A Note on Outer-Independent 2-Rainbow Domination in Graphs

: Let G be a graph with vertex set V ( G ) and f : V ( G ) → { ∅ , { 1 } , { 2 } , { 1,2 }} be a function. We say that f is an outer-independent 2-rainbow dominating function on G if the following two conditions hold: ( i ) V ∅ = { x ∈ V ( G ) : f ( x ) = ∅ } is an independent set of G . ( ii ) ∪ u ∈ N ( v ) f ( u ) = { 1,2 } for every vertex v ∈ V ∅ . The outer-independent 2-rainbow domination number of G , denoted by γ oir 2 ( G ) , is the minimum weight ω ( f ) = ∑ x ∈ V ( G ) | f ( x ) | among all outer-independent 2-rainbow dominating functions f on G . In this note, we obtain new results on the previous domination parameter. Some of our results are tight bounds which improve the well-known bounds β ( G ) ≤ γ oir 2 ( G ) ≤ 2 β ( G ) , where β ( G ) denotes the vertex cover number of G . Finally, we study the outer-independent 2-rainbow domination number of the join, lexicographic, and corona product graphs. In particular, we show that, for these three product graphs, the parameter achieves equality in the lower bound of the previous inequality chain.


Introduction
Over the last decade, many variants associated with classical domination parameters in graphs have been defined and studied.In particular, variants related to domination and independence in graphs have attracted the attention of many researchers.
One of the most analysed ideas, and from which many parameters have been defined, is considering dominating sets whose complements form independent sets.Some recent references about some of these remarkable variants can be observed in [1,2] for total outerindependent domination, in [3,4] for outer-independent double Roman domination, in [5][6][7][8] for outer-independent (total) Roman domination, and in [9][10][11][12] for outer-independent (total) 2-rainbow domination.
This note mainly deals with providing new results about one of the aforementioned parameters: the outer-independent 2-rainbow domination number (OI2RD number) of a graph.Given a graph G, we say that a function f : V(G) → {∅, {1}, {2}, {1, 2}} is an outer-independent 2-rainbow dominating function (OI2RD function) on G if the following two conditions hold.
, {2}, {1, 2}}.We will identify an OI2RD function f with the subsets V ∅ , V {1} , V {2} , and V {1,2} of V(G) associated with it, and so we will use the unified notation f (V ∅ , V {1} , V {2} , V {1,2} ) for the function and these associated subsets.The OI2RD number of G, denoted by γ oi r2 (G), is the minimum weight As previously mentioned, this parameter has been studied by different researchers.For instance, in [9,10] interesting tight bounds were obtained for general graphs and for the particular case of trees.Moreover, in [9] graphs with small and large OI2RD numbers were characterized.Finally, in [11] the authors studied the OI2RD number for the Cartesian products of paths and cycles.
The note is organised as follows.In Section 2, we provide new tight bounds which improve the well-known bounds β(G) ≤ γ oi r2 (G) ≤ 2β(G) given in [9], where β(G) denotes the vertex cover number of G. Finally, in Section 3 we provide closed formulas for this parameter in the join, lexicographic, and corona product graphs.

Additional Definitions and Tools
In this note, we consider that all graphs are simple and undirected, meaning that they have only undirected edges with no loops and no multiple edges between two fixed vertices.Given a graph The domination number of G, denoted by γ(G), is the minimum cardinality among all dominating sets of G.A dominating set D with |D| = γ(G) is defined as a γ(G)-set.This classical parameter has been extensively studied.From now on, for a parameter ρ(G) of a graph G, by ρ(G)-set we mean a set of cardinality ρ(G).
Two of the best-known variants of dominating sets, which they are also related to each other, are the independent sets and the vertex cover sets.
The minimum cardinality among all vertex cover sets of G, denoted by β(G), is the vertex cover number of G.In 1959, Gallai established the following well-known relationship.
Theorem 1 ([13]).If G is a nontrivial graph, then Finally, we state the following useful tool.For the remainder of the paper, definitions will be introduced whenever a concept is needed.Proposition 1.Let G be a graph with no isolated vertex.Then, there exists a , which completes the proof.

New Bounds on the Outer-Independent 2-Rainbow Domination Number
Kang et al. [9] showed that, for any graph G with no isolated vertex, The following theorem shows that the bounds given in (1) have room for improvement, since |S s (G)| ≥ 0 and γ(G) ≤ β(G).Theorem 2. For any graph G with no isolated vertex, Proof.We first prove the lower bound.Let f (V ∅ , V {1} , V {2} , V {1,2} ) be a γ oi r2 (G)-function which satisfies Proposition 1.Hence, V(G) \ V ∅ is a vertex cover and S s (G) ⊆ V {1,2} , which implies that Now, we proceed to prove the upper bound.Let D be a γ(G)-set and S a β(G)-set.Let g(W ∅ , W {1} , W {2} , W {1,2} ) be a function defined as follows.
We claim that g is an OI2RD function on G.If W ∅ = ∅, then we are done.Hence, we assume that W ∅ = ∅.Notice that W ∅ is an independent set of G because S is a vertex cover set of G.We only need to prove that g(N(x)) = ∪ u∈N(x) g(u) = {1, 2} for every x ∈ W ∅ .Let v ∈ W ∅ .Since S and D are both dominating sets of G, we deduce that either In both cases, and by definition of g, we obtain that g(N(v)) = {1, 2}.Thus, g is an OI2RD function on G, as required. Therefore, The following result, which is a direct consequence of Theorem 2, the upper bound given in (1), and the fact that γ(G) ≤ β(G), provides a necessary condition for the graphs that satisfy the equality γ oi r2 (G) = 2β(G).

Proposition 2.
Let G be a graph with no isolated vertex.If The converse of proposition above does not hold.For instance, the graph G given in Figure 1 satisfies As a second consequence of Theorem 2 we can derive the next proposition.
Proposition 3. Let G be a graph with no isolated vertex.If S s (G) is a dominating set of G, then Therefore, Theorem 2 leads to the equality, which completes the proof.
The next theorem improves the upper bound given in Theorem 2 for the case where G is a tree.Theorem 3.For any nontrivial tree T, Proof.Let S be a β(T)-set such that S(T) ⊆ S. Now, we construct a partition {I, D} of S as follows.Let u ∈ S(T) and S u i = {w ∈ S : d(w, u) = i}, where d(w, u) represents the distance between w and u.Now, we need to introduce some necessary definitions.Let (u) be the eccentricity of u, and, for any vertex x = u, the Parent[x] is the vertex adjacent to x on the unique x − u path.
Let I = ∪ With this property in mind and the fact that V(T) \ S is an independent set, it is easy to deduce that the function f , defined below, is an OI2RD function on T.
From Theorems 2 and 3, we obtain that for any nontrivial tree T, The following result is a direct consequence of the previous inequality chain.
Proposition 4. If T is a tree such that S(T) = S s (T), then

The Cases of the Join, Lexicographic, and Corona Product Graphs
In this section, we consider the OI2RD number of three well-known product graphs (join − +, lexicographic − •, and corona − ).If G 1 and G 2 are any two graphs with no isolated vertex, then

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The join graph G 1 + G 2 is the graph with vertex set For instance, the graph G given in Figure 1 is isomorphic to the join graph N 2 + N 5 , where N r is the empty graph of r vertices.

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The corona product graph G 1 G 2 is the graph obtained from G 1 and G 2 , by taking one copy of G 1 and n(G 1 ) copies of G 2 and joining by an edge every vertex from the i th -copy of G 2 with the i th -vertex of G 1 .Figure 2 shows the graph P 4 P 3 .
and D = ∪ (u) i=0 D i , where I 0 = {u} and D 0 = ∅ and for i ≥ 1 we define I i and D i as follows.For every v ∈ S u i , define the class v⊆ S u i such that v, v ∈ v if and only if Parent[v]=Parent[v ].From i = 1 to eccentricity (u), we consider the next cases for every v ⊆ S u i , where we fix v ∈ v. (i) Parent[v] ∈ S. In this case, we set v ⊆ I i .(ii) Parent[v] / ∈ S (notice that i ≥ 2 and Parent[Parent[v]] ∈ S).If Parent[Parent[v]] ∈ I i−2 , then we set v ⊆ D i , otherwise we set v ⊆ I i .It is clear that {I, D} is a partition of S. By condition (ii) in the construction above, it follows that N(x) ∩ I = ∅ and N(x) ∩ D = ∅ for every vertex x ∈ V(T) \ (S ∪ L(T)).