8.
Theorem 2.10. For any graph G with no isolated vertex, Furthermore, the equality holds if and only if ∆(G) = n(G) − i(G).
. Therefore, the upper bound follows. Now, if ∆(G) = n(G) − i(G), then from the bounds given in Theorem 2.8 we deduce We now characterize all trees attaining the lower bound presented in Theorem 2.8. To this end, we will use the following notation. Given a tree T , a vertex v ∈ V (T ) with deg(v) = ∆(T ) and any vertex v i ∈ N(v), denote by T i (v) the sub-tree of T rooted in v i obtained from T by removing the edge {v, v i }. For each i ∈ {1, . . . , ∆(T )} such that T i (v) is non-trivial, denote by w i the neighbour of v i in T i (v) with the largest degree.
The distance between two vertices x, y ∈ V (T ) will be denoted by d(x, y).
If there exists w ∈ V (T ) such that d(v, w) = 4, then for u ∈ N(w) with d(v, u) = 3 the set {v, u} is a 2-packing and deg(u) is non-trivial} is a 2-packing and this gives (iii). Now, let T be a non-trivial tree satisfying (i), (ii) and (iii), let S be a 2-packing of T and let v be a vertex with deg(v) = ∆(T ). Since ∂ 2p (T ) ≥ ∆(T ) − 1, we proceed to show that ∂ 2p (T ) ≤ ∆(T ) − 1. To this end, we differentiate the following three cases.
Case 1: v ∈ S. If there exists u ∈ S\{v}, then d(u, v) ≥ 3 and condition (i) gives that deg(u) = 1 (see Figure 4). Therefore, In such a case, u belongs to a non-trivial rooted sub-tree T j (v), j = i and d(u, v) ≥ 2 (see Figure 4). Clearly, S ∩ V (T j (v)) = {u} and from (ii) we obtain ∂(S)  Some upper bounds of the 2-packing differential can be also expressed in terms of the total domination number. A set S ⊆ V (G) is a total dominating set of a graph G without isolated vertices if every vertex v ∈ V (G) is adjacent to at least one vertex in S. Let D t (G) be the set of total dominating sets of G. The total domination number of G is defined to be, γ t (G) = min{|S| : S ∈ D t (G)}.
Lemma 2.12. Let G be a graph with no isolated vertex. Let S be a ∂ 2p (G)-set such that |S| is maximum among all ∂ 2p (G)-sets, The following statements hold.
Proof. Suppose that there exists a vertex v ∈ S ′′ such that N(v) ⊆ S ′′ . Notice that S v = S∪{v} is a 2-packing of G and that ∂(S v ) ≥ ∂(S) = ∂ 2p (G), which is a contradiction because |S v | > |S|. Hence, for every v ∈ S ′′ , it follows that N(v) ∩ N(S) = ∅. Now, we define W ′ as a set of minimum cardinality among the sets W ⊆ N(S) satisfying that N(x) ∩ W = ∅ for every x ∈ S ∪ S ′ ∪ S ′′ .
Theorem 2.14. For any graph G with no isolated vertex, Proof. Let S be a ∂ 2p (G)-set such that |S| is maximum among all ∂ 2p (G)-sets, and let m ′ be the size of the subgraph of G induced by V (G) \ S. Notice that every vertex in N(S) has exactly one neighbour in S, which implies that Observe that Lemma 2.12 leads to γ t (G) ≤ 2|S| + |V (G) \ N[S]|. Hence, To conclude this section, we derive some bounds in terms of the order of G.

Nordhaus-Gaddum type relations
Nordhaus and Gaddum [20] in 1956 proposed lower and upper bounds, in terms of the order of the graph, on the sum and the product of the chromatic number of a graph and its complement. Since then, several inequalities of a similar type have been proposed for other graph parameters. In this section we derive some Nordhaus-Gaddum type relations for the 2-packing differential.
A By Theorem 2.15 and the lower bound given in Theorem 2.8, we deduce the following result.
Theorem 3.1. Given a graph G, the following statements hold. • If ∆(G) = n(G) − 2 and δ(G) ≥ 3, then 4) if and only if G is an efficient open domination graph with ρ o (G) = 2.
• If ∆(G) ≤ n(G) − 4 and δ(G) ≥ 3, then • 4 Perfect differential versus 2-packing differential In this section we show some relationships between the 2-packing differential and the perfect differential of a graph. Given a set S ⊆ V (G) and a vertex v ∈ S, the external private neighbourhood epn(v, S) of v with respect to S is defined to be epn(v, We define the perfect differential of a set S ⊆ V (G) as ∂ p (S) = |N p (S)| − |S|. The perfect differential of a graph, introduced by Cabrera Martínez and Rodríguez-Velázquez in [5], is defined as  [5] Given a nontrivial graph G, the following inequality chain holds.
if and only if γ p (G) = n(G) ∆(G)+1 . The following theorems show some relationships between the 2-packing differential and the perfect differential.
Theorem 4.2. For any nontrivial graph G, the following statements hold.
The difference ∂ p (G)−∂ 2p (G) can be as large as desired. Consider the graph obtained by attaching to each vertex of a 5-cycle a set of k independent vertices. The resulting graph G has order n(G) = 5k + 5 and satisfies ∂ p (G) = 5k − 5 (the set of vertices of the 5-cycle is a ∂ p (G)-set) and ∂ 2p (G) = k + 1 (each single vertex in the 5-cycle is a ∂ 2p (G)set). Moreover, the set of vertices of the 5-cycle is a minimum perfect dominating set, so γ p (G) = 5 and ∂ p (G) = n(G) − 2γ p (G) takes the smallest possible value.
Regarding the case in which both parameters agree, we have the following result. (i) ∂ 2p (G) = ∂ p (G).
The bound above is tight. For instance, for any integer t ≥ 2, the double star S t,t satisfies that ∂ p (S t,t ) = 2t − 2, ∂ 2p (S t,t ) = t, n(S t,t ) = 2t + 2 and ∆(S t,t ) = t + 1. Hence, the bound is achieved by any double star S t,t with t ≥ 2.

Unique response Roman domination versus 2-packing differential
In this section we establish a Gallai-type theorem which states the relationship between the 2-packing differential and the unique response Roman domination number. Cockayne, Hedetniemi and Hedetniemi [7] defined a Roman dominating function on a graph G to be a function f : V (G) −→ {0, 1, 2} satisfying the condition that every vertex u for which f (u) = 0 is adjacent to at least one vertex v for which f (v) = 2. For The unique response version of Roman domination was introduced by Rubalcaba and Slater in [23] and studied further in [9,14,29]. A function f : The unique response Roman domination number of G, denoted by µ R (G), is the minimum weight among all unique response Roman dominating functions on G. Theorem 5.1 allows us to derive results on the unique response Roman domination number from results on the 2-packing differential and vice versa. For instance, from Corollary 2.2 and Theorem 5.1 we deduce the following result.
Theorem 5.2. The problem of finding the unique response Roman domination number of a graph is NP-hard, even for regular bipartite graphs.
Theorem 5.1 suggests the challenge of obtaining new results on the unique response Roman domination number from the approach of 2-packing differentials. As an example, the following table summarizes some of those results obtained here. The first column describes the result that combined with Theorem 5.1 leads to the result on the second column.

From
Result for graphs with no isolated vertex The equality holds if and only if there exists a ρ(G)-set S such that deg(v) = ∆(G) for every v ∈ S.
The equality holds if and only if there exists a ρ(G)-set S such that deg(v) = δ(G) for every v ∈ V (G) \ S.
The equality holds if and only if ∆(G) = n(G) − i(G).
The equality holds if and only if G is an efficient closed domination graph and γ t (G) = 2γ(G). .

The case of lexicographic product graphs
In this final section we present the behaviour of the 2-packing differential under the lexicographic product operation. We first recall the following definition. Let G and H be two graphs. The lexicographic product of G and H is the graph G • H whose vertex set is V (G • H) = V (G) × V (H) and (u, v)(x, y) ∈ E(G • H) if and only if ux ∈ E(G) or u = x and vy ∈ E(H). Notice that for any u ∈ V (G) the subgraph of G • H induced by {u} × V (H) is isomorphic to H. For simplicity, we will denote this subgraph by H u . For a basic introduction to the lexicographic product of two graphs we suggest the books [11,13]. One of the main problems in the study of G • H consists of finding exact values or tight bounds of specific parameters of these graphs and express them in terms of known invariants of G and H. In particular, we cite the following works on domination theory of lexicographic product graphs: (total) domination [18,21,28], Roman domination [25], weak Roman domination [27], rainbow domination [26], super domination [8], doubly connected domination [1], secure domination [15], double domination [3] and total Roman domination [6,4].
The following claim, which states the distance formula in the lexicographic product of two graphs, is one of our main tools. Remark 6.1. [11] For any connected graph G of order n(G) ≥ 2 and any graph H, the following statements hold.
Given a set W ⊆ V (G) × V (H), the projection of W on V (G) will be denoted by P G (W ). The following corollary is a direct consequence of the previous remark. In the following result we show general lower and upper bounds for the 2-packing differential of a lexicographic product graph, in terms of some parameters of both factors.
We define the following parameter.