Relating the Outer-Independent Total Roman Domination Number with Some Classical Parameters of Graphs

For a given graph G without isolated vertex we consider a function f:V(G)→{0,1,2}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f: V(G) \rightarrow \{0,1,2\}$$\end{document}. For every i∈{0,1,2}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i\in \{0,1,2\}$$\end{document}, let Vi={v∈V(G):f(v)=i}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V_i=\{v\in V(G):\; f(v)=i\}$$\end{document}. The function f is known to be an outer-independent total Roman dominating function for the graph G if it is satisfied that; (i) every vertex in V0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V_0$$\end{document} is adjacent to at least one vertex in V2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V_2$$\end{document}; (ii) V0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V_0$$\end{document} is an independent set; and (iii) the subgraph induced by V1∪V2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V_1\cup V_2$$\end{document} has no isolated vertex. The minimum possible weight ω(f)=∑v∈V(G)f(v)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega (f)=\sum _{v\in V(G)}f(v)$$\end{document} among all outer-independent total Roman dominating functions for G is called the outer-independent total Roman domination number of G. In this article we obtain new tight bounds for this parameter that improve some well-known results. Such bounds can also be seen as relationships between this parameter and several other classical parameters in graph theory like the domination, total domination, Roman domination, independence, and vertex cover numbers. In addition, we compute the outer-independent total Roman domination number of Sierpiński graphs, circulant graphs, and the Cartesian and direct products of complete graphs.


Introduction and Preliminaries
This work mainly deals with showing some existent interconnections between several classical graph theory topics like domination, independence, covers, and Roman domination. These topics have attracted the attention of several researches in the last few decades, and a high number of significant contributions are nowadays well known. Each new topic or parameter that is described naturally gives more insight into the classical ones, and it is usually welcome by the research community. A quick search in databases like MathSciNet or One of the attempts on improving the ideas of Roman domination in graphs was first presented in [18] through some more general settings, and after formally, specifically and better studied in [2,3]. The main idea of such improvement comes with the addition of a total domination property. That is, a Roman dominating function f (V 0 , V 1 , V 2 ) is called a total Roman dominating function on G, if the subgraph induced by V 1 ∪ V 2 has no isolated vertices, i.e., V 1 ∪ V 2 is a total dominating set of G. The total Roman domination number of G is analogously defined, and denoted by γ tR (G).
Another improvements of the Roman domination concept are that ones connecting them with independent sets, and thereby, with vertex cover sets. A (total) Roman dominating function f (V 0 , V 1 , V 2 ) is called an outer-independent (total) Roman dominating function (OIRDF and OITRDF for short) if V 0 is an independent set of G. Notice that, this is equivalent to say that V 1 ∪V 2 is a vertex cover set of G. In connection with this last fact, it would be even more natural to call such function a covering (total) Roman dominating function instead of outer-independent (total) Roman dominating function. However, to keep the already stated terminology, we prefer to use that on OIRD and OITRD functions. The outer-independent (total) Roman domination number of G is the minimum possible weight among all outer-independent (total) Roman dominating functions on G, and is denoted by (γ oitR (G)) γ oiR (G). The parameter γ oiR (G) was introduced in [1], while γ oitR (G) was first presented in [7].
Once defined all the concepts above, we are prepared to begin with our exposition. Our main objective is then to present several relationships between all the parameters mentioned above, by making emphasis on γ oitR (G), which is the center of our investigation. We must remark that this parameter has also been recently studied in [17,19]. For instance, in [17], some computational and approximation results on this parameter were presented, and in [19], some Nordhauss-Gaddum results for it were proved. In our work, we bound γ oitR (G) in terms of the other above mentioned parameters, give several chains of inequalities involving many of these parameters, and finally, we present exact values of it for a number of remarkable families of graphs G which have been recently attracting the attention of several researchers. From now on, for a parameter p(G) of a graph G, by a p(G)-set, or a p(G)-function, we mean a set of cardinality p(G) or a function of weight p(G), respectively.

Some Primary and Basic Results
To be used as examples in several places, for showing the tightness or not of several bounds and relationships, we give some preliminary results in this subsection concerning a few basic families of graphs. Some of them are classical ones in studies of domination in graphs, although in such cases, the computations are straightforward to see, and thus left to the reader. Remark 1.1. For any complete bipartite graph K r,s , with 3 ≤ r ≤ s, the following observations hold. • The wheel graph W n is a graph of order n formed by connecting a single universal vertex to all vertices of a cycle of order n − 1.

Remark 1.2.
For any wheel graph W n , with n ≥ 4, the following observations hold.
• α(W n ) = n−1 The family F p,q of graphs, which we next construct, shall also be useful for our purposes. We begin with p star graphs S 1,t1 , . . . , S 1,tp , with centers c 1 , . . . , c p , respectively, such that t 1 , . . . , t p ≥ 3; and q complete bipartite graphs K r1,r 1 , . . . , K rq,r q with 4 ≤ r i ≤ r i /2 for every i ∈ {1, . . . , q}. Next, for every i ∈ {1, . . . , q}, we add r i pendant vertices to exactly two vertices, say x i , y i , of each complete bipartite graph K ri,r i belonging to the bipartition set of cardinality r i . Finally, to obtain a graph G ∈ F p,q , we add an extra vertex w, and join w with an edge to exactly one leaf, say z i , of each star S 1,ti , and to exactly one vertex, say w i , of the bipartition set of cardinality r i of each complete bipartite graph K ri,r i . Figure 1 shows a fairly representative example of a graph in F 2,2 .
The following remark gives the values of some domination parameters of a graph in F p,q . Some of these values can be straightforwardly computed, and thus left to the reader's discretion. For any graph G ∈ F p,q the following claims hold.
Proof. (a) and (b) can be easily observed. For (c), we note that the function f 3 defined as follows, to get a γ tR (G)-function with weight 3p + 6q. For item (e), we first note that the set , for (f), we observe that the function f 6 given as, where W is the union of all bipartition sets of cardinality r i of each complete bipartite graph K ri,r i minus the vertices x i , y i , is a γ oiR (G)-function of weight 2p+2q +1+ q i=1 r i . Finally, for item (g), we consider the function f 7 defined as with W as defined above. With some not so hard arguments, we note that such function f 7 is a γ oitR (G)-function of weight 3p + 3q + q i=1 r i .

Bounds and Relationships with Other Parameters
Cabrera Martínez, Kuziak and Yero [7] in 2019, established the following result for any connected nontrivial graph, although it also holds for any graph with no isolated vertex.
To improve these bounds above, we need to introduce the next results. Also, we recall that a graph is claw-free if it does not contain K 1,3 as an induced subgraph.
Proof. Let S be a vertex cover of G. Hence, S is also a dominating set. If the subgraph induced by S has an isolated vertex v, then since G has minimum degree three, the vertex v has at least three neighbors not in S. Since V (G) \ S is an independent set, we have that v together with these three neighbors induce a K 1,3 , which is not possible. Therefore, the subgraph induced by S has no isolated vertices or equivalently, S is a total dominating set of G.
With the results above in mind, we state the following theorem, which improves the bounds given in Theorem 2.1.

Theorem 2.4.
For any graph G with no isolated vertex, order n and maximum degree Δ ≥ 2, Moreover, for any claw-free graph G of minimum degree δ ≥ 3, Proof. We first proceed to prove the first upper bound. Let D be a γ t (G)-set and S an α(G)-set.
is an independent set and V 1 ∪ V 2 = D ∪ S is a total dominating set. We only need to prove that every vertex in V 0 has a neighbor in Since S is a vertex cover and D is a total dominating set, we deduce that N (x) ⊆ S and N (x) ∩ D = ∅, respectively. Hence N (x) ∩ D ∩ S = ∅, or equivalently, N (x) ∩ V 2 = ∅. Thus, f is an OITRDF on G, as desired, and so, Next, we proceed to prove the lower bound. Let f (V 0 , V 1 , V 2 ) be a γ oitR (G)-function. Notice first that (V 1 \ L(G)) ∪ V 2 is a vertex cover. Moreover, S(G) ⊆ V 1 ∪ V 2 and |V 1 ∩ S(G)| ≤ |V 1 ∩ L(G)|. Therefore, Now, we notice that every vertex in V 2 has at most Δ − 1 neighbors in V 0 as V 1 ∪ V 2 is a total dominating set. Hence, |V 0 | ≤ (Δ − 1)|V 2 |. Taking into account the inequality above, and the fact that
We now consider G is claw-free. The next proof follows along the lines of the first part of this proof. We assume D is a γ(G)-set and S is an α(G)set. We claim that the function f (V 0 , V 1 , V 2 ) (as above) is an OITRDF on G.
It is clear that V 0 is an independent set as V 0 ⊆ V (G) \ S. Since G is claw-free graph, by Lemma 2.3, we have that S is also a total dominating set. Hence, V 1 ∪ V 2 = D ∪ S is a total dominating set as well. We only need to prove that every vertex in V 0 has a neighbor in V 2 . Let x ∈ V 0 = V (G)\(D∪S). Since S is a vertex cover and D is a dominating set, we deduce that The bounds above are tight. For instance, for the graph G shown in Fig. 2 we have that γ oitR (G) = 7 = α(G) + γ t (G). Also, the complete graph K n satisfies that γ oitR (K n ) = n = α(K n ) + n−α(Kn) Δ−1 . Furthermore, in [7], the authors showed that the corona graph G ∼ = G 1 N r satisfies that γ oitR (G) = 2|S(G)| = α(G) + |S(G)|. In addition, for the case of wheel graphs W n , since they have no leaves, they clearly have no support vertices, and so |S(W n )| = 0. For such graphs, the lower bound above is tight when n ≥ 7, since n−α(Wn) Δ(Wn)−1 = 1 and, by Remark 1.   Figure 2. A graph G with γ oitR (G) = 7 = α(G)+γ t (G) with the positive labels of a γ oitR (G)-function α(W n ) + 1. Other graphs that show the tightness of the upper bound of Theorem 2.4 are the complete bipartite graphs K r,s , which can be seen using Remark 1.1. For the tightness of the bound concerning claw-free graphs, we consider for instance the complete graph K n (n ≥ 4), which is claw-free, and satisfies that γ oitR (K n ) = n = α(K n ) + γ(K n ).
The next result, which also improves the upper bound given in Theorem 2.1, is an immediate consequence of Theorem 2.4 and Observation 2.2 (ii).

Theorem 2.5. For any graph G with no isolated vertex,
With respect to the equality in the bound γ oitR (G) ≤ α(G) + 2γ(G) above, we can deduce the following connection. To this end, we need to say that a graph G is called a Roman graph if γ R (G) = 2γ(G). Proposition 2.6. If G is a graph such that γ oitR (G) = α(G) + 2γ(G), then G is a Roman graph.
Notice that the opposed to the result above is not necessarily true. For instance, any complete bipartite graph K r,s , with 3 ≤ r ≤ s, is a Roman graph, but it does not satisfy the equality since, by Remark 1.1, α(K r,s ) + 2γ(K r,s ) = r + 4 = r + 2 = γ oitR (K r,s ).
Concerning the outer-independent Roman domination number and the outer-independent total domination number, which are closely related to γ oitR (G), in [7], the authors showed the following results.
Theorem 2.7. [7] The following statements hold for any graph G of order n ≥ 3 with no isolated vertex.
(i) γ t,oi (G) We now provide a result which improves the upper bounds given in Theorem 2.7.

Relating the Outer-Independent Total Roman domination
Proof. First, we proceed to prove that γ oitR (G) ≤ γ t,oi (G) + γ(G). Let D be a γ t,oi (G)-set and S a γ(G)-set. Let f (V 0 , V 1 , V 2 ) be a function defined by We claim that f is an OITRDF on G. Notice that V 0 ⊆ V (G) \ D is an independent set and V 1 ∪ V 2 is a total dominating set as D is a outer-independent total dominating set of G. Now, we prove that every vertex in V 0 has a neighbor in Since D is also a vertex cover and S is a dominating set, we deduce that N (x) ⊆ D and N (x) ∩ S = ∅, respectively.
The following result shows a class of graphs which satisfy the equality γ oitR (G) = γ oiR (G). Theorem 2.9. For any claw-free graph G of minimum degree δ ≥ 3, Proof. Let f (V 0 , V 1 , V 2 ) be a γ oiR (G)-function. Since V 0 is an independent set, we have that V 1 ∪V 2 is a vertex cover of G. As every vertex cover in a claw-free graph of minimum degree δ ≥ 3 is also a total dominating set by Lemma 2.3, we deduce that f is also an OITRDF. Hence, γ oitR (G) ≤ ω(f ) = γ oiR (G). Theorem 2.7 (ii) completes the proof.

MJOM
+ γ(K 1,n−1 ). In contrast with the example above, if we consider G ∈ F p,q (as defined in Section 1), we note that there are graphs achieving a strict inequality in all the steps of the inequality chain (1). That is, for any graph G ∈ F p,q , (using Remark 1.3) it follows that Theorem 2.10 [5]. For any connected graph G of minimum degree δ, Next, we provide a new upper bound for the outer-independent total Roman domination number, which is an immediate consequence of Theorems 2.8 and 2.10 . Notice that this result improves Theorem 2.5 for the graphs G that satisfy the inequality α(G) ≤ γ(G) + δ − 1.
The bound above is tight for the case of star graphs, and in connection with this fact, we pose the following question.
Open question: Is it the case that γ oitR (G) = 2α(G) + γ(G) − δ + 1 if and only if G is a star graph?

Exact Formulas for Some Families of Graphs
This section is centered into giving the exact value of the outer-independent total Roman domination number of some significant families of graphs, that have been frequently studied in the literature in connection with several domination related invariants.

Sierpiński Graphs
Sierpiński graphs were introduced in [15]. However, they were named in this way a little further in [16]. For integers n ≥ 2 and p ≥ 3, the Sierpiński graph   Figure 3. The graphs S 2 4 , S 3 4 and S 2 5 (from left to right), and a labeling for an OITRDF of minimum weight in each graph