On the Secure Total Domination Number of Graphs

A total dominating set D of a graph G is said to be a secure total dominating set if for every vertex u ∈ V(G) \ D, there exists a vertex v ∈ D, which is adjacent to u, such that (D \ {v}) ∪ {u} is a total dominating set as well. The secure total domination number of G is the minimum cardinality among all secure total dominating sets of G. In this article, we obtain new relationships between the secure total domination number and other graph parameters: namely the independence number, the matching number and other domination parameters. Some of our results are tight bounds that improve some well-known results.


Introduction
The following approach to the protection of a graph was proposed by Cockayne et al. [1].Suppose that one or more entities are stationed at some of the vertices of a graph G and that an entity at a vertex can deal with a problem at any vertex in its closed neighbourhood.In general, an entity could consist of an observer, a robot, a guard, a legion, and so on.Informally, we say that G is protected under a given placement of entities if there exists at least one entity available to handle a problem at any vertex.The simplest cases of graph protection are those in which you can locate at most one entity per vertex.In such a case, the set of vertices containing the entities is said to be a dominating set.
In a graph G = (V(G), E(G)), a vertex dominates itself and its neighbours.A subset S ⊆ V(G) is said to be a dominating set of G if S dominates every vertex of G, while S is said to be a total dominating set if every vertex v ∈ V(G) is dominated by at least one vertex in S \ {v}.As usual, the neighbourhood of a vertex v ∈ V(G) will be denoted by N(v).Now, a set S ⊆ V(G) is said to be a secure (total) dominating set if S is a (total) dominating set and for every v ∈ V(G) \ S there exists u ∈ N(v) ∩ S such that (S ∪ {v}) \ {u} is a (total) dominating set.In the case of secure (total) domination, the graph is deemed protected by a (total) dominating set and when an entity moves (to deal with a problem) to a neighbour not included in the (total) dominating set, the new set of entities obtained from the movement of the entity is a (total) dominating set which protects the graph as well.
The minimum cardinality among all dominating sets of G is the domination number of G, denoted by γ(G).The total domination number, the secure domination number and the secure total domination number of G are defined by analogy, and are denoted by γ t (G), γ s (G) and γ st (G), respectively.
In this work we study the relationships between the secure total domination number and other graph parameters.The article is organized as follows.In Section 2 we define key terms and additional notation.In Section 3 we show that γ st (G) ≤ α(G) + γ(G), where α(G) denotes the independence number of G. Since γ(G) ≤ α(G), this result improves the bound γ st (G) ≤ 2α(G) obtained in [14].Section 4 is devoted to the study of relationships between the secure total domination number and other domination parameters.In particular, we outline some known results that become tools to derive new ones.Finally, in Section 5 we obtain several bounds on the secure total domination number in terms of the matching number and other graph parameters.

Some Additional Concepts and Notation
All graphs considered in this paper are finite and undirected, without loops or multiple edges.The minimum degree of a graph G will be denoted by δ(G) and the maximum degree by ∆(G).As usual, the closed neighbourhood of a vertex v ∈ V(G) is denoted by . By analogy with the notation used for vertices, for a set S ⊆ V(G), its open neighbourhood is the set N(S) = ∪ v∈S N(v), and its closed neighbourhood is the set N[S] = N(S) ∪ S. We also define the following sets associated with v ∈ V(G).
• The internal private neighbourhood of v relative to S is defined by ipn(v, S) = {u ∈ S : N(u) ∩ S = {v}}.
• The external private neighbourhood of v relative to S is defined by epn(v, S) = {u ∈ V(G) \ S : N(u) ∩ S = {v}}.
• The private neighbourhood of v relative to S is defined by The subgraph induced by S ⊆ V(G) will be denoted by S , while the graph obtained from G by removing all the vertices in S ⊆ V(G) (and all the edges incident with a vertex in S) will be denoted by G − S. If H is a graph, then we say that a graph G is H-free if G does not contain any copy of H as an induced subgraph.
We denote the set of leaves of a graph G by L(G), and the set of support vertices (vertices adjacent to leaves) by S(G).The set of isolated vertices of V(G) \ (S(G) ∪ L(G)) will be denoted by I G .
We will use the notation C n , N n and P n for cycle graphs, empty graphs and path graphs of order n, respectively.
Let f : V(G) → {0, 1, 2} be a function.For any i ∈ {0, 1, 2} we define the subsets of vertices V i = {v ∈ V(G) : f (v) = i} and we identify f with the three subsets of V(G) induced by f .Thus, in order to emphasize the notation of these sets, we denote the function by f (V 0 , V 1 , V 2 ).Given a set X ⊆ V(G), we define f (X) = ∑ v∈X f (v), and the weight of f is defined to be ω The weak Roman domination number, denoted by γ r (G), is the minimum weight among all weak Roman dominating functions on G.By analogy we define the total weak Roman domination number, which is denoted by γ tr (G).The weak Roman domination number was introduced by Henning and Hedetniemi [15] and studied further in several works including [7,8,10,16,17], while the total weak Roman domination number was recently introduced in [12].
A dominating set of cardinality γ(G) will be called a γ(G)-set.A similar agreement will be assumed when referring to optimal sets associated with other parameters used in the article.As usual, we will use the acronyms TDS and STDS to refer to total dominating sets and secure total dominating sets, respectively.
A TDS X is said to be a total outer-connected dominating set if the subgraph induced by V(G) \ X is connected.The total outer-connected domination number of G, denoted by γ toc (G), is the minimum cardinality among all total outer-connected dominating sets of G.This parameter was introduced by Cyman in [18] and studied further in [19][20][21].
An independent set of a graph G is a subset of vertices such that no two vertices in the subset represent an edge of G.The maximum cardinality among all independent sets is the independence number of G, denoted by α(G).Analogously, two edges in a graph G are independent if they are not adjacent in G.A set of pairwise independent edges of G is called a matching of G.The matching number α (G) , sometimes known as the edge independence number, is the maximum cardinality among all matchings of G.
For the remainder of the paper, definitions will be introduced whenever a concept is needed.

Secure Total Domination & Independence
Klostermeyer and Mynhardt [9] in 2008, established the following upper bound.
Theorem 1. [9] For any graph G with no isolated vertex, In 2017 Duginov [14] answered the following open question posed by Klostermeyer and Mynhardt [9] p. 282: Is there a graph G such that γ st (G) = 3α(G) − 1, where α(G) ≥ 2? Duginov provided a negative answer to this question by confirming the suspicions of Klostermeyer and Mynhardt that γ st (G) ≤ 2α(G).Theorem 2. [14] For any graph G with no isolated vertex, We now proceed to improve the bound above.
Proof.Let D be a γ(G)-set.By Lemma 1 there exists an α(G)-set I such that ipn(x, D ∪ I) = ∅ for every x ∈ I.We define the set S ⊆ V(G) as follows. (a) For every vertex x ∈ D ∩ I, It is readily seen that S is a TDS.Now, let v ∈ V(G) \ S. Since I ⊆ S is also a dominating set, there exists a vertex u ∈ N(v) ∩ I ⊆ N(v) ∩ S. To conclude that S is STDS, we only need to prove that S = (S \ {u}) ∪ {v} is a TDS.To this end, we differentiate two cases for any w ∈ N(u).Now, since S is a TDS, we have that every vertex in V(G) \ N(u) is dominated by some vertex in S and, according to the two cases above, we can conclude that S is a TDS.Therefore, S is a STDS and so The bound above is tight.For instance, it is achieved for any corona product graph G = H 1 H 2 , where H 1 is an arbitrary graph and H 2 is the disjoint union of k complete nontrivial graphs.Notice that 2, where γ st (G) = 8, α(G) = 6 and γ(G) = 2.

Secure Total Domination & Other Kinds of Domination
For any graph G with no isolated vertex, V(G) is a secure total dominating set, which implies that Theorem 4. [11] Let G be a graph of order n.Then Since every secure total dominating set is a total dominating set, it is clear that γ t (G) ≤ γ st (G).All graphs satisfying the equality were characterized by Klostermeyer and Mynhardt in [9].Theorem 5. [9] If G is a connected graph, then the following statements are equivalent.
• G has two universal vertices.
The result above is an important tool to characterize all graphs with γ st (G) = 3.To begin with, we need to state the following basic tool.Proposition 1.If H is a spanning subgraph (with no isolated vertex) of a graph G, then Proof.Let E − = {e 1 , . . ., e k } be the set of all edges of G not belonging to the edge set of H. Let H 0 = G and, for every i ∈ {1, . . ., k}, let X i = {e 1 , . . ., e i } and Let G be the family of graphs H of order n ≥ 3 such that the subgraph induced by three vertices of H contains a path P 3 and the remaining n − 3 vertices have degree two and they form an independent set.Figure 1 shows a graph belonging to G. • γ st (G) = 3.
• G has at most one universal vertex and there exists H ∈ G which is a spanning subgraph of G.
Proof.Let D be a γ st (G)-set and assume that |D| = 3.By Theorem 5, G has at most one universal vertex.Let D = {u, v, w} and notice that D contains a path P 3 , as D is a total dominating set of G. Since D is a STDS of G, we observe that |N(z) ∩ D| ≥ 2 for every z ∈ V(G) \ D. Hence, in this case, G contains a spanning subgraph belonging to G.
Conversely, since G has at most one universal vertex, by Theorem 5 we have that γ st (G) ≥ 3.Moreover, it is readily seen that γ st (H) ≤ 3 for any H ∈ G. Hence, if H ∈ G is a spanning subgraph of G, by Proposition 1 it follows that γ st (G) ≤ 3. Therefore, γ st (G) = 3.
We now consider the relationship between γ s (G) and γ st (G).Theorem 7. [9] Let G be a graph with no isolated vertex.
A natural question is if the bound γ st (G) ≤ 2γ s (G), due to Klostermeyer and Mynhardt, can be improved with γ st (G) ≤ γ s (G) + γ(G).The example given in Figure 2 shows that, in general, this inequality does not hold.In Theorem 10 we will show some cases in which γ st (G) ≤ γ s (G) + γ(G).To this end, we need to outline the following two known results.Theorem 8. [12] The following inequalities hold for any graph G with no isolated vertex.
Although the problem of characterizing all graphs with γ tr (G) = γ st (G) remains open, some particular cases were described in [12].
From Theorems 8 and 9 (ii), and using the fact that γ r (G) ≤ γ s (G), we can show that the bound γ st (G) ≤ 2γ s (G) established in Theorem 7 can be improved for any {K 1,3 , K 1,3 + e}-free graph.
Theorem 10.For any {K 1,3 , K 1,3 + e}-free graph G with no isolated vertex, The previous bounds are tight.They are achieved, for instance, for the wheel graph G ∼ = N 1 + C 4 and for G ∼ = N 2 + P 3 , which is the join of N 2 and P 3 .For these two graphs we have that To derive a consequence of Theorem 10 we need to state the following result due to Burger et al. [6].
The following result shows us a relationship between the secure total domination number and the total outer-connected domination number.
Theorem 13.Let G be a graph of order n.If Proof.We assume that γ toc (G) , which completes the proof.
The bound above is tight.For instance, it is achieved for the wheel graph G ∼ = N 1 + C 4 and for G ∼ = N 2 + P 3 .In both cases γ st (G) = 3 and γ toc (G) = 2.
The following result was obtained by Favaron et al. in [20].
Theorem 14. [20] For any graph G of order n, diameter diam(G) ≤ 2 and minimum degree δ(G) ≥ 3, The following result is a direct consequence of combining the result above and Theorem 13.
Theorem 15.For any graph G of order n, diameter two and minimum degree δ(G) ≥ 3, The bound above is achieved for the wheel graph G ∼ = N 1 + C 4 and for G ∼ = N 2 + P 3 .As we already know, in both cases γ st (G) = 3.

Secure Total Domination & Matching
To begin this section, we proceed to introduce new definitions and terminology.Given a matching M of a graph G, let V M be the set formed by the end-vertices of edges belonging to M. Given a vertex v ∈ V M , we say that v ∈ V M is the partner of v if vv ∈ M. Observe that if v is the partner of v, then v is the partner of v .
A maximum matching is a matching of cardinality α (G).The following lemmas show some properties of maximum matchings.Lemma 2. Let M be a maximum matching of a graph G.The following statements hold.
If there exists a vertex w ∈ N(u) ∩ (V(G) \ V M ), then the set M ∪ {uw} is a matching of G of cardinality greater than |M|, which is a contradiction.Hence, N(u) ⊆ V M and (i) follows.Now, we suppose that there exists u ∈ V(G) \ V M and a vertex v ∈ N(u) ∩ V M .Let v be the partner of v.If there exists a vertex w ∈ N(v ) ∩ (V(G) \ (V M ∪ {u})), then the set M \ {vv } ∪ {uv, v w} is a matching of G of cardinality greater than |M|, which is a contradiction.Hence, N(v ) ⊆ V M ∪ {u} and (ii) follows.Lemma 3.For any graph G with L(G) = ∅, there exists a maximum matching M such that for each vertex x ∈ S(G) there exists y ∈ L(G) such that xy ∈ M. The next result provides a relationship between the secure total domination number, the matching number and some special vertices of a graph.Theorem 16.For any graph G with minimum degree δ(G) = 1, which is adjacent to v. Let S = (S \ {u}) ∪ {v}.We will see that S is a TDS of G as well.Since S is a TDS of G, every vertex w ∈ V(G) \ N(u) is adjacent to some vertex belonging to S .Let w ∈ N(u) and observe that |N(w)| ≥ 2 as u / ∈ S(G).
If w ∈ V(G) \ V M , then by Lemma 2 (i) we have that N(w) ⊆ V M .Hence there exists a vertex in V M \ {u} ⊆ S which is adjacent to w, as |N(w)| ≥ 2. Now, if w ∈ V M \ {u }, where u is the partner of u, then w is adjacent to its partner, which belongs to S .Finally, if w = u , then by Lemma 2 (ii) we have that Thus, S is a TDS of G, as desired.Therefore, S is a STDS and so γ st (G) The bound above is tight.For instance, it is achieved for the graph shown in Figure 3.In this case, From now on we consider the case of graphs with minimum degree δ(G) ≥ 2. is a dominating set).We only need to prove that S is a TDS of G. Since S is a TDS of G, every vertex in V(G) \ N(v * ) has at least one neighbour in S .Now, let u ∈ N(v * ) and consider the following two cases.
by Lemma 2 (i) we deduce that there exists some vertex in N(u) ∩ S .Cockayne et al. in [8] obtained the following bound on the secure domination number in terms of the order and the matching number.Theorem 18. [8] If a graph G of order n does not have isolated vertices, then Therefore, by Theorems 10 and 18 we deduce the following result.
The k-domination number of G, denoted by γ k (G), is another well-known parameter [3].The following theorem is a contribution of DeLaViña et al. in [22].Theorem 20. [22] Let k be a positive integer.For any graph G with minimum degree δ(G) ≥ 2k − 1, Since every γ 2 (G)-set is a secure dominating set of G, it is immediate that γ s (G) ≤ γ 2 (G), and so Theorems 10 and 20 lead to the following result.

Lemma 1 .Theorem 3 .
For any graph G and any set D ⊆ V(G), there exists an α(G)-set I such that for any x ∈ I, ipn(x, D ∪ I) = ∅.Proof.Let I be an α(G)-set, D ⊆ V(G) and D I = {x ∈ I : ipn(x, D ∪ I) = ∅}.We can assume that we have taken I in such a way that |D I | is minimum among all α(G)-sets.Suppose that there exists u ∈ D I , and consequently, let v ∈ ipn(u, D ∪ I).Observe that v ∈ D \ I and I = (I ∪ {v}) \ {u} is an α(G)-set.Since ipn(v, D ∪ I ) = ∅ and ipn(x, D ∪ I ) = ipn(x, D ∪ I) for every x ∈ I \ {v}, we can conclude that I is an α(G)-set satisfying that |D I | < |D I |, which is a contradiction.Therefore, D I = ∅, which completes the proof.Since γ(G) ≤ α(G), the following result improves Theorem 2. For any graph G with no isolated vertex,

Case 1 .
u ∈ I \ D. If w / ∈ D, then there exists some vertex in D ⊆ S which dominates w, as D is a dominating set.If w ∈ D, then by Lemma 1 we have that w / ∈ ipn(u, D ∪ I).Hence, there exists some vertex in (D ∪ I) \ {u} ⊆ S which dominates w.Case 2. u ∈ I ∩ D. If w ∈ D, then by Lemma 1 we have that w / ∈ ipn(u, D ∪ I), and so there exists some vertex in (D ∪ I) \ {u} ⊆ S which dominates w.From now on, suppose that w / ∈ D. If w / ∈ epn(u, D ∪ I), then w is dominated by some vertex in (D ∪ I) \ {u} ⊆ S .If w ∈ epn(u, D ∪ I) then, as all vertices in epn(u, D ∪ I) form a clique and by (b) in the definition of S, w is dominated by some vertex in S \ {u} ⊆ S .
All graphs achieving this trivial bound were characterized by Benecke et al. as follows.

Figure 1 .Theorem 6 .
Figure 1.A graph H belonging to G. The set of black-coloured vertices forms a γ st (H)-set Theorem 6.Given a graph G, the following statements are equivalent.

Proof.
Let M be a maximum matching of G such that |V M ∩ L(G)| is maximum.It is easy to see that the maximality of M leads to S(G) ⊆ V M .Suppose that there exists a support vertexx such that xx ∈ M and x / ∈ L(G).Let y ∈ N(x) ∩ L(G).Notice that the set M = M \ {xx } ∪ {xy} is a maximum matching of G and |V M ∩ L(G)| > |V M ∩ L(G)|,which is a contradiction.Therefore, the result follows.

Figure 3 .
Figure 3.The set of black-coloured vertices forms a γ st (G)-set.
is a secure (total) dominating set if and only if there exits a (total) weak Roman dominating function f