New Bounds on the Double Total Domination Number of Graphs

Let G be a graph of minimum degree at least two. A set D⊆V(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D\subseteq V(G)$$\end{document} is said to be a double total dominating set of G if |N(v)∩D|≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|N(v)\cap D|\ge 2$$\end{document} for every vertex v∈V(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v\in V(G)$$\end{document}. The minimum cardinality among all double total dominating sets of G is the double total domination number of G. In this article, we continue with the study of this parameter. In particular, we provide new bounds on the double total domination number in terms of other domination parameters. Some of our results are tight bounds that improve some well-known results.


Introduction
Throughout this article, we consider G = (V (G), E(G)) as a simple graph of minimum degree at least two. A set D ⊆ V (G) is said to be a dominating set if every vertex in V (G)\D is adjacent to at least one vertex in D. The domination number of G, denoted by γ (G), is the minimum cardinality among all dominating sets of G. Among  1 the most studied domination variants can be found those where the dominating sets are defined by an added condition that is imposed on every vertex of G. In this article, we continue with the study of one of these variants, namely double total domination number.
Given a graph G, a set D ⊆ V (G) is said to be a double total dominating set if |N (v) ∩ D| ≥ 2 for every v ∈ V (G). The double total domination number of G, denoted by γ ×2,t (G), is the minimum cardinality among all double total dominating sets of G. We define a γ ×2,t (G)-set as a double total dominating set of cardinality γ ×2,t (G). We remark that this parameter is also called in the literature as 2-tuple total domination number or total 2-domination number.
The double total domination number has been well studied by the research community. For instance, in [1,4,17,19,21,22], some combinatorial results were presented, and in [2,3,18,23], some studies on certain product graphs were carried out. Our goal is to making some new remarkable contributions to this domination parameter. In particular, we provide new bounds on the double total domination number of a graph of minimum degree at least two. Such bounds can also be seen as relationships between the double total domination number and several classical domination parameters like double domination number, 2-domination number, independence number, among others. Some of our results are tight bounds that improve some well-known results.

Additional Definitions and Previous Results
Given a graph G, we denote by n(G) = |V (G)| its order, by m(G) = |E(G)| its size, and by Δ(G) and δ(G) its maximum and minimum degrees, respectively. Given (u, v) between two vertices u and v, in a connected graph G, equals the minimum length of a (u, v)-path in G. The diameter of G, denoted by diam(G), is the maximum distance among all pairs of vertices of G. We say that a graph G is H -free if G does not contain the graph H as an induced subgraph. Given two disjoint graphs G 1 and G 2 , the join graph G 1 + G 2 is the graph obtained from G 1 and G 2 , with vertex set }. We will use the notation P n , C n , K n , and N n for path graphs, cycle graphs, complete graphs, and empty graphs of order n, respectively. We end this subsection with some definitions and terminology on domination invariants which will be further used.
The maximum cardinality among all independent sets is the independence number of G and is denoted by α(G). We define an α(G)-set as an independent set of cardinality α(G). Moreover, a set W ⊆ V (G) is said to be a vertex cover of G if V (G)\W is an independent set of G. The minimum cardinality among all vertex covers of G is the vertex cover number of G and is denoted by β(G). We define a β(G)-set as a vertex cover of cardinality β(G). In 1959, Gallai [10] proved that α(G) + β(G) = n(G) for every nontrivial graph G. We remark that the previous relation is a well-known result in graph theory.
The total domination number of G, denoted by γ t (G), is the minimum cardinality among all total dominating sets of G. We define a γ t (G)-set as a total dominating set of cardinality γ t (G) (see [16,20]).
, is the minimum cardinality among all 2-dominating sets of G. We define a γ 2 (G)-set as a 2-dominating set of cardinality γ 2 (G).
Combinations between domination parameters are probably the most used features in different versions of domination invariants. For instance, the double domination is just a combination of the total domination and the 2- is the minimum cardinality among all double dominating sets of G. We define a γ ×2 (G)-set as a double dominating set of cardinality γ ×2 (G). For more information about 2-domination and double domination in graphs, we suggest the chapter [11] due to Hansberg and Volkmann and the recent works [7,9]. A double dominating set D ⊆ V (G) is a total outer-independent dominating set of G if V (G)\D is an independent set. The total outer-independent domination number of G, denoted by γ t,oi (G), is the minimum cardinality among all total outer-independent dominating sets of G. We define a γ t,oi (G)-set as a total outer-independent dominating set of cardinality γ t,oi (G) (see [5,6,8,24]).
By definition, all the previous parameters are, in one way or another, related to each other. The following result shows the most natural relationships that exist between them.

Proposition 1 The following inequality chains hold for any graph G with
For instance, for the graphs shown in Fig. 1 we have the following.
We suggest the books [12,15] in case the reader is not familiar with another some basic concepts, notation and terminology of graphs.

Bounds and Relationships with Other Parameters
The following upper bounds for the double total domination number were established by Bermudo et al. in [1] and independently by Bonomo et al. in [4].
The following statements hold for any graph G with δ(G) ≥ 2.
Since γ 2 (G) ≤ γ ×2 (G), the next theorem improves the upper bound given in Proof Let D be a γ ×2 (G)-set and X a γ 2 (G)-set. We next define a set S ⊆ V (G) of minimum cardinality among the sets satisfying the following properties.
From (i), we deduce that S is a double dominating set of G and by the minimality of |S| we have that |S| ≤ |D ∪ X | + |D ∩ X | = |D| + |X |. To conclude that S is a double total dominating set of G, we only need to prove that To show clear examples where the difference between the bounds given by Theorems 2 and 1 can be as large as desired, we can take the graph H r defined as follows.
For any integer r ≥ 2, the graph H r is defined as the graph obtained from the path P r and the graph H given in Fig. 2 by taking one copy of P r and r copies of H and identifying the i th vertex of P r with the vertex v ∈ V (H ) in the i th copy of H for every i ∈ {1, 2, . . . , r }. Figure 2 shows the graph H 3 . Notice that for the graph H r , it follows that |V (H r )| = 6r , γ ×2 (H r ) = 3r and γ 2 (H r ) = 2r . Therefore, if we take r large enough, then the difference between the bounds given by Theorems 2 and 1 can be as large as desired.
We next proceed to show that the bound given in Theorem 1-(ii) has room for improvement. To this purpose, we need to state the following result due to Cabrera Martínez and Rodríguez-Velázquez [9].

Theorem 4 For any graph G with
From Theorem 2 and Proposition 1, we deduce the bound γ ×2,t (G) ≤ γ ×2 (G) + n(G) − α(G). We next show that this previous inequality can be improved for two specific families of graphs. Emphasize that K 1,3 + e is the graph obtained by adding an edge to the complete bipartite graph K 1,3 .

Theorem 5
The following statements hold for any graph G.
Proof Let G be a K 1,3 -free graph with δ(G) ≥ 3 and S a β(G)-set. First, we shall prove that S is a double dominating set of G. Notice that S is a 2-dominating set of G because V (G)\S is an independent set. Now, if there exists a vertex v ∈ S such that N (v) ∩ S = ∅, then the set formed by v and any three neighbours induces a K 1,3 , which is a contradiction. Therefore, every vertex in S has at least one neighbour in S, which implies that S is a double dominating set of G, as required. Now, suppose that δ(G) ≥ 4. If there exists a vertex v ∈ S such that |N (v)∩ S| = 1, then the set formed by v and any three vertices in N (v)\S induces a K 1,3 , which is a contradiction. Hence, every vertex in S has at least two neighbours in S, which implies that S is a double total dominating set of G. Therefore, γ ×2,t (G) ≤ |S| = β(G) = n(G) − α(G), which completes the proof of (i).
Finally, we suppose that G is a {K 1,3 , K 1,3 + e}-free graph with δ(G) = 3. Let D be a γ (G)-set. Now, we define W ⊆ V (G) as a set of minimum cardinality among all sets W ⊆ V (G) such that the following conditions are satisfied.   G is a {K 1,3 , K 1,3 + e}-free graph and uu / ∈ E(G). Moreover, applying the same reasoning with the vertices v , v, u, w, we deduce that w ∈ N (v) because wu / ∈ E(G), which is a contradiction. Hence, W is a double total dominating set of G, and so, Therefore, the proof is complete.
Next, we discuss the relationship between the double total domination number and the total outer-independent domination number.

Theorem 6
The following statements hold for any graph G with δ(G) ≥ 2. By definition, it is straightforward that D is a double total dominating set of G. In addition, every vertex in D 0 has at least one neighbour in D \D. Hence, and by the minimality of D , we deduce that |D \D | ≤ |D 0 | ≤ |D| − δ(G). Therefore, Therefore, (i) holds. Finally, (ii) follows by Theorem 5-(i) and Proposition 1-(ii), which completes the proof.
The bound given in (i) is achieved for the graph G ∼ = K 2 + N r , where r ∈ Z + . In this case, we have that γ ×2,t (G) = 3 and γ t,oi (G) = δ(G) = 2.

Theorem 7
The following statements hold for any connected graph G with δ(G) ≥ 2.
be a diametrical path of G (in this case, k = diam(G)) and S = {v 0 , v 5 , . . . , v 5 k/5 }. Hence, d(x, y) ≥ 5 for any different vertices x, y ∈ S. Now, let D ⊂ D be a set of cardinality |S| such that |N [x] ∩ D| = 1 for every x ∈ S. Observe that by the definitions of D and S, the set D is well defined. In addition, and by the definitions of S and D , we have that d(x, y) ≥ 3 for any different vertices x, y ∈ D . With all of the above in mind, let us proceed to prove that D = D\D is a total dominating set of G. Suppose to the contrary that there exists v ∈ V (G) such that which is a contradiction. Therefore, D is a total dominating set of G, which implies that Hence, the proof of (i) is complete. In order to prove (ii), we assume that G is a K 1,3 -free graph with diam(G) = 2. This implies that |N (x) ∩ W | ≥ 2 for every x ∈ W \{u, v}, and that N (x) ∩ W = ∅ for every x ∈ V (G)\W . We only need to prove that |N (x) ∩ W | ≥ 2 for every x ∈ V (G)\W . Suppose to the contrary that there exists x ∈ V (G)\W such that |N (x ) ∩ W | = 1. As diam(G) = 2, there exists a vertex w ∈ N (v) ∩ N (u) such that N (x ) ∩ W = {w}, and then, the set formed by w, u, v and x induces a K 1,3 , which is a contradiction. Hence, every vertex in V (G)\W has at least two neighbours in W . Thus, W is a double total dominating set of G, which implies that Therefore, the proof is complete.
The bounds above are tight. For instance, the bound given in (i) is achieved by the graph G 1 shown in Figure 1. In this case, we have that γ ×2,t (G 1 ) = 6, γ t (G 1 ) = 4 and diam(G 1 ) = 6. Moreover, the bound given in (ii) is achieved for the cycle C 5 .
We continue with a new lower bound for the double total domination number in terms of the order, the maximum degree, and the total domination number of G.
Let D be a γ ×2,t (G)-set and let g be a double dominating function of minimum weight among all double dominating functions on G. Observe that the function g , defined by g (x) = 1 whenever x ∈ D and g (x) = 0 otherwise, is a double dominating function on G. Hence, ω(g) ≤ ω(g ) = |D| = γ ×2,t (G). Moreover, notice that From (1) and the previous two inequality chains, we deduce that , which completes the proof.
. Therefore, the proof is complete.

Open Problems
(a) In the light of Theorem 3 and the bound obtained in Theorem 2, it is natural to ask whether γ ×2,t (G) ≤ γ ×2 (G) + α(G). Moreover, it is straightforward that the bound holds whenever γ 2 (G) ≤ α(G). However, we conjecture that this inequality is satisfied for every graph G with δ(G) ≥ 2. (b) After numerous attempts, we have not been able to find examples of graphs that reach the bounds given in Theorems 2, 4 and 5 . In this sense, we propose to find examples of graphs for which the equalities are reached, or to improve these bounds.
(c) We have shown that γ ×2,t (G) ≥ γ t (G) + (diam(G) + 1)/5 for any connected graph G with δ(G) ≥ 2. We propose the problem of characterizing all connected graphs for which this equality holds, or providing necessary or sufficient conditions to achieve it.