On the Existence of Independent [j,k]-Dominating Sets in the Generalized Corona of Graphs

Abstract

In 2013 the concept of $[j,k]$-dominating sets in graphs have been introduced as an extension of $[1,k]$-dominating sets. Regarding studying $[j,k]$-dominating sets, the research is also interesting if the additional independence requirement is assumed, because the existence of independent $[1,k]$-dominating sets is an $\mathcal{NP}$-complete problem in general case. Inspired by this topic, in this paper we study the problem of the existence of independent $[j,k]$-dominating sets in a special, multi-argument graph product. We give the complete characterization of the generalized corona $G\circ h_n$ with an independent $[j,k]$-dominating set, for arbitrary positive integers j and k, which generalize the known result for a corona of two graphs.

1. Introduction

In general, we use the standard terminology and notation of graph theory. For concepts not defined here, see [1].

Let G be a finite, simple, undirected graph with the vertex set $V\left(G\right)$ and the edge set $E\left(G\right)$. By $d_G\left(x\right)$, we denote the degree of a vertex x.

Let G be a graph such that $V\left(G\right)=\{x_1,x_2,\cdots ,x_n\}$, $n\ge 1$ and let $h_n={\left(H_i\right)}_{i\in \mathcal{I}=\{1,2,\cdots ,n\}}$ be a sequence of arbitrary graphs. The generalized corona of a graph G and the sequence $h_n$ is the graph $G\circ h_n$ such that $V(G\circ h_n)=V\left(G\right)\cup {\bigcup }_{i=1}^{n}V\left(H_i\right)$ and $E(G\circ h_n)=E\left(G\right)\cup {\bigcup }_{i=1}^{n}E\left(H_i\right)\cup {\bigcup }_{i=1}^n\{x_{i}y:y\in V\left(H_i\right)\}.$ If $H_i\cong H$, for $i\in \{1,\cdots ,n\}$; then, we obtain the definition of corona of two graphs G and H.

A subset $D\subseteq V\left(G\right)$ is dominating, if every vertex of G is either in D or it is adjacent to at least one vertex of D. The minimum cardinality of all dominating sets of G is the domination number $\gamma \left(G\right)$. For domination-related concepts not defined here, see [2].

A subset $S\subseteq V\left(G\right)$ is called an independent set of G if no two vertices from S are adjacent. A subset of $V\left(G\right)$ being independent and dominating simultaneously is an independent dominating set (also named a kernel) of G. The minimum cardinality of an independent dominating set is the independent domination number $i\left(G\right)$. An independent set with minimum cardinality is called an $i\left(G\right)$-set. The theory of kernels was introduced for digraphs by Neumann and Morgenstern in 1953 in the context of game theory, see [3], and next studied because of its relations with distinct graph problems, for example, in the list colorings and perfectness. Note that Berge was a pioneer in studying independent dominating sets in digraphs; see [4]. Next this theory was generalized into two directions in the distance sense by considering so-called $(k,l)$-kernels, which were introduced by Borowiecki and Kuzak in [5], or kernels by the monochromatic paths, introduced by Sands, Sauer and Woodrow in [6]. Details can be found in a number of papers; for example, see the results of Galeana-Sanchez et al., Kucharska, Szumny et al. and others [7,8,9,10,11]. Different types of kernels have been introduced by adding some new restrictions to the classical definition of kernels—for example, restrained independent dominating sets [12].

In undirected graphs, the problem of the existence of independent dominating sets is trivial because every maximal independent set is dominating. The problem is more complicated and interesting if we add a restriction related to the domination or independence.

Different variants and extensions of dominating sets, for example, total, perfect, efficient, secondary domination, have been introduced and studied in both combinatorial and algoritmic senses; see [13,14,15,16,17,18]. Dominating sets have also various applications in many practical problems, such as electrical networks, monitoring communication and others; for example, see [19].

Some considered types of independent dominating sets in graphs are based on multiple domination introduced by Fink and Jacobson in [20]. Let k be a positive integer. A subset $D\subseteq V\left(G\right)$ is k-dominating if every vertex from $V\left(G\right)\backslash D$ has at least k neighbors in D. If $k=1$, then we obtain a dominating set in a classical sense. If $k=2$, then we obtain a definition of 2-dominating sets. In [21], A. Włoch introduced the concept of independent 2-dominating sets (named as $(2$-d)-kernels), and next, their some properties were studied in [16,22,23,24,25,26]. Based on the definition of $(2$-d)-kernel, Nagy introduced independent k-dominating sets as a subset being independent and k-dominating; details can be found in [27,28].

Another type of dominating sets related to multiple domination is $[j,k]$-dominating, which we study in this paper.

Let us recall that the set of all vertices which are adjacent to $x\in V\left(G\right)$ is named the neighborhood of x and denoted by $N_G\left(x\right)$. Let j and k be non-negative integers. A subset $S\subseteq V\left(G\right)$ is a $[j,k]$-dominating set (in short: a $[j,k]$-dset) if for every vertex $v\in V\left(G\right)\backslash S$, $j\le |N_G\left(v\right)\cap S|\le k$. In other words, every vertex $v\in V\left(G\right)\backslash S$ is adjacent to at least j but not more than k vertices in S. Clearly, for $j=0$, the set S does not have to be dominating. Moreover, every $[j,k]$-dset is a j-dominating set and a dominating set in a classical sense, for all $j\ge 1$. The concept of the $[j,k]$-dset was introduced in 2013 by Chellali et al. in [29]. In earlier work, Dejter in [30] considered quasiperfect dominating sets, which are basically $[1,2]$-dsets. Moreover, in [29], it was noted that $[j,k]$-dsets are related to several other types of dominating sets, such as nearly perfect sets ($[0,1]$-dsets), perfect dominating sets ($[1,1]$-dsets) and nearly 2-perfect sets ($[0,2]$-dsets); see, for example, [15,31]. In recent years, $[j,k]$-dsets were broadly studied in the literature; see [32,33,34,35,36,37].

A $[j,k]$-dset which is independent we call an independent $[j,k]$-dominating set (in short: an independent $[j,k]$-dset). Clearly, an independent $[j,k]$-dset is an independent dominating set in a classical sense for all $j\ge 1$. If $j=1$ and $k=1$, we obtain a definition of an efficient dominating set, which were introduced by Bange, Barkauskas and Slater in [13] and later studied, for example, in [38,39,40]. In [41], independent $[1,k]$-dsets were considered. It was noted that not every graph has an independent $[1,k]$-dset, and it was shown that the related decision problem is $NP$-complete. Moreover, the trees with an independent $[1,k]$-dset were characterized.

The graphs products are operations which enable one to construct new classes of graphs. We can describe their properties based on their components. In the context of existence-independent $[1,k]$-dsets, the corona of two graphs was studied. In [41], the following theorem was proved, which gives sufficient and necessary conditions for the existence of an independent $[1,k]$-dset.

Theorem 1

([41]). Let k be a positive integer and $G\circ H$ be a corona of graphs G and H. Then, $G\circ H$ has an independent $[1,k]$-dset if and only if each component of G is an isolated vertex or $i\left(H\right)\le k$.

In this paper, we generalize this result by studying the existence of independent $[j,k]$-dsets in the generalized corona $G\circ h_n$ of a graph G and a sequence $h_n$ of graphs. The product $G\circ h_n$ was studied with respect to distinct kinds of domination and together with the independence restriction; see for details [16]. In this paper, we will give a complete characterization of the generalized corona $G\circ h_n$ with an independent $[j,k]$-dset, for arbitrary positive integers j and k.

2. Main Results

In this section, we study and completely solve the problem of the independent $[j,k]$-dominating sets in $G\circ h_n$. We begin with the existence of independent $[1,k]$-dsets, $k\ge 1$.

Let us begin with proving lemma, which will be useful in next theorems.

Lemma 1.

Let G be an arbitrary graph with $n\ge 1$ vertices and $h_n={\left(H_i\right)}_{i\in \mathcal{I}}$ be a sequence of n arbitrary graphs. Moreover, let k be a positive integer and $S=\{x_i\in V\left(G\right):i\left(H_i\right)>k\}$. If the graph $G\circ h_n$ has an independent $[1,k]$-dset J, then $S\subseteq J$.

Proof. 

Let J be an independent $[1,k]$-dset. By contradiction, let us suppose that there exists $i\in \mathcal{I}$ such that $x_i\in S$ and $x_i\notin J$. Since the set J is dominating and independent, it contains at least $i\left(H_i\right)$ vertices from the set $V\left(H_i\right)$. It means that the vertex $x_i$ is adjacent to at least $i\left(H_i\right)\ge k+1$ vertices from the set J, a contradiction. □

Firstly, we provide theorem which provides the complete characterization of $G\circ h_n$ with an independent $[1,1]$-dset in the case where $H_i\ne \varnothing$, for all $i\in \{1,\dots ,n\}$.

Theorem 2.

Let G be an arbitrary graph with $n\ge 1$ vertices and $h_n={\left(H_i\right)}_{i\in \mathcal{I}}$ be a sequence of n nonempty graphs. The graph $G\circ h_n$ has an independent $[1,1]$-dset if and only if, for every $i\in \mathcal{I}$, $\gamma \left(H_i\right)=1$ or $x_i$ is an isolated vertex of graph G.

Proof. 

Let us assume that for every $i\in \mathcal{I}$, it holds that $\gamma \left(H_i\right)=1$ or $x_i$ is an isolated vertex of graph G. We will prove that graph $G\circ h_n$ has an independent $[1,1]$-dset, say, $J={\bigcup }_{i\in {\mathcal{I}}_1}\left\{x_i\right\}\cup {\bigcup }_{i\in \mathcal{I}\backslash {\mathcal{I}}_1}{J}_i$, where ${\mathcal{I}}_1=\{j\in \mathcal{I}:d_G\left(x_i\right)=0\}$ and $J_i$ is the minimum dominating set of a graph $H_i$. If $x_i\in V\left(G\right)$ is an isolated vertex of G, then $x_i\in J$ and $V\left(H_i\right)\cap J=\varnothing$. Then, the vertices from the set $V\left(H_i\right)$ are adjacent to exactly one vertex from the set J. Otherwise, an arbitrary minimum dominating set $J_i$ of graph $H_i$ is included in the set J. Since $\gamma \left(H_i\right)>1$, the vertices from the set $(V\left(H_i\right)\backslash J_i)\cup \left\{x_i\right\}$ have exactly one neighbor in the set $J_i$. Hence, the set J is an independent $[1,1]$-dset of the graph $G\circ h_n$.

Conversely, let us assume that the graph $G\circ h_n$ has an independent $[1,1]$-dset J and there exists $i\in \mathcal{I}$ such that $x_i$ is not an isolated vertex of G and $\gamma \left(H_i\right)\ne 1$. Since $H_i$ is nonempty graph, $\gamma \left(H_i\right)>1$. This means that $i\left(H_i\right)>1$. Let $S=\{x_i\in V\left(G\right):i\left(H_i\right)>1\}$. Then, $x_i\in S$. By Lemma 1, we obtain that $x_i\in J$. Since $x_i$ is not an isolated vertex of G, there exists $x_j\in V\left(G\right)$, $i\ne j$, such that $x_i{x}_j\in E\left(G\right)$. The independence of the set J implies that $x_j\notin J$. Since the vertices from the set $V\left(H_j\right)$ must be dominated by the set J, $V\left(H_j\right)\cap J\ne \varnothing$. Consequently, we obtain that $|N_{G\circ h_n}\left(x_j\right)\cap J|\ge 2$, which means that the vertex $x_j$ has at least two neighbors in the set J, a contradiction. □

The next theorem completely characterizes the generalized corona $G\circ h_n$ having an independent $[1,k]$-dset, $k\ge 1$, where graphs from the sequence $h_n$ are nonempty.

Theorem 3.

Let G be an arbitrary graph with $n\ge 1$ vertices and $h_n={\left(H_i\right)}_{i\in \mathcal{I}}$ be a sequence of n nonempty graphs. Moreover, let k be a positive integer and $S=\{x_i\in V\left(G\right):i\left(H_i\right)>k\}$. The graph $G\circ h_n$ has an independent $[1,k]$-dset if and only if following conditions hold:

(1)

S is an independent set of a graph G and

(2)

$|N_G\left(x_j\right)\cap S|+i\left(H_j\right)\le k$ for all vertices $x_j\in N_G\left(S\right)$.

Proof. 

Let us suppose that $G\circ h_n$ has an independent $[1,k]$-dset J. We will show that conditions (1) and (2) must hold.

First, assume that condition (1) does not hold. Since S is not an independent set, there exist vertices $x_j\in S$ and $x_k\in S$ such that $x_j{x}_k\in E(G\circ h_n)$. By Lemma 1 we obtain that $x_j\in J$ and $x_k\in J$. This means that J is not an independent set, a contradiction.

Assume now that condition (2) does not hold. Then, there exists a vertex $x_j\in N_G\left(S\right)$ such that $|N_G\left(x_j\right)\cap S|+i\left(H_j\right)>k$. Since $N_G\left(x_j\right)\cap S\subseteq S$ and by Lemma 1, we obtain that $N_G\left(x_j\right)\cap S\subseteq J$. Then, the vertex $x_j$ is adjacent to at least $|N_G\left(x_j\right)\cap S|$ vertices from the set J. Clearly, $x_j\notin J$. Since the vertices from the set $V\left(H_j\right)$ must be dominated by the independent set J, at least $i\left(H_j\right)$ vertices from the set $V\left(H_j\right)$ belong to J. Then, the vertex $x_j$ has at least $|N_G\left(x_j\right)\cap S|+i\left(H_j\right)\ge k+1$ neighbors in J, a contradiction.

We obtain that if the generalized corona $G\circ h_n$ has an independent $[1,k]-$ dset, $k\ge 1$, then it satisfies conditions (1) and (2).

Conversely, let us assume that conditions (1) and (2) hold. We will show that the graph $G\circ h_n$ has an independent $[1,k]$-dset $J=S\cup {\bigcup }_{\{i:i\left(H_i\right)\le k\}}{S}_i$, where $S_i$ is an arbitrary $i\left(H_i\right)-set$ of graph $H_i$. We will prove that J is a $[1,k]$-dset.

If $i\left(H_i\right)>k$, then $x_i\in S$. Consequently, $x_i\in J$ and $V\left(H_i\right)\cap J=\varnothing$. This means that every vertex from the set $V\left(H_i\right)$ is adjacent to exactly one vertex from the set J.

If $i\left(H_i\right)\le k$, then $S_i\subseteq J$, where $S_i$ is an arbitrary $i\left(H_i\right)$-set of $H_i$. Since graph $H_i$ is nonempty, $i\left(H_i\right)\ge 1$. Then, every vertex from the set $V\left(H_i\right)\backslash J$ has at most $i\left(H_i\right)\le k$ neighbors in the set J. Moreover, the vertex $x_i$ is adjacent to $|N_G\left(x_i\right)\cap S|+i\left(H_i\right)$ vertices from the set J. If $x_i\notin N_G\left(S\right)$, then it has exactly $i\left(H_i\right)$ neighbors in the set J. If $x_i\in N_G\left(S\right)$, then from condition (2) we know that the vertex $x_i$ has at most k neighbors in J.

We obtain that J is a $[1,k]-$ dset of $G\circ h_n$. Moreover, the definition of the generalized corona implies the independence of the set J. Hence, J is an independent $[1,k]$-dset of the graph $G\circ h_n$, which ends the proof. □

From Theorem 3, we obtain the following corollary.

Corollary 1.

Let G be an arbitrary graph with $n\ge 1$ vertices and $h_n={\left(H_i\right)}_{i\in \mathcal{I}}$ be a sequence of n nonempty graphs. If for every $i\in \mathcal{I}$ it holds that $i\left(H_i\right)\le k$ or $x_i$ is an isolated vertex of G, then graph $G\circ h_n$ has an independent $[1,k]$-dset.

Proof. 

Let us assume that $i\left(H_i\right)\le k$ or $x_i$ is an isolated vertex of G for every $i\in \mathcal{I}$. Let $S=\{x_i\in V\left(G\right):i\left(H_i\right)>k\}$. If $x_i\in S$, then $d_G\left(x_i\right)=0$. This means that S is an independent set and $N_G\left(S\right)=\varnothing$. By Theorem 3 we obtain that $G\circ h_n$ has an independent $[1,k]$-dset. □

Now we are ready to give equivalent conditions for the existence of an independent $[1,2]$-dset in generalized corona.

Corollary 2.

Let G be an arbitrary graph with $n\ge 1$ vertices and $h_n={\left(H_i\right)}_{i\in \mathcal{I}}$ be a sequence of n nonempty graphs. Let $S=\{x_i\in V\left(G\right):i\left(H_i\right)>2\}$. The graph $G\circ h_n$ has an independent $[1,2]$-dset if and only if the following conditions hold:

(1)

S is an independent $[0,1]$-dset of a graph G and

(2)

$i\left(H_j\right)=1$ for all $x_j\in N_G\left(S\right)$.

Proof. 

We will show that conditions (1) and (2) from Theorem 3 are equivalent to conditions (1) and (2) from Corollary 2. If conditions (1) and (2) from Corollary 2 hold, the result is obvious. Conversely, by Theorem 3 we know that $|N_G\left(x_j\right)\cap S|+i\left(H_j\right)\le 2$ for all $x_j\in N_G\left(S\right)$. The graph $H_j$ is nonempty, so $i\left(H_j\right)\ge 1$. Hence, $1\le i\left(H_j\right)\le 2$. Since $|N_G\left(x_j\right)\cap S|\ge 1$, we obtain $i\left(H_j\right)=1$ and $|N_G\left(x_j\right)\cap S|=1$. Then, $|N_G\left(x_k\right)\cap S|\le 1$ for every $x_k\in V\left(G\right)\backslash S$. Since S is an independent set, it is an independent $[0,1]$-dset. □

It is worth emphasizing that if $H_i\cong H$ for every $i\in \mathcal{I}$, then we obtain Theorem 1, which is the result from [41].

Now we give necessary and sufficient conditions for the existence an independent $[1,k]$-dset in $G\circ h_n$, where graphs from the sequence $h_n$ are arbitrary.

Theorem 4.

Let G be an arbitrary graph with $n\ge 1$ vertices and $h_n={\left(H_i\right)}_{i\in \mathcal{I}}$ be a sequence of n arbitrary graphs. Moreover, let k be a positive integer and $S=\{x_i\in V\left(G\right):i\left(H_i\right)>k\}$. The graph $G\circ h_n$ has an independent $[1,k]$-dset if and only if the following conditions hold:

(1)

S is an independent set of a graph G,

(2)

the subgraph induced by the set ${\bigcup }_{\{i:x_i\notin N_G\left[S\right]\}}(V\left(H_i\right)\cup \left\{x_i\right\})$ has an independent $[1,k]$-dset $J^{*}$ and

(3)

$|N_G\left(x_i\right)\cap S|+|N_G\left(x_i\right)\cap J^{*}|+i\left(H_i\right)\le k$ for all $x_i\in N_G\left(S\right)$.

Proof. 

Let us assume that a graph $G\circ h_n$ has an independent $[1,k]$-dset J. We will show that conditions (1)–(3) must hold. The independence of the set S follows from Lemma 1. Otherwise, the set J is not independent. Hence, condition (1) holds. We will show that the subgraph induced by the set ${\bigcup }_{\{i:x_i\notin N_G\left[S\right]\}}(V\left(H_i\right)\cup \left\{x_i\right\})$ has an independent $[1,k]$-dset. If $x_i\in S$, then $x_i\in J$. We obtain $N_G\left(S\right)\cap J=\varnothing$. Otherwise, J is not independent. Let $J^{*}=J\cap {\bigcup }_{\{i:x_i\notin N_G\left[S\right]\}}(V\left(H_i\right)\cup \left\{x_i\right\})$. Since J is an independent $[1,k]$-dset of the graph $G\circ h_n$ and $N_G\left(S\right)\cap J=\varnothing$, $J^{*}$ is an independent $[1,k]$-dset of the subgraph induced by the set ${\bigcup }_{\{i:x_i\notin N_G\left[S\right]\}}(V\left(H_i\right)\cup \left\{x_i\right\})$. Thus, condition (2) holds. Now let us suppose that there exists $x_i\in N_G\left(S\right)$ such that $|N_G\left(x_i\right)\cap S|+|N_G\left(x_i\right)\cap J^{*}|+i\left(H_i\right)>k$. Since $x_i\notin J$, at least $i\left(H_i\right)$ vertices from the set $V\left(H_i\right)$ belong to the set J. Otherwise, the vertices from the set $V\left(H_i\right)$ are not dominated by the set J. Then, the vertex $x_i$ has at least $|N_G\left(x_i\right)\cap S|+|N_G\left(x_i\right)\cap J^{*}|+i\left(H_i\right)$ neighbors in the set J, a contradiction. Hence, condition (3) holds.

Conversely, let us assume that conditions (1)–(3) hold. We will show that the graph $G\circ h_n$ has an independent $[1,k]$-dset $J=S\cup J^{*}\cup {\bigcup }_{\{i:x_i\in N_G\left(S\right)\}}{S}_i$, where $S_i$ is an arbitrary $i\left(H_i\right)$-set of $H_i$. The definition of the generalized corona implies the independence of the set J. It is sufficient to show that J is a $[1,k]$-dset. Firstly, if $i\left(H_i\right)>k$, then $x_i\in S$. By Lemma 1, we obtain $x_i\in J$. Then, $V\left(H_i\right)\cap J=\varnothing$. The vertices from the set $V\left(H_i\right)$ have exactly one neighbor in the set J, i.e., the vertex $x_i$. Secondly, if $i\left(H_i\right)\le k$ and $x_i\notin N_G\left[S\right]$. Since $N_G\left(S\right)\cap J=\varnothing$ and $J^{*}\subseteq J$, and from condition (2), we obtain that vertices from the set $V\left(H_i\right)\cup \left\{x_i\right\}$ belong to the set J or are $[1,k]$-dominated by the set J. Finally, if $i\left(H_i\right)\le k$ and $x_i\in N_G\left[S\right]$, then $S_i\subseteq J$. Hence, $x_i\notin J$. Then every vertex from the set $V\left(H_i\right)\backslash J$ is dominated by the set J and has at most $i\left(H_i\right)\le k$ neighbors in the set J. Moreover, the vertex $x_i$ is dominated, for example, by vertices from the set $S_i$. By condition (3), we obtain that $x_i$ is adjacent to at most k vertices from the set J. Hence, J is an independent $[1,k]$-dset of the graph $G\circ h_n$, which ends the proof. □

Now we study the existence of independent $[j,k]$-dsets, $2\le j\le k$, in the generalized corona $G\circ h_n$. Let us denote ${\mathcal{I}}_0=\{j\in \mathcal{I}:|V\left(H_j\right)|=0\}$. Clearly, ${\mathcal{I}}_0\subseteq \mathcal{I}$. Let us begin with proving an helpful lemma.

Lemma 2.

Let J be an independent $[j,k]$-dset, $2\le j\le k$, in the generalized corona $G\circ h_n$. If $i\in \mathcal{I}\backslash {\mathcal{I}}_0$, then $x_i\notin J$.

Proof. 

Let J be an independent $[j,k]$-dset. By contradiction let us assume that there exists $i\in \mathcal{I}\backslash {\mathcal{I}}_0$ such that $x_i\in J$. Since J is independent, $V\left(H_i\right)\cap J=\varnothing$. This means that vertices from the set $V\left(H_i\right)$ have exactly one neighbor in the set J, i.e., the vertex $x_i$, a contradiction. □

Now we give the necessary and sufficient conditions for the existence independent $[j,k]$-dsets, $2\le j\le k$, in the generalized corona of graphs in the case where $H_i\ne \varnothing$, for all $i\in \{1,\dots ,n\}$.

Theorem 5.

Let G be an arbitrary graph with $n\ge 1$ vertices and $h_n={\left(H_i\right)}_{i\in \mathcal{I}}$ be a sequence of n nonempty graphs. Let $2\le j\le k$. The graph $G\circ h_n$ has an independent $[j,k]$-dset if and only if all graphs $H_i$, $i\in \mathcal{I}$ have an independent $[j,k]$-dset $J_i$ such that $j\le |J_i|\le k$.

Proof. 

Let us assume that all graphs $H_i$, $i\in \mathcal{I}$, have an independent $[j,k]$-dset $J_i$ such that $j\le |J_i|\le k$. We will show that the set $J={\bigcup }_{i=1}^n{J}_i$ is an independent $[j,k]$-dset in the graph $G\circ h_n$. From the definition of the generalized corona, we obtain that J is an independent set. Moreover, every vertex $x_i\in V\left(G\right)$ has exactly $|J_i|$ neighbors in the set J. Hence, J is an independent $[j,k]$-dset of $G\circ h_n$.

Now, conversely, let us suppose that the graph $G\circ h_n$ has an independent $[j,k]$-dset J. Note that ${\mathcal{I}}_0=\varnothing$. By Lemma 2, we obtain that $V\left(G\right)\cap J=\varnothing$. Then, all graphs $H_i$, $i\in \mathcal{I}$, must have an independent $[j,k]$-dset $J_i$. Since $x_i\notin J$, $i\in \mathcal{I}$, and $G\circ h_n$ has an independent $[j,k]$-dset, the vertex $x_i$ is adjacent to at least j and at most k vertices from the set J. Hence, $j\le |J_i|\le k$, which ends the proof. □

The next theorem characterizes generalized corona $G\circ h_n$ having an independent $[j,k]$-dset, $2\le j\le k$, where graphs from the sequence $h_n$ are arbitrary.

Theorem 6.

Let G be an arbitrary graph with $n\ge 1$ vertices and $h_n={\left(H_i\right)}_{i\in \mathcal{I}}$ be a sequence of n arbitrary graphs. Let $2\le j\le k$. The graph $G\circ h_n$ has an independent $[j,k]$-dset if and only if the following conditions hold:

(1)

every graph $H_i$, $i\in \mathcal{I}\backslash {\mathcal{I}}_0$, has an independent $[j,k]$-dset $J_i$,

(2)

the subgraph induced by the set $\{x_l;l\in {\mathcal{I}}_0\}$ has an independent $[j,k]$-dset $J^{*}$ and

(3)

$j\le |N_G\left(x_i\right)\cap J^{*}|+|J_i|\le k$ for all $i\in \mathcal{I}\backslash {\mathcal{I}}_0$.

Proof. 

Let us assume that the graph $G\circ h_n$ has an independent $[j,k]$-dset J. Firstly, by Lemma 2 we obtain that $x_i\notin J$ for every $i\in \mathcal{I}\backslash {\mathcal{I}}_0$. Then, the graph $H_i$ must have an independent $[j,k]$-dset $J_i$. Hence, the condition $\left(1\right)$ holds. Secondly, from Lemma 2 we know that if $x_i\in V\left(G\right)\cap J$, then $i\in {\mathcal{I}}_0$. Let $J^{*}=V\left(G\right)\cap J$. Since J is an independent $[j,k]$-dset in generalized corona $G\circ h_n$, $J^{*}$ is an independent $[j,k]$-dset in the subgraph induced by the set $\{x_l;l\in {\mathcal{I}}_0\}$. Thus, condition (2) holds. Finally, we will show that condition (3) must hold. Since $x_i\notin J$, the vertex $x_i$ is $[j,k]$-dominated by the set J, i.e., $j\le |N_{G\circ h_n}\left(x_i\right)\cap J|\le k$. Note that $N_{G\circ h_n}\left(x_i\right)=N_G\left(x_i\right)\cup V\left(H_i\right)$, $N_G\left(x_i\right)\cap J=N_G\left(x_i\right)\cap J^{*}$ and $V\left(H_i\right)\cap J=J_i$. Hence, $N_{G\circ h_n}\cap J=(N_G\left(x_i\right)\cap J^{*})\cup J_i$. Since the sets $N_G\left(x_i\right)\cap J^{*}$ and $J_i$ are disjoint, $|N_{G\circ h_n}\cap J|=|N_G\left(x_i\right)\cap J^{*}|+|J_i|$. We obtain that condition (3) hold.

Conversely, let us suppose that conditions (1)–(3) hold. We will show that the generalized corona $G\circ h_n$ has an independent $[j,k]$-dset $J=J^{*}\cup {\bigcup }_{i\in \mathcal{I}\backslash {\mathcal{I}}_0}{J}_i$, where $J_i$ is an independent $[j,k]$-dset of graph $H_i$ and $J^{*}$ is an independent $[j,k]$-dset in the subgraph induced by the set $\{x_l;l\in {\mathcal{I}}_0\}$. From the definition of $G\circ h_n$, we obtain that J is independent. It is sufficient to show that J is a $[j,k]$-dset. Let $y\in V(G\circ h_n)\backslash J$. If $y\in V\left(H_i\right)$, $i\in \mathcal{I}$, then the vertex y is $[j,k]$-dominated by the set $J_i$. If $y=x_i\in V\left(G\right)$, $i\in {\mathcal{I}}_0$, then $x_i$ is $[j,k]$-dominated by the set $J^{*}$. If $y=x_i\in V\left(G\right)$, $i\in \mathcal{I}\backslash {\mathcal{I}}_0$, then it is dominated by the vertices from the set $V\left(H_i\right)\cap J=J_i$ and $N_G\left(x_i\right)\cap J^{*}$. From condition (3), we obtain that the vertex $x_i$ is $[j,k]$-dominated. Hence, the set J is an independent $[j,k]$-dset of the graph $G\circ h_n$, which ends the proof. □

3. Discussion and Conclusions

In the paper, we discussed and solved the problems of the existence of independent $[j,k]$-dominating sets in the generalized corona of graphs. The presented results conclude the topic of independent $[j,k]$-dominating sets in this product. We showed that the asymmetry of generalized corona of graphs is not an obstacle in deriving complete characterizations of independent $[j,k]$-dominating sets in this product. The problems analyzed in this paper are still unsolved for other graphs and their products, and the results presented in this paper can give a motivation for studying this field. In particular, based on the methods we used, it is natural to consider independent $[j,k]$-dominating sets in other multi-argument graph products and different one-argument graph operations.