Outer-independent k-rainbow domination

ABSTRACT An outer-independent k-rainbow dominating function of a graph G is a function f from to the set of all subsets of such that both the following hold: (i) whenever v is a vertex with , and (ii) the set of all with is independent. The outer-independent k-rainbow domination number of G is the invariant , which is the minimum sum (over all the vertices of G) of the cardinalities of the subsets assigned by an outer-independent k-rainbow dominating function. In this paper, we initiate the study of outer-independent k-rainbow domination. We first investigate the basic properties of the outer-independent k-rainbow domination and then we focus on the outer-independent 2-rainbow domination number and present sharp lower and upper bounds for it.


Introduction
In general, we follow the notation and graph theory terminology in [1]. Specifically, let G = (V(G), E(G)) be a finite simple graph.  (u), and so on.) We denote by P n and C n the path and cycle on n vertices, respectively. The distance between two vertices u and v in a connected graph G is the length of a shortest uv-path in G. The diameter of a graph G, denoted by diam(G), is the greatest distance between two vertices of G. For a vertex v in a rooted tree T, let C(v) and D(v) denote the set of children and descendants of v, respectively and let D[v] = D(v) ∪ {v}. Also, the depth of v, depth(v), is the largest distance from v to a vertex in D(v). The maximal subtree at v is the subtree of T induced by D [v], and is denoted by T v . By K p,q we denote a complete bipartite graph with partite sets of cardinalities p and q. A star is a K 1,q and a double star DS q,p , where q ≥ p ≥ 1, is a tree containing exactly two non-leaf vertices which one is adjacent to p leaves and the other is adjacent to q leaves. By X we denote the induced subgraph of a graph G with vertex set X ⊆ V(G).
A set I ⊆ V(G) is independent if no two vertices in I are adjacent. The maximum cardinality of an independent set in G equals the independence number β 0 (G). A vertex cover of a graph G is a set of vertices that covers all the edges. The minimum cardinality of a vertex cover is denoted by α 0 (G). The following theorem due to Gallai.

Theorem 1.1 ([2]): Let G be a graph. A subset I of V(G) is independent if and only if V(G) − I is a vertex cover of G.
In particular, β 0 (G) = |V(G)| − α 0 (G). (G). The domination number γ (G) equals the minimum cardinality of a dominating set in G. For many applications, it is not possible to use an arbitrary dominating set D of G. One possible form of restriction is based on imposing some conditions on the set V(G) − D. Here we concentrate on the property of being outer-independent, i.e. V(G) − D is independent. Results on outer-independent domination parameters can be found e.g. in [3][4][5][6][7].
For a positive integer k we denote the set {1, 2, . . . , k} by [k]. The power set (that is, the set of all subsets) of [k] is denoted by 2 [k] . Let G be a graph and let f be a function that assigns to each vertex a subset of [k]; that is, f : . . , k} is fulfilled. Given a graph G, the minimum weight of a k-rainbow dominating function is called the k-rainbow domination number of G, which we denote by γ k r (G). The concept of rainbow domination was introduced in [8] and has been studied extensively [9][10][11][12][13][14][15].
Here we introduce and study a new variant of a k-rainbow dominating function. A k-rainbow dominating function f : The outerindependent k-rainbow domination number γ k oir (G) is the minimum weight of an OIkRD-function on G. An OIkRDfunction of weight γ k oir (G) is called a γ k oir -function. Since any OIkRD-function is a kRD-function, we have In this paper, we initiate the study of outer-independent k-rainbow domination. We first investigate the basic properties of the outer-independent k-rainbow domination and then we focus on the outer-independent 2-rainbow domination number and present sharp lower and upper bounds for it.

Observation 2.2:
For any graph G of order n, min{n, k} ≤ γ k oir (G) ≤ n. In particular, γ k oir (G) = n whenever k ≥ n.

Observation 2.3: For any OIkRD
Notice also that the outer-independent domination is the same as the outer-independent 1-rainbow domination if we view an outer-independent dominating set D as an outer-independent 1-rainbow dominating function f defined by f (v) = {1} when v ∈ D and f (v) = ∅ otherwise. Therefore here we concentrate on the case when a graph G is connected and n − 1 ≥ k ≥ 2.
Now we characterize all connected graphs of order n ≥ k + 1 attaining the lower bound in Observation 2.2. Theorem 2.1: Let k ≥ 2 be a positive integer and let G be a connected graph of order n ≥ k + 1. Then γ k oir (G) = k if and only if G = H G ∨ K n−h , where H G is a graph of order h ≤ k. Proof: First assume that G satisfies (I) and (II). It follows from Theorem 2.1 and (I) that γ k oir (G) ≥ k + 1. Now define the function f : with ω(f ) = k + 1 and thus γ k oir (G) = k + 1. Conversely, assume that γ k oir (G) = k + 1. It follows from Theorem 2.1 that G satisfies (I). Now we show that G satisfies (II). Let f be a γ k oir -function on G. Choose f so that Since ω(f ) = k + 1, there exists a colour, say k, which appears exactly twice and each other colour appears exactly once. Hence there are two vertices, say z and w, such otherwise, is an OIkRDfunction of G with weight less that γ k oir (G) which is a contradiction. Thus A and B are non-empty sets. Since f is an It follows from γ k oir (G) = k + 1 that |A| + |B| ≤ k + 1 and so (i) holds. Since for each vertex u ∈ A, f (u) has a colour which is not appeared in other vertices, every vertex in V(G) − (A ∪ B) is adjacent to every vertex of A. Also since the colour k appears exactly in f (z) and f (w), each vertex of V(G) − (A ∪ B) must be adjacent to one of the vertices z and w. Hence (ii) holds.
-function which contradicts the choice of f. Thus (iv) holds and the proof is complete. .
Finally, the right side inequality follows by Observation 2.2.

Proposition 2.2: For any graph
Proof: Let C be any minimum vertex cover set of G and define the function f : Clearly f is an OIkRD-function on G which immediately implies the required.
The bounds in Proposition 2.2 are attainable. Let G be a graph such that each vertex is either a leaf or a support vertex and let each support vertex of G is adjacent to at least k + 1 leaves. Then clearly S(G) is a minimum vertex cover set and the function f : when u is a support vertex and f (u) = ∅ when u is a leaf, is an OIkRD-function on G of minimum weight. Thus γ k oir (G) = kα 0 (G). We will say that a graph G is a vertex cover outer independent k-rainbow graph, a VCOI-k-rainbow graph for short, if γ k oir (G) = kα 0 (G).
Proof: Assume that G is a VCOI-k-rainbow graph and let D be a minimum vertex cover set of G. Then the func- . Thus all inequalities in this chain must be equality and so Conversely, assume that there exists a γ k . By Proposition 2.2 we deduce that γ k oir (G) = kα 0 (G) and this implies that G is a VCOI-k-rainbow graph.

Proposition 2.4: Let H be an induced subgraph of a graph G. Then
Then ω(f ) = a 1 + a 2 + · · · + a k .

Theorem 2.3: Let G be a graph of order at least two and
Proof: Let f be a γ k oir -function on G, and Assume without loss of generality that a 1 ≥ a 2 ≥ · · · ≥ a k .
Define g : Clearly g is an OIkRD-function on G. This fact and Observation 2.4 lead to

Outer-independent 2-rainbow domination number
In this section, we focus on outer-independent 2rainbow domination. An OI2RD-function f on a graph G can be represented by the ordered 4-tuple In this representation, its weight is ω(f ) = |V 1 | + |V 2 | + 2|V 1,2 |.

Outer-independent 2-rainbow domination versus domination parameters
is an independent set. The outer-independent Roman domination number γ oiR (G) is the minimum weight of an OIRD-function on G. Outer-independent Roman domination was introduced by Abdollahzadeh Ahangar et al. in [3].
is an outer-independent 2-rainbow dominating function on a graph G and so Abdollahzadeh Ahangar et al. proved the following bounds on γ oiR (G).
Next results are immediate consequences of Propositions 3.1, 3.2 and inequality (2).

Corollary 3.2:
If G is a connected triangle-free graph of order n ≥ 2 and maximum degree , then γ 2 oir (G) ≤ n − + 1. This bound is sharp for all stars K 1,n−1 , n ≥ 2.

Corollary 3.3:
Let G be a connected graph of order n. If G has girth g < ∞, then γ 2 oir (G) ≤ n + g 2 − g.
In the following, we provide an upper bound on γ oiR (G) in terms of γ 2 oir (G) for arbitrary graphs G.

Theorem 3.1:
For any graph G, γ oiR (G) ≤ 3 2 γ 2 oir (G). This bound is sharp for the family F of graphs illustrated in Figure 1.
oir (G)-function and without loss of generality |V 1 | ≥ |V 2 |. Then g = The notion of outer-independent Italian domination in graphs was introduced in [16]. An outer-independent Italian dominating function (OIID-function) on a graph G is a function f : V(G) → {0, 1, 2} such that every vertex v ∈ V(G) with f (v) = 0 has at least two neighbours assigned 1 under f or one neighbour w with f (w) = 2 and the set of all vertices assigned 0 under f is independent. The weight of an OIID-function f is the value ω(f ) = u∈V(G) f (u). The minimum weight of an OIIDfunction on a graph G is called the outer-independent Italian domination number γ oiI (G) of G. Clearly, if f = (V 0 , V 1 , V 2 , V 1,2 ) is a γ 2 oir (G)-function, then the function g = (V 0 , V 1 ∪ V 2 , V 2 ) is an outer-independent Italian dominating function of G and so In [16], the authors proved that γ oiI (C n ) = n 2 for n ≥ 3 and γ oiI (K p,q ) = q for p ≥ q ≥ 2. Using these we obtain the next results.

Theorem 3.2:
For any connected graph G of order n ≥ 2 with minimum degree δ and maximum degree , Next result is an immediate consequence of Theorem 3.2 and inequality (3).

Corollary 3.4:
For any connected graph G of order n ≥ 2 with minimum degree δ and maximum degree ,

This bound is sharp for cycles.
Fan et al. [16] proved the following Nordhaus-Gaddum type result for outer-independent Italian domination number.

Theorem 3.3: For any graph G on n vertices,
As an immediate consequence we have:

Corollary 3.5: For any graph G on n vertices,
The upper bound is sharp for K 2 and the lower bound is sharp for a graph G obtained from K 10 with vertex set {u

Trees
Here we present sharp upper and lower bounds on the outer-independent 2-rainbow domination number of trees. First we show that the outer-independent 2rainbow domination number and the outerindependent Italian domination number of a tree are equal.

Theorem 3.4: For any tree T, γ oiI (T) = γ 2 oir (T).
Proof: Consider a γ oiI -function f = (V 0 , V 1 , V 2 ) on T and let U 1 , U 2 , . . . , U s be the components of the graph V 1 ∪ V 2 . Let T f be the graph whose vertex set is {U 1 , U 2 , . . . , U s } and two vertices U i and U j are adjacent if and only if there are vertices u i ∈ U i , u j ∈ U j and u ij ∈ V 0 such that u i u ij u j is a path in T. If V 1 = ∅, then we may assume that V 1 ∩ U 1 = ∅. Define g : V(G) → 2 [2] as follows: where the distance U 1 and U j in T f , is even, and g(x) = {2} otherwise. Clearly g is an OI2RD-function on T with weight ω(f ) and so γ 2 oir (T) ≤ γ oiI (T). The result now immediately follows from (3).
Using the results given in [16] and Theorem 3.4, we obtain the following results.

Theorem 3.5:
For any tree T of order n ≥ 2,

where (T) is the number of leaves of T. This bound is sharp for stars and paths.
As a consequence of Propositions 2.4 and 3.5 we obtain the following result.

Corollary 3.6: For any connected graph G of order n,
In the sequel we will use the following observation.

Observation 3.1:
Let G be a graph.
Next we present an upper bound on outerindependent 2-rainbow domination number of a tree T in terms of the order and its number of support vertices. For any tree T, let s(T) denote the number its support vertices.

Theorem 3.6: If T is a tree of order at least 3, then
This bound is sharp for all paths P 2k (k ≥ 1).

Proof:
The proof is by induction on n(T). It is easy to verify that the statement is true for n(T) ≤ 4. Hence, let n(T) ≥ 5 and assume that every  s(T). Henceforth, we assume that every support vertex of T is adjacent to at most two leaves.
Let v 1 v 2 . . . v k be a diametrical path in T such that deg T (v 2 ) is as large as possible and root T at v k . We consider the following cases. . Let diam(T) ≥ 5. We distinguish the followings.
If v 4 is a support vertex, then any γ 2 oir (T − T v 2 )function can be extended to an OI2RD-function of T by assigning {1, 2} to v 2 and ∅ to the leaves adjacent to v 2 and it follows from the induction hypothesis on T − T v 2 and the facts n(T − T v 2 ) = n − 3 and s(T − T v 2 ) = s(T) − 1 that Assume next that v 4 is not a support vertex. We consider the following situations.
(i) v 4 has a child w 2 with depth one. Let T = T − T w 2 and let f be a γ 2 oir (T )-function such that f (v 2 ) = {1, 2}. Without loss of generality, we may assume that |f (v 4 )| ≥ 1 and that 1 ∈ f (v 4 ). If deg T (w 2 ) = 2, then the function f can be extended to an OI2RDfunction of T by assigning ∅ to w 2 and {2} to the leaf-neighbour of w 2 , and using the induction hypothesis on T and the facts n(T ) = n(T) − 2 and s(T ) = s(T) − 1 we obtain If deg T (w 2 ) = 3, then the function f can be extended to an OI2RD-function of T by assigning a {1, 2} to w 2 and ∅ to its leaf-neighbours, and using the induction hypothesis on T and the facts n(T ) = n(T) − 3 and s(T ) = s(T) − 1 we obtain (ii) v 4 has a child w 3 with depth two different from Assume that deg T (v 5 ) = 2. Let T = T − T v 3 and f be a γ 2 oir -function. By Observation 3.1 (item (2) By the choice of diametrical path, we deduce that every child of v 3 with depth one has degree two. Consider the following subcases.
First suppose that v 3 is a strong support vertex or adjacent to a support vertex except v 2 . Let T = T − {v 1 , v 2 } and f a γ 2 oir (T )-function. By Observation 3.1, we may assume without loss of generality that 1 ∈ f (v 3 ). Now f can be extended to an OI2RD-function of T by assigning a {2} to v 1 and an ∅ to v 2 . Now we deduce from the induction hypothesis on T and the facts n(T ) = n(T) − 2 and s(T ) ≤ s(T) that  (2)), we may assume that 1 ∈ f (v 3 ). Now f can be extended to an OI2RD-function of T by assigning a {2} to v 1 and an ∅ to v 2 , and we deduce from the induction hypothesis and the facts n(T ) = n(T) − 2 and s(T ) = s(T) that γ 2 oir (T) ≤ γ 2 oir (T ) + 1 ≤ n+s 2 and this completes the proof.
Next result in an immediate consequence of Theorems 3.4 and 3.6 since the number of support vertices of a tree on n ≥ 3 vertices is at most n 2 .
A constructive instruction of trees attaining the bound given in Corollary 3.7 is given in [16].
We close this section by establishing a lower bound on the outer-independent 2-rainbow domination number of a tree in terms of the order and the total outerindependent domination number. Recall that a set S of vertices of a graph G is a total outer-independent dominating set if every vertex from V(G) has a neighbour in S and the complement of S is an independent set. The total outer-independent domination number γ oit (G) of G is the smallest possible cardinality of any total outer-independent dominating set of G. The total outer-independent domination was introduced in [17,18]. It was observed in [17] that γ oit (P n ) = 2 i fn = 2, 2n/3 if n ≥ 3. Furthermore, this bound is sharp for paths P 6k (k ≥ 1).

Proof:
The proof is by induction on the order n(T).
If n(T) = 2 or n(T) = 3, then γ 2 oir (T) = γ oit (T) = 2 and the bound is sharp. Let n(T) ≥ 4 and assume that for any tree T of order n(T ) < n(T), γ 2 oir (T ) ≥ γ oit (T ) − n(T ) 6 + 1. Let T be a tree of order n(T) and f = (V 0 , V 1 , V 2 , V 1,2 ) be a γ 2 oir (T)-function. Since for stars and double stars T we have γ oit (T) = 2, so the statement holds. Hence, we assume that T has diameter at least four. If V 0 = ∅, then by the fact that n(T) 6 ≥ 1 we have γ 2 oir (T) = n(T) ≥ γ oit (T). Therefore we assume that V 0 = ∅. First suppose that T has a support vertex x which is adjacent to two or more leaves. Let u, v be two leaves