On the game domination number of graphs with given minimum degree

In the domination game, introduced by Bre\v{s}ar, Klav\v{z}ar and Rall in 2010, Dominator and Staller alternately select a vertex of a graph $G$. A move is legal if the selected vertex $v$ dominates at least one new vertex -- that is, if we have a $u\in N[v]$ for which no vertex from $N[u]$ was chosen up to this point of the game. The game ends when no more legal moves can be made, and its length equals the number of vertices selected. The goal of Dominator is to minimize whilst that of Staller is to maximize the length of the game. The game domination number $\gamma_g(G)$ of $G$ is the length of the domination game in which Dominator starts and both players play optimally. In this paper we establish an upper bound on $\gamma_g(G)$ in terms of the minimum degree $\delta$ and the order $n$ of $G$. Our main result states that for every $\delta \ge 4$, $$\gamma_g(G)\le \frac{30\delta^4-56\delta^3-258\delta^2+708\delta-432}{90\delta^4-390\delta^3+348\delta^2+348\delta-432}\; n.$$ Particularly, $\gamma_g(G)<0.5139\; n$ holds for every graph of minimum degree 4, and $\gamma_g(G)<0.4803\; n$ if the minimum degree is greater than 4. Additionally, we prove that $\gamma_g(G)<0.5574\; n$ if $\delta=3$.


Introduction
In this note, our subject is the domination game introduced by Brešar, Klavžar and Rall in [4]. The domination game, introduced by Brešar, Klavžar and Rall [4], is played on a simple undirected graph G = (V, E) by two players, named Dominator and Staller, respectively. They take turns choosing a vertex from V such that a vertex v can be chosen only if it dominates at least one new vertex -that is, if we have a u ∈ N[v] for which no vertex from N[u] was selected up to this turn of the game. The game is over when no more legal moves can be made; equivalently, when the set D of vertices chosen by the two players becomes a dominating set of G. The aim of Dominator is to finish the game as soon as possible, while that of Staller is to delay the end of the game. The game domination number γ g (G) is the number of turns in the game when the first turn is Dominator's move and both players play optimally. Analogously, the Staller-start game domination number γ ′ g (G) is the length of the game when Staller begins and the players play optimally.

Results
Although the subject is quite new, lots of interesting results have been obtained on the domination game (see [2,3,4,5,6,7,9,13,14]). Note that also the total version of the domination game was introduced [11] and studied [12] recently.
Concerning our present work, the bounds proved for the game domination number γ g (G) are the most important preliminaries. The following fact was verified in [4] and [13] as well.

Conjecture 1
If G is an isolate-free graph of order n, then γ g (G) ≤ 3n/5 holds.
Conjecture 1 has been proved for the following graph classes: • for trees of order n ≤ 20 (Brešar, Klavžar, Košmrlj and Rall [3]); • for caterpillars -that is, for trees in which the non-leaf vertices induce a path (Kinnersley, West and Zamani [13]); • for trees in which no two leaves are at distance four apart (Bujtás [6,7]).
Moreover, in a manuscript in preparation, Henning and Kinnersley prove Conjecture 1 for graphs of minimum degree at least 2 [10].
On the other hand, upper bounds weaker than 3n/5 were obtained for some wider graph classes. For trees, the inequality γ g (G) ≤ 7n/11 was established by Kinnersley, West and Zamani in [13] and it was recently improved to γ g (G) ≤ 5n/8 by the present author in [7]. For the most general case, Kinnersley, West and Zamani proved [13] that the game domination number of any isolate-free graph G of order n satisfies γ g (G) ≤ ⌈7n/10⌉. In Section 2 we improve this upper bound by establishing the following claim.
Proposition 1 For any isolate-free graph G of order n, In fact, in a manuscript under preparation [8] we will prove the stronger inequality γ g (G) ≤ 0.64n, but the proof of Proposition 1 may be of interest because of its simplicity and gives illustration for the proof technique applied in the later sections.
One of our main results gives an upper bound smaller than 0.5574n on the game domination number of graphs with minimum degree 3.
Theorem 1 For any graph G of order n and with minimum degree 3, For graphs all of whose vertices are of degree greater than 3, we prove an upper bound in terms of the order and the minimum degree.
Theorem 2 If G is a graph on n vertices and its minimum degree is δ(G) ≥ d ≥ 4, then As the coefficient in this upper bound equals 37/72 < 0.5139 for d = 4, and equals 2102/4377 < 0.4803 for d = 5, the following immediate consequences are obtained.

Corollary 1
(i) For any graph G of order n and with minimum degree δ(G) = 4, the inequality γ g (G) ≤ 37n/72 holds.
(ii) For any graph G of order n and with minimum degree δ(G) ≥ 5, the inequality γ g (G) ≤ 2102n/4377 holds.
Particularly, these statements show that the coefficient 3/5 in Conjecture 1 can be significantly improved if only those graphs with δ(G) ≥ 4 are considered.
On the other hand, note that Theorem 1 and Theorem 2 establish new results only for 3 ≤ δ(G) ≤ 21. Although it was not mentioned in the earlier papers, the upper bound in (1) together with the well-known theorem (see e.g., [1]) for each δ ≥ 2. For integers 3 ≤ δ(G) = d ≤ 21, it is easy to check that our bound is better than the above one in (2).
Our proof technique is based on a value assignment to the vertices where the value of a vertex depends on its current status in the game. We will consider a greedy strategy of Dominator, where the greediness is meant concerning the decrease in the values. Our main goal is to estimate the average decrease in a turn achieved under this assumption. We have been introduced this type of approach in the conference paper [6] and in the paper [7]. The frame of this technique and the basic observations are contained here in Section 2. Then, in Section 3 and Section 4 we specify the details and prove our Theorem 1 and Theorem 2 respectively. In the last section we make some additional notes concerning the Staller-start version of the game.

Preliminaries
Here we introduce the notion of the residual graph, define the color assignment to the vertices and give a general determination for the phases of the game. Then, we take some simple observations which will be used in the later sections.
Colors Consider any moment of the process of a domination game on the graph G * = (V, E), and denote by D the set of vertices chosen up to this point of the game. As it was introduced in [6] and [7], we distinguish between the following three types of vertices.
Residual graph Clearly, a red vertex v and all its neighbors are already dominated in the game. Hence the choice of v would not be a legal move in the later turns and further, the status of v remains red. So, red vertices do not influence the continuation of the game and they can be deleted. Similarly, edges connecting two blue vertices can be omitted too. This graph, obtained after the deletion of red vertices and edges between two blue vertices, is called residual graph, as it was introduced in [13]. At any point of the game, the set of vertices chosen up to this point is denoted by D and the residual graph is denoted by G. When it is needed, we use the more precise notations D i and G i for the current D and G just before the ith turn.
Phases of the game The phases will be defined for the Dominator-start game that is, for each odd integer j the jth turn belongs to Dominator. The Staller-start version will be treated later by introducing a Phase 0 for the starting turn. In our proofs, nonnegative values p(v) are assigned to the vertices, and the value p(G) of the residual graph is just the sum of the values of the vertices. Also, we assume that Dominator always chooses greedily. More precisely, for each odd j, in the jth turn he plays a vertex which results the possible maximum p(G j ) − p(G j+1 ). This difference is called the decrease in the value of p(G) and also referred to as the gain of the player.  (iii) If Phase i ends after the e i th turn but the game is not over yet, then the (e i + 1)st turn is the beginning of Phase i ′ , where i ′ is the smallest integer with i < i ′ such that (Ci ′ ) is fulfilled in the (e i + 1)st turn.
(iv) If Phase i is followed by Phase i ′ and i + 2 ≤ i ′ holds, we say that Phases i + 1, . . . i ′ − 1 are skipped; moreover, their starting and end points are interpreted to be the same as the end of Phase i.

Further notations
The colors white, blue and red will be often abbreviated to W, B and R, respectively. For example, a B-neighbor is a blue neighbor, and the notation v: W→B/R means that vertex v changed from white to either blue or red in the turn considered. Moreover, d W (v) and d B (v) stand for the number of W-neighbors and B-neighbors of v, respectively.
We cite the following observations (in a slightly modified form) from [7]: The following statements are true for every residual graph G in a domination game started on G * .
then v has only white neighbors and definitely has at least one. That is, At the end of this section, we provide a simple example for applying the tools introduced above. We prove Proposition 1, which states that for any isolate-free graph G of order n, hold.
Proof of Proposition 1. First, we consider the Dominator-start game on G * = (V, E), which is a simple graph without isolated vertices. In every residual graph G, let the value p(v) of a vertex v be equal to 2, 1 and 0, when v is white, blue and red, respectively. Hence, we start with p(G * ) = 2n and assume that Dominator always selects a vertex which results in a maximum decrease in p(G). The game is divided into two phases, which are determined due to Definition 1 with the following conditions: (C1) Dominator gets at least 4 points.

Phase 1.
If Staller selects a W-vertex, then it becomes red and causes at least 2-point decrease in the value of the residual graph. In the other case, Staller selects a B-vertex v which has a W-neighbor u. Then, the changes v: B→R and u: W→B/R together result in a decrease of at least 1 + 1 = 2. Hence, in each of his turns Staller gets at least 2 points. By condition (C1), Dominator always gets at least 4 points. As Dominator begins the phase, the average decrease in p(G) must be at least 3 in a turn.
Phase 2. When Phase 2 starts, Dominator cannot seize 4 or more points by playing any vertex of G j . This implies the following properties of the residual graph: Indeed, if v had two W-neighbors u 1 and u 2 , then Dominator could choose v and the changes v: W→R and u 1 , u 2 : W→B/R would give a gain of at least 2 + 2 · 1 = 4 points, which is a contradiction.
We have seen that d W (v) ≤ 1. Now, assuming two W-neighbors v and u, the choice of v would result in the changes v, u: W→R, which give a gain of at least 4 points to the player. This is a contradiction again.
• For every B-vertex v, d(v) = 1. By Lemma 1(ii), d(v) ≥ 1. Now, assume that v has two different W-neighbors u 1 and u 2 . As we have shown, d W (u 1 ) = d W (u 2 ) = 0 must hold and consequently, if Dominator plays v, then both u 1 and u 2 turn to red. This gives a gain of at least 1 + 2 · 2 = 5 points, which cannot be the case at the endpoint of Phase 1.
• Each component of G j is a P 2 and contains exactly one white and one blue vertex. By the claims above, each component is a star with a white center and blue leaves. If it contained at least two leaves then Dominator could play the center and get at least 4 points.
Therefore, at the beginning of Phase 2 the residual graph consists of components of order 2. As follows, in each turn an entire component becomes red and p(G) decreases by exactly 3 points.
In the game, the value of the residual graph decreased from 2n to zero, and the average decrease in a turn was proved to be at least 3. Consequently, the number of turns required is not greater than 2n/3, which proves γ g (G * ) ≤ 2n/3.
If Staller starts the game, his first move definitely decreases p(G) by at least 3 points as there are no isolated vertices. Then, the game is continued as in the Dominator-start game, and the average decrease remains at least 3 points. Thus, γ ′ g (G * ) ≤ 2n/3 holds.

Graphs of minimum degree 3
In this section we prove the upper bound stated on the game domination number of graphs with minimum degree 3. Also, this proof serves as an introduction to the details of the idea applied in the next section to prove our main theorem.
Proof of Theorem 1. We consider a graph G = (V, E) of minimum degree 3 and define the value assignments of types A1.1, A1.2 and A1.3 as they are given in Table 1. Hence, the game starts with p(G * ) = 34n. First, assume that Dominator begins the game and determine Phases 1-4 due to Definition 1 with the following specified conditions: (C1) Dominator gets at least 88 points due to the assignment A1.1.

Phase 1.
Here, we apply the value assignment A1.1. In each of his turns, Staller either selects a white vertex and gets at least 34 points; or he plays a blue vertex v which has a white neighbor u and then the color changes v: B→R and u: W→B/R give at least 16+18=34 points. By condition (C1), each move of Dominator yields a gain of at least 88 points. As Dominator begins the game, we have the following estimation on the average decrease of p(G) in a turn.
if v is white, and by at least if v is blue.
First, assume that the played vertex v is white, d W (v) = k and the W-neighbors of v are u 1 , . . . u k . For each 1 ≤ i ≤ k, the W-vertex u i becomes either a B-vertex of degree at most d W (u i ) − 1 or an R-vertex. As 1 ≤ d W (u i ) ≤ 2, p(u i ) decreases by at least 34 − (7 + 3d W (u i ) − 3) = 30 − 3d W (u i ) in either case. Then, the decrease in p(G) is not smaller than where 0 ≤ k ≤ 3 must hold. This establishes statement (i).
In the other case, v is blue with d(v) = k and its W-neighbors are u 1 , . . . u k . As v has only white neighbors and definitely has at least one and no more than 3, 1 ≤ k ≤ 3 holds; moreover, 0 ≤ d W (u i ) ≤ 2 is true for all 1 ≤ i ≤ k. When v is played, u i becomes red if d W (u i ) = 0, otherwise it will be a blue vertex of degree at most d W (u i ). Therefore, the decrease in p(u i ) is at least 34−(7+3d W (u i )) = 27−3d W (u i ) and that in p(v) is exactly 7 + 3k. Then, the sum of the decreases cannot be smaller than as stated in (ii).
To prove (iii), it suffices to consider the minimum of 43 + 24k in case (i), which is 43; and that of 7 + 24k in case (ii), which is 31 because of the condition k ≥ 1.
By Lemma 4(iii), Staller gets at least 31 points, and by Condition (C2), Dominator gets at least 91 points in each of their turns. Hence, we have the following estimation.

Lemma 5 In Phase 2, the average decrease of p(G) in a turn is at least 61 points.
As shown by the next lemma, the W-degrees are more strictly bounded from the end of Phase 2 than earlier.

Lemma 6 After the end of Phase 2, throughout the game, each white vertex has at most 1 white neighbors, and each blue vertex has at most 2 white neighbors.
Proof By condition (C2), at the end of Phase 2 Dominator can seize only less than 91 points by choosing any vertex of G. By Lemma 4(i), the selection of a W-vertex v with d W (v) = 2 causes a decrease of at least 43 + 24 · 2 = 91 points in p(G). Hence, each W-vertex has either zero or exactly one W-neighbor. Now, assume that v is a B-vertex with three W-neighbors, say u 1 , u 2 and u 3 . We have already seen that the inequalities 0 ≤ d W (u i ) ≤ 1 hold for i = 1, 2, 3. Then, as it was shown in the proof of Lemma 4(ii), the choice of v would decrease p(G) by at least which is a contradiction.  (ii) If the played vertex v is blue and has exactly one white neighbor u, then the changes v: B 1 →R and u: W→B 1 /R cause a decrease of at least 9 + 25 = 34 points in p(G). Additionally, u has at least one B-neighbor different from v, whose value is decreased by at least 4 points. Consequently, the total decrease is at least 38 points. Similarly, if v is blue and has two W-neighbors u 1 and u 2 , then the total decrease in p(G) is at least 13 + 2 · 25 + 2 · 4 = 71 points.
(iii) As the four cases above cover all possible moves which can be made in Phase 3, p(G) is decreased by at least 38 points in each turn.
As consequences of Condition (C3) and Lemma 7(iii), Dominator gets at least 84 points and Staller gets at least 38 points in each of his turns. Hence, we have the desired average.

Lemma 8 In Phase 3, the average decrease of p(G) in a turn is at least 61 points.
When Dominator cannot get at least 84 points in a turn, the structure of the residual graph must be very simple. Proof Let G i be the residual graph obtained at the end of Phase 3. Due to Lemma 7(i), the presence of a W-vertex v with d W (v) = 1 provides an opportunity for Dominator to get at least 84 points. Then, the ith turn would belong to Phase 3, which is a contradiction. Consequently, in G i each W-vertex has only B-neighbors.
Next, assume that we have a B-vertex v which has two W-neighbors u 1 and u 2 in G i . As we have seen, in G i d W (u 1 ) = d W (u 2 ) = 0 must hold and moreover, both u 1 and u 2 have at least two B-neighbors. Therefore, if v is selected by Dominator, the changes v: B 2 →R and u 1 , u 2 : W→R with the change in the values of B-neighbors, all together yield at least 13 + 2 · 34 + 4 · 4 = 97 point decrease in p(G i ), which is a contradiction again. Hence, each B-vertex has at most one W-neighbor.
Since each W-vertex v has the same degree in the residual graph G i as it had in G * , it has at least three B-neighbors in G i . In addition, each B-vertex is a leaf in G i . This implies that at the end of Phase 3 every component is a star with the structure stated. Finally, for the Staller-start version of the game we define Phase 0, which contains only the first turn and the values are counted due to A1.1. Observe that Staller's any choice results in at least 34 + 3 · 18 = 88 point decrease in p(G * ). Then, Phase 1 might be skipped if (C1) is not true for G 1 , but otherwise the game continues as in the Dominator-start version and our lemmas remain valid. Therefore, by the 27 point overplus arising in Phase 0, for γ ′ g (G * ) we obtain a slightly better bound, This completes the proof of Theorem 1.

Graphs with minimum degree greater than 3
Here we prove Theorem 2 and Corollary 1.
Proof of Theorem 2. First, we consider the Dominator-start game on a graph G * = (V, E) of order n, whose minimum degree is δ(G * ) ≥ d ≥ 4. The proof and the game starts with the value assignment A2.1 to the vertices as shown in Table 2. Later, we use a more subtle distinction between the types of blue vertices due to assignments A2.2, A2.3 and A2.4 (see Table 2. We will see that the value p(G) of the residual graph cannot increase when we change to an assignment with a higher index. We consider a graph G * = (V, E) with a minimum degree δ(G) ≥ d ≥ 4 and define the value assignments of types A2.1, A2.2, A2.3 and A2.4 as they are given in Table 2.
The values of a, b, x 1 , x 2 , x 3 and s are defined in terms of the parameter d. We aim to prove that s is a lower bound on the average decrease of p(G) in a turn, if Dominator follows the prescribed greedy strategy.
Concerning the values above and the change between assignments, we take the following observations.

Lemma 11
For every fixed integer d ≥ 4: (ii) For every 1 ≤ i < j ≤ 4 and every residual graph G, p(G) does not increase if the value assignment A2.i is changed to A2.j (assuming that A2.j is defined for G).
Proof The proof of (i) is based on a simple counting and estimation. Table 3 shows the differences and their exact values for d = 4, 5, 6. The comparison of coefficients verifies our statements for d ≥ 7. Table 3: Values of the differences for the proof of Lemma 11 Once (i) is proved, Table 2 shows that no vertex has greater value by A2.j than by A2.i, whenever j > i holds.
Note that later we will use further relations between a, b, x 1 , x 2 , x 3 and s but these are equations, which can be verified by simple counting, so the details will be omitted.
The game is divided into five phases due to Definition 1 with the following five conditions: (C1) Dominator gets at least 5a − 4b points due to the assignment A2.1.
Thus, the game starts on G * = G 1 with p(G 1 ) = a · n, and ends with a residual graph whose value equals 0. Recall that Dominator plays a purely greedy strategy. Our goal is to prove that the average decrease in p(G) is at least s points in a turn.
Phase 1 In each turn, the player either selects a W-vertex which turns red and hence p(G) decreases by at least a points; or he selects a B-vertex v which has a W-neighbor u. In the latter case the changes v: B→R and u: W→B/R together yield a decrease of at least b + (a − b) = a points. Therefore, Staller gets at least a points in each of his turns in Phase 1. By condition (C1), Dominator seizes at least 5a − 4b points and therefore, in any two consecutive turns p(G) decreases by at least 6a − 4b = 2s points. As Dominator starts, the following statement follows.

Lemma 12
In Phase 1, the average decrease of p(G) in a turn is at least s points.
Concerning the structure of the residual graph obtained at the end of this phase, we prove the following properties.
Lemma 13 At the end of Phase 1, Proof At the end of the phase, we have a residual graph G i in which Dominator cannot get 5a − 4b or more points. Assuming a W-vertex v with W-neighbors u 1 , u 2 , u 3 and u 4 , Dominator could play v and the changes v: W→R and u 1 , u 2 , u 3 , u 4 : W→B/R would result in a decrease of at least a + 4(a − b) = 5a − 4b points, which is a contradiction. In the other case, the choice of a B-vertex which has five Wneighbors would yield a gain of at least b + 5(a − b) = 5a − 4b points, which is a contradiction again.
Phase 2 In this phase we apply the value assignment A2.2. By Lemma 13(ii), each B-vertex has degree smaller than or equal to 4. Moreover by the definition of A2.2 and by Lemma 11, in the jth turn the value of a B-vertex u decreases by at least (d G j (u) − d G j+1 (u))x 1 points.

Lemma 14
In Phase 2, the average decrease of p(G) in a turn is at least s points.
Proof If a W-vertex v is played, each of its neighbors has a decrease of at least x 1 points in its value, no matter whether this change on the neighbor is B i →B i−1 or B 1 →R or W→B i /R. Then, playing a W-vertex results in at least a + d · x 1 point decrease in p(G).
In the other case, when the played vertex v is blue, the decrease in its value is at least b − 3x 1 . As v has a W-neighbor u, whose W-degree is at most 3, the change u: W→B i /R (i ≤ 3) yields further at least a − (b − x 1 ) points gain; and since u has at least d − 4 B-neighbors different from v, the total decrease in p(G) is at least This yields that Staller gets at least a + (d − 6)x 1 points whenever a white or a blue vertex is played by him.
Lemma 17 At the end of Phase 3, Proof At the end of Phase 3 we have a residual graph G i , in which the choice of any vertex decreases p(G i ) by strictly less than 3a − 2b + 2x 1 + (3d − 2)x 2 points. (i) Playing a W-vertex v, which has two W-neighbors say u 1 and u 2 , results in the changes v: W→R and u 1 , . This means a decrease of at least in p(G i ). This cannot be the case; so each W-vertex has either zero or exactly one W-neighbor in G i . (ii) Now suppose that a B-vertex v with W-neighbors u 1 , u 2 and u 3 is played in G i . We have already seen that d W (u j ) ≤ 1 holds for every W-vertex u j in G i . Then, we have the changes v: B 3 →R and u 1 , u 2 , u 3 : W→B 1 /R. Further, each vertex from {u 1 , u 2 , u 3 } has at least d − 2 B-neighbors different from v. Hence, the total gain of the player would be at least This contradiction proves (ii).
Phase 4 First, we change to assignment A2.3. By Lemma 17(ii), in any residual graph of Phase 4, the W-vertices induce a subgraph consisting of isolated vertices and P 2 -components; moreover, each blue vertex has at most 2 (white) neighbors. Moreover, by Table 2 and Lemma 3, if a B-vertex loses y W-neighbors in a turn, its value is reduced by at least yx 3 points.

Lemma 18 In Phase 4, the average decrease of p(G) in a turn is at least s points.
Proof If Staller selects a W-vertex v, each neighbor u of v has a decrease of at least x 3 in its value. Hence, the total decrease in p(G) is not smaller than b−x 1 −x 2 +dx 3 .
If Staller selects a B-vertex v, the change is either v: B 2 →R or v: B 1 →R, it means at least (b − x 1 − x 2 − x 3 )-point gain. As d(v) ≥ 1, we necessarily have a W-neighbor u of v whose change is u: W→B 1 /R. Further, u has at least d − 2 B-neighbors different from v. Therefore, the decrease in p(G) is at least Hence, in any case, Staller gets at least a + (d − 2)x 3 points in a turn of his own. By (C4), Dominator gets at least 2a + (2d − 2)x 3 point in each of his turns and as follows, the average gain is at least To prove (ii) we suppose for a contradiction that a B-vertex v has two Wneighbors u 1 and u 2 . By (i), these neighbors are "single-white" vertices and they turn to red if v is played; in addition both u 1 and u 2 has at least d − 1 B-neighbors different from v. Hence, selecting v Dominator could seize at least points, which is a contradiction again.
Phase 5 By Lemma 19(ii), the residual graphs occurring in this phase have simple structure, each of their components is a star of order at least d + 1 whose center is white and the leaves are blue. Then, in each turn of Phase 5 exactly one such a star component becomes completely red, no matter whether a white or a blue vertex is played. Then, the value of the residual graph is decreased by at least a + d(b − x 1 − x 2 − x 3 ) = s points in each single turn.
Lemma 20 In Phase 5, the average decrease of p(G) in a turn is at least s points.
By Lemmas 12,14,16,18 and 20, the average decrease per turn in the residual graph is at least s for the entire game. As p(G 1 ) = an and the changes between assignments nowhere caused increase in p(G), the domination game where Dominator plays the described greedy strategy yields a game with at most an/s turns. This establishes Theorem 2.

Concluding remarks on the Staller-start game
In our main theorem, we do not give upper bound on γ ′ g (G) for graphs with δ(G) ≥ d ≥ 4. It is quite clear from the proof that we can establish the same upper bound on γ ′ g (G) as proved for γ g (G). Moreover, a slight improvement on it is also possible. We close the paper with this complicated formula.