Independent double Roman domination in graphs

TL;DR

This paper introduces the concept of independent double Roman domination in graphs, explores its properties and bounds, and characterizes possible pairs of domination numbers in trees.

Contribution

It defines the independent double Roman domination number, establishes bounds relating it to other domination parameters, and characterizes realizable pairs in trees.

Findings

Established bounds on IDRDN in terms of graph order, degree, and edge cover

Connected IDRDN with independent domination and Roman domination numbers

Characterized all feasible pairs of IDN and IDRDN in non-trivial trees

Abstract

An independent double Roman dominating function (IDRDF) on a graph $G=(V,E)$ is a function $f:V(G)\rightarrow \{0,1,2,3\}$ having the property that if $f(v)=0$, then the vertex $v$ has at least two neighbors assigned $2$ under $f$ or one neighbor $w$ with assigned $3$ under $f$, and if $f(v)=1$, then there exists $w\in N(v)$ with $f(w)\geq2$ such that the positive weight vertices are independent. The weight of an IDRDF is the value $\sum_{u\in V}f(u)$. The independent double Roman domination number $i_{dR}(G)$ of a graph $G$ is the minimum weight of an IDRDF on G. We initiate the study of the independent double Roman domination and show its relationships to both independent domination number (IDN) and independent Roman $\{2\}$-domination number (IR2DN). We present several sharp bounds on the IDRDN of a graph $G$ in terms of the order of $G$, maximum degree and the minimum size of edge cover. Finally, we show that, any ordered pair $(a,b)$ is realizable as the IDN and IDRDN of some non-trivial tree if and only if $2a + 1 \le b \le 3a$.

Equations

$$i_{dR}(P_{n})=\left\{\begin{array}[]{lll}n&\mbox{if $n\equiv 0$ \emph{(}mod\ 3\emph{)},}\vspace{1.5mm}\\ n+1&\mbox{otherwise.}\end{array}\right.$$

$$i_{dR}(C_{n})=\left\{\begin{array}[]{lll}n&\mbox{if $n\equiv 0,2,3,4$ \emph{(}mod\ 6\emph{)},}\vspace{1.5mm}\\ n+1&\mbox{otherwise.}\end{array}\right.$$

$$i_{dR}(G)=\left\{\begin{array}[]{lll}3&\mbox{if\ $m_{1}=1$,}\vspace{1.5mm}\\ 2m_{1}&\mbox{otherwise.}\end{array}\right.$$

$$\frac{3}{2}{i}_{\{R2\}}(G)\le i_{dR}(G)\le 2i_{\{R2\}}(G)$$

$$i_{dR}(G)\le w(g)=3|V_{\{1,2\}}^f|+2|V_{\{1\}}^f|+2|V_{\{2\}}^f|$$

$$=2(2|V_{\{1,2\}}^f|+|V_{\{1\}}^f|+|V_{\{2\}}^f|)-|V_{\{1,2\}}^f|$$

$$=2i_{r2}(G)-|V_{\{1,2\}}^f|\le 2i_{r2}(G).$$

$$i_{dR}(G)\ge i_{\{R2\}}(G)+i(G)$$

$$i(G)\le |V_2^f|+|V_3^f|.$$

$$g(v)=\{\begin{array}{ll}f(v)-1& ifv\in V_2^f\cup V_3^f\\ f(v)& ifv\in V_0^f.\end{array}$$

$$i_{\{R2\}}(G)\le \sum _{v\in V(G)}g(v)=|v_2^f|+2|V_3^f|.$$

$$i_{dR}(G)\le i_{\{R2\}}(G)+{\beta }^{\prime }(G)$$

$$g(v)=\{\begin{array}{ll}f(v)+1& \text{mbox}ifv\in V_1^f\cup V_2^f\\ 0& \text{mbox}ifv\in V_0^f.\end{array}$$

$$i_{dR}(G)\le \sum _{v\in V(G)}g(v)=2|V_1^f|+3|V_2^f|=i_{\{R2\}}(G)+|V_1^f|+|V_2^f|=i_{\{R2\}}(G)+n-|V_0^f|.$$

$$i_{dR}(G)\le i_{\{R2\}}(G)+n-{\alpha }^{\prime }(G)=i_{\{R2\}}(G)+{\beta }^{\prime }(G).$$

$$i_{dR}(G)=2|V_2|+3|V_3|\ge 2(|V_2|+|V_3|)\ge 2i(G).$$

$$i_{dR}(G)+(2\delta -1)\rho (G)\le 2n$$

$$|B|=n-|A\cup R|=n-|A|-|R|\le n-(\delta +1)|R|$$

$$f(v)=\{\begin{array}{ll}3& \text{mbox}ifv\in R\\ 0& \text{mbox}ifv\in A\\ 2& \text{mbox}ifv\in B.\end{array}$$

$$i_{dR}(G)\le w(f)=3|R|+2|B|=3|R|+2n-(2\delta +2)|R|=2n-(2\delta -1)\rho (G).$$

$$i_{dR}(G)\ge \frac{2n}{\Delta }+\frac{\Delta -2}{\Delta }i(G)$$

$$\Delta i_{dR}(G)=\Delta (2|V_2^f|+3|V_3^f|)$$

$$=\Delta (|V_2^f|+|V_3^f|)+2\Delta |V_3^f|+\Delta |V_2^f|$$

$$\ge \Delta (|V_2^f|+|V_3^f|)+2|V_0^f|$$

$$=(\Delta -2)(|V_2^f|+|V_3^f|)+2n.$$

$$f^{\prime }(z)=\{\begin{array}{ll}0& \text{mbox}ifz\in (L_u\cup A\cup V_0^f\setminus \{u\})\\ 1& otherwise,\end{array}$$

$$g(z)=\{\begin{array}{ll}0& \text{mbox}ifz\in (V_0\cup \{v\}\setminus \{u,w\})\\ 1& otherwise.\end{array}$$

$${\gamma }_{\{R2\}}(T)=\gamma (T)+1\text{if}\text{and}\text{only}\text{if}T\in \mathcal{T}\cup \mathfrak{F}.$$

$$i_{\{R2\}}(T)=i(T)+1\text{if}\text{and}\text{only}\text{if}T\in \mathcal{T}\cup \mathfrak{F}.$$

Full text

2010 Mathematical Subject Classification: 05C69

Keywords: Independent double Roman domination, independent Roman $\{2\}$-domination, independent domination, graphs.

1 Introduction and terminologies

Let $G=(V,E)$ be a simple graph with the vertex set $V=V(G)$ and the edge set $E=E(G)$. For any vertex $v\in V$, the open neighborhood of $v$ is the set $N(v)=\{u\in V|uv\in E\}$ and the closed neighborhood of $v$ is the set $N[v]=N(v)\cup\{v\}$. For a set $S\subseteq V$, the open neighborhood of $S$ is $N(S)=\bigcup_{v\in S}N(v)$ and the closed neighborhood of $S$ is $N[S]=N(S)\cup S$. We use [10] as a reference for terminology and notation which are not defined here.

Let $f$ be a function that assigns a subset of $\{1,2\}$ to each vertex of $G$, that is, $f:V(G)\rightarrow\mathcal{P}\{1,2\}$ where $\mathcal{P}\{1,2\}$ is the power set of $\{1,2\}$. If for each vertex $v\in V(G)$ such that $f(v)=\emptyset$, we have $\bigcup_{u\in N(v)}f(u)=\{1,2\}$, then $f$ is called a $2$-rainbow dominating function (2RDF) of $G$. The weight of a 2RDF $f$ is defined as $f(V(G))=\sum_{v\in V(G)}|f(v)|$. For simplicity, a 2RDF $f$ on a graph $G$ will be represented by the ordered partition $f=(V_{\emptyset}^{f},V_{\{1\}}^{f},V_{\{2\}}^{f},V_{\{1,2\}}^{f})$ of $V(G)$ induced by $f$, where $V_{\emptyset}^{f}=\{u\in V(G)|f(u)=\emptyset\}$, $V_{\{1\}}^{f}=\{u\in V(G)|f(u)=\{1\}\}$, $V_{\{2\}}^{f}=\{u\in V(G)|f(u)=\{2\}\}$ and $V_{\{1,2\}}^{f}=\{u\in V(G)|f(u)=\{1,2\}\}$. A function $f:V(G)\rightarrow\mathcal{P}\{1,2\}$ is called an independent $2$-rainbow dominating function (I2RDF) of $G$ if $f$ is a 2RDF and no two vertices in $V(G)\setminus V_{\emptyset}^{f}$ are adjacent. The independent $2$-rainbow domination number (I2RDN) $i_{r2}(G)$ is the minimum weight of an I2RDF of $G$ (see [6]). The $2$-rainbow domination was introduced by Bresar et al. in [3], and has been studied by several authors, for example, see [4] and [11].

A function $f:V(G)\rightarrow\{0,1,2\}$ is a Roman dominating function (RDF) on $G$ if every vertex $u\in V$ for which $f(u)=0$ is adjacent to at least one vertex $v$ for which $f(v)=2$. The weight of a RDF is the value $f(V(G))=\sum_{v\in V(G)}f(v)$. The Roman domination number $\gamma_{R}(G)$ is the minimum weight of a RDF on $G$. The Roman domination was introduced by Cockayne et al. in [7]. Since 2004, so many papers have been published on this topic, where several new variations were introduced: weak Roman domination, maximal Roman domination, mixed Roman domination, and recently, Roman $\{2\}$-domination ([5]) and double Roman domination ([2]). A Roman $\{2\}$-dominating function (R2DF) is a function $f:V(G)\rightarrow\{0,1,2\}$ with the property that for every vertex $v\in V$ with $f(v)=0$, $f(N(v))\geq 2$, that is, there is a vertex $u\in N(v)$, with $f(u)=2$, or there are two vertices $x,y\in N(v$) with $f(x)=f(y)=1$. The weight of a R2DF is the value $f(V(G))=\sum_{v\in V(G)}f(v)$, and the minimum weight of a R2DF is called the Roman $\{2\}$-domination number and denoted by $\gamma_{\{R2\}}(G)$.

A double Roman dominating function (DRDF) on a graph $G$ is a function $f:V(G)\rightarrow\{0,1,2,3\}$ having the property that if $f(v)=0$, then the vertex $v$ has at least two neighbors assigned $2$ under $f$ or one neighbor $w$ with $f(w)=3$, and if $f(v)=1$, then there exists $w\in N(v)$ such that $f(w)\geq 2$. The weight of a DRDF is the value $f(V(G))=\sum_{u\in V}f(u)$. The double Roman domination number (DRDN) $\gamma_{dR}(G)$ of a graph $G$ is the minimum weight of a DRDF on G. For simplicity, a DRDF $f$ on a graph $G$ may be represented by the ordered partition $f=(V_{0},V_{1},V_{2},V_{3})$ of $V(G)$ induced by $f$, where $V_{i}=\{u\in V(G)|f(u)=i\}$ for $0\leq i\leq 3$. A DRDF $f=(V_{0},V_{1},V_{2},V_{3})$ is called independent if $V_{1}\cup V_{2}\cup V_{3}$ is an independent set in $G$. The independent double Roman domination number (IDRDN) $i_{dR}(G)$ is the minimum weight of an independent double Roman dominating function (IDRDF) on $G$.

In this work, we mainly present lower and upper bounds on IDRDN of graphs, as for example by the well-known result of Gallai (concerning the maximum matching and the minimum edge cover) we prove that $i_{dR}(G)\leq i_{\{R2\}}(G)+\beta^{\prime}(G)$ in which $G$ is a graph of order $n$ with no isolated vertices and $\beta^{\prime}(G)$ is the maximum size of an edge cover of $G$. We also prove that $2i(T)+1\leq i_{dR}(T)\leq 3i(T)$ for all trees $T$ of order $n\geq 2$ and show that all values between the lower and upper bounds are realizable.

2 Preliminary results

In this section, we obtain some basic results and give the exact formulas for the IDRDNs for some well-known graphs. We first show $i_{dR}$ is well-defined for all graphs.

Proposition 1**.**

Every graph $G$ has an IDRDF.

Proof.

Let $S$ be a maximal independent set of $G$. Then, every vertex in $V-S$ has at least one neighbor in $S$. Now the function $f:V(G)\rightarrow\{0,1,2,3\}$ which assigns $3$ to the vertices in $S$ and [math] to the other ones is an IDRDF of $G$. ∎

In fact, Proposition 1 guarantees that the IDRDF and therefore the IDRDN $i_{dR}(G)$ exists for all graphs $G$.

Since in any IDRDF $f=(V_{0},V_{1},V_{2},V_{3})$ the set $V_{1}\cup V_{2}\cup V_{3}$ is an independent set in $G$, so by the definition, $V_{1}=\emptyset$ for any $i_{dR}(G)$-function. It turns out to be useful in dealing with some results in this paper.

Observation 2**.**

In any IDRDF $f:V(G)\rightarrow\{0,1,2,3\}$, $f(v)\neq 1$ for all $v\in V(G)$.

The DRDN of $P_{n}$ and $C_{n}$ were given in [1]. The IDRDNs of $P_{n}$ and $C_{n}$ can be defined similar to those given in [1] as follows.

Proposition 3**.**

For $n\geq 1$,

[TABLE]

For $n\geq 3$,

[TABLE]

Proof.

Consider the path $v_{1}\cdots v_{n}$. It is easy to see that the function $f:V(G)\rightarrow\{0,2,3\}$ defined by $f(v_{3i+2})=3$ and $f(v_{j})=0$ for other vertices if $n=0$ (mod $3$), and $f(v_{3i+2})=f(v_{n})=3$ and $f(v_{j})=0$ for the other vertices if $n=1,2$ (mod $3$) is an IDRDF of $P_{n}$ with the weight $\gamma_{dR}(P_{n})$. Since $\gamma_{dR}(P_{n})\leq i_{dR}(P_{n})$, it follows that $i_{dR}(P_{n})=\gamma_{dR}(P_{n})$.

For the cycle $C_{n}$, if we assign $2$ to the vertices with the even index and [math] to others when $n$ is even, if $n\equiv 3$ (mod $6$), then we assign $3$ to the vertices with the index $3i$ and [math] to the others, if $n\equiv 1$ (mod $6$) we assign $3$ to the vertices $v_{3i}$, $2$ to $v_{1}$ and [math] to others, and finally if $n\equiv 5$ (mod $6$), then we assign $3$ to the vertices $v_{3i}$ and $v_{1}$, and [math] to others. We also have $i_{dR}(C_{n})=\gamma_{dR}(C_{n})$. ∎

In what follows the IDRDNs of the complete graphs and complete $r(\geq 2)$-partite graphs are given.

Observation 4**.**

(i)* Let $G=K_{m_{1},\cdots,m_{r}}$ be a complete $r(\geq 2)$-partite with size $m_{1}\leq\cdots\leq m_{r}$. Then,*

[TABLE]

(ii)* $i_{dR}(K_{n})=3.$*

(iii)* $i_{dR}(G)=3$ if and only if $\Delta(G)=n-1$.*

3 Independent double Roman and Independent Roman $\{2\}$-domination

In this section, we establish some relationships between the IDRDN and IR2DN in graphs.

Proposition 5**.**

For any graph $G$,

[TABLE]

and these bounds are sharp.

Proof.

Let $f=(V_{0},V_{1},V_{2})$ be an $i_{\{R2\}}(G)$-function with $i_{\{R2\}}(G)=|V_{1}|+2|V_{2}|$. Then, $g=(V^{\prime}_{0}=V_{0},V^{\prime}_{2}=V_{1},V^{\prime}_{3}=V_{2})$ is an IDRDF of $G$. Therefore $i_{dR}(G)\leq 2|V_{1}|+3|V_{2}|\leq 2(|V_{1}|+2|V_{2}|)=2i_{\{R2\}}(G)$.

The graph $\overline{K_{n}}$ and a graph of order at least $4$ with two independent vertices $u$ and $v$ such that all the other vertices are adjacent to both $u$ and $v$ are graphs that achieve the upper bound.

In order to prove the lower bound we let $f=(V_{0}^{f},V_{2}^{f},V_{3}^{f})$ be an $i_{dR}(G)$-function. If $g=(V^{\prime}_{0}=V_{0}^{f},V^{\prime}_{1}=V_{2}^{f},V^{\prime}_{2}=V_{3}^{f})$, then $g$ is an IR2DF of $G$ with $w(g)=|V_{2}^{f}|+2|V_{3}^{f}|\leq\frac{2}{3}(2|V_{2}^{f}|+3|V_{3}^{f}|)=\frac{2}{3}i_{dR}(G)$.

Using Part (iii) of Observation 4, we have $\frac{3}{2}i_{\{R2\}}(G)=i_{dR}(G)=3$ for all graphs $G$ with $\Delta(G)=n-1$. So, the lower bound is sharp. ∎

As an immediate result we have,

Corollary 6**.**

For every graph $G$, $i_{\{R2\}}(G)<i_{dR}(G)$.

Proposition 7**.**

For any graphs $G$, $i_{dR}(G)\leq 2i_{r2}(G)$. This bound is sharp.

Proof.

Let $f=(V_{\emptyset}^{f},V_{\{1\}}^{f},V_{\{2\}}^{f},V_{\{1,2\}}^{f})$ be an $i_{r2}(G)$-function. Define the function $g$ on $G$ by $g(x)=3$ if $x\in V_{\{1,2\}}^{f}$, $g(x)=2$ if $x\in V_{\{1\}}^{f}\cup V_{\{2\}}^{f}$, and $g(x)=0$ if $x\in V_{\emptyset}^{f}$. Clearly $g$ is an IDRDF of $G$ with the weight $i_{r2}(G)$, and therefore

[TABLE]

[TABLE]

[TABLE]

The upper bound holds with the equality for the complete $r(\geq 2)$-partite graphs $G=K_{m_{1},\cdots,m_{r}}$ with $m_{1}\leq\cdots\leq m_{r}$, where $(m_{1}\geq 2)$. ∎

Theorem 8**.**

Let $G$ be a connected graph. Then,

[TABLE]

and this bound is sharp.

Proof.

The bound clearly holds for $K_{1}$ and $K_{2}$. So, we assume that $G$ is of order $n\geq 3$. In view of Observation 2 we let $f=(V_{0}^{f},V_{2}^{f},V_{3}^{f})$ be an $i_{dR}(G)$-function. We observe that $V_{2}^{f}\cup V_{3}^{f}$ is an independent dominating set in $G$ and therefore,

[TABLE]

We define $g:V(G)\rightarrow\{0,1,2\}$ by

[TABLE]

Therefore, $g$ is an IR2DF. Moreover,

[TABLE]

Together inequalities (1) and (2) imply that $i(G)+i_{dR}(G)\leq 2|V_{2}^{f}|+3|V_{3}^{f}|=i_{dR}(G)$.

To see the bound is sharp it suffices to consider the complete graph $K_{n}$ or any graph $G$ with $\Delta(G)=|V(G)|-1$ or the $r$-partite graph $G=K_{m_{1},\cdots,m_{r}}$ with $m_{1}\leq\cdots\leq m_{r}$, where $m_{1}\geq 2$. ∎

We recall that a matching $M$ of graph $G$ is a subset of the set of edges $E(G)$, such that no vertex in $V(G)$ is incident to more than one edge in $M$ in the other words one can say that no two edges in $M$ have a common vertex. A matching $M$ is said to be maximum if, $|M|\geq|M^{\prime}|$, for any other matching $M^{\prime}$ of $G$. We also recall that an edge cover $Q$ of a graph $G$ is a set of edges such that each vertex in $G$ is incident to at least one edge in $Q$. By $\alpha^{\prime}(G)$ and $\beta^{\prime}(G)$ we mean the maximum cardinality of a matching and the minimum cardinality of an edge cover in $G$, respectively.

We make use of the following classic result due to Gallai.

Lemma 9**.**

([10])* If $G$ is a graph of order $n$ and with no isolated vertices, then $\alpha^{\prime}(G)+\beta^{\prime}(G)=n$.*

Theorem 10**.**

Let $G$ be a graph with no isolated vertices. Then,

[TABLE]

and this bound is sharp.

Proof.

Let $f:V(G)\rightarrow\{0,1,2\}$ be an $i_{\{R2\}}(G)$-function. We define $g:V(G)\rightarrow\{0,2,3\}$ by

[TABLE]

This equation shows that $g$ is an IDRDF of $G$. Therefore,

[TABLE]

It is easy to see that at least one vertex incident to each edge of a maximum matching belongs to $V_{0}^{f}$. Therefore, $\alpha^{\prime}(G)\leq|V_{0}^{f}|$. Now Lemma 9 and the inequality (3) imply that

[TABLE]

For sharpness consider the complete bipartite graph $K_{p,p}$ in which $p\geq 2$. Then, $i_{dR}(K_{p,p})=2p=p+p=i_{\{R2\}}(K_{p,p})+\beta^{\prime}(K_{p,p})$. ∎

4 Independent double Roman and independent domination

We first give some lower and upper bounds on the IDRDN in terms of the independent domination number.

Proposition 11**.**

For any graph $G$, $2i(G)\leq i_{dR}(G)\leq 3i(G)$. These bounds are sharp.

Proof.

For the lower bound, in view of Observation 2 we let $f=(V_{0},V_{2},V_{3})$ be an $i_{dR}(G)$-function. Let $S$ be a minimum independent dominating set in $G$. Note that $(\emptyset,\emptyset,S)$ is an IDRDF. This yields the upper bound $i_{dR}(G)\leq 3i(G)$.

Furthermore, $V_{2}\cup V_{3}$ is an independent dominating set in $G$. Thus, $i(G)\leq|V_{2}|+|V_{3}|$. Taking into account this, we obtain the lower bound as follows.

[TABLE]

For the upper bound, let $\mathcal{F}$ be the family of graphs $G$ with $\Delta(G)=|V(G)|-1$. Then $i(G)=1$ and $i_{dR}(G)=3$. For the lower bound, let $\mathcal{H}$ be a family of graphs $G$ such that $G$ has a minimum maximal independent set with at least $2$ vertices like $U$ such that every vertices of $G-U$ is adjacent to $2$ vertices of $U$. Then $i_{dR}(G)=2i(G)$. For example, let $G=K_{m_{1},\cdots,m_{r}}$ be a complete $r(\geq 2)$-partite graph with $m_{1}\leq\cdots\leq m_{r}$ where $m_{1}\geq 2$. ∎

Recall that a set $R\subseteq V(G)$ is a packing set of $G$ if $N[x]\cap N[y]=\emptyset$ holds for any two distinct vertices $x,y\in R$. The packing number $\rho(G)$ is the maximum cardinality of a packing set in $G$. Let $\delta$ denote the minimum degree of the graph $G$. A classical result shows that: for any graph $G$, $\rho(G)\leq\gamma(G)\leq i(G)$.

Proposition 12**.**

If $G$ is a connected graph of order $n$, then

[TABLE]

and this bound is sharp.

Proof.

Let $R$ be a maximum packing set of $G$ and $A=N(R)$. Let $B=V(G)-(A\cup R)$. Each vertex in $A$ has exactly one neighbor in $R$ and each vertex in $R$ has at least $\delta$ neighbors in $A$. Therefore, $\delta|R|\leq|[R,V-R]|=|A|$. Therefore,

[TABLE]

Now we define $f:V(G)\rightarrow\{0,1,2,3\}$ by,

[TABLE]

It is easy to see that $f$ is an IDRDF of $G$. Therefore,

[TABLE]

This bound is sharp for the complete graph $K_{n}$, for $n\geq 2$. ∎

Proposition 13**.**

For any graph $G$ of order $n$ with maximum degree $\Delta$,

[TABLE]

and this bound is sharp.

Proof.

Let $f=(V_{0}^{f},V_{2}^{f},V_{3}^{f})$ be an $i_{dR}(G)$-function of $G$. Using Observation 2 we may assume that $V_{1}^{f}=\emptyset$. Let $S=V_{0}^{f}\bigcap N(V_{3})$ and $T=V_{0}^{f}\bigcap N(V_{2})$. Since each vertex in $V_{3}^{f}$ dominates at most $\Delta$ vertices of $S$, we have $|S|\leq\Delta|V_{3}^{f}|$. Since each vertex in $V_{2}^{f}$ dominates at most $\Delta$ vertices of $T$ and since each vertex in $T$ has at least two neighbors in $T$, we have $2|T|\leq|E(V_{2},T)|\leq\Delta|V_{2}^{f}|$ yielding $|T|\leq\frac{\Delta}{2}|V_{2}^{f}|$. Hence, $|V_{0}^{f}|=|S|+|T|\leq\Delta|V_{3}^{f}|+\frac{\Delta}{2}|V_{2}^{f}|$. Now we have

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Since $V_{2}^{f}\cup V_{3}^{f}$ is an independent dominating set of $G$, it follows that $\Delta i_{dR}(G)\geq(\Delta-2)i(G)+2n$.

The bound is sharp for the cycles $C_{n}$ where $n\equiv 0,2,3,4\ (mod\ 6)$ and for the paths $P_{n}$ where $n\equiv 0\ (mod\ 3)$. ∎

5 Trees

We make use the following result to show that the IDRDNs of trees are bounded from below and above just in terms of the independent domination number. The result may be important in its own right.

Theorem 14**.**

For any tree $T$ of order $n\geq 2$, $i_{\{R2\}}(T)\geq i(T)+1$.

Proof.

If $T$ is a star, then $i_{\{R2\}}(T)=2=i(T)+1$. Let $T$ be a double star $T_{r,s}$ in which $1\leq r\leq s$. If $r=1$, then $i_{\{R2\}}(T)=3=2+1=i(T)+1$. If $r\geq 2$, then $i_{\{R2\}}(T)=2+r=i(T)+1$. So, we assume from now on that $diam(T)\geq 4$. Let $P$ be a diametral $r,s$-path of $T$. We root the tree $T$ at $r$. Let $f$ be an $i_{\{R2\}}(T)$-function of $T$. We now deal with two cases depending on $f$.

Case 1. Suppose that there exists a vertex $x$ for which $f(x)=2$. It is easy to observe that $S=\{v\in V(T)|f(v)\neq 0\}$ is an independent dominating set in $T$. Therefore, $i(T)\leq|S|\leq w(f)-1=i_{\{R2\}}(T)-1$.

Case 2. Suppose that $f(x)=0$ or $1$ for all $x\in V(T)$. Since $f$ is an IR2DF, it follows that $f$ assigns $1$ to all leaves and [math] to all support vertices. Let $u$ be a support vertex and $L_{u}$ be the set of all leaves adjacent to $u$. If $x\in N(u)-L_{u}$ has the weight $f(x)=1$, then every vertex $v\in N(x)-\{u\}$ has a neighbor $w\neq u$ with $f(w)=1$ and $f(v)=0$ by the properties of the IR2DF $f$. Let $A=\{x\in N(u)-L_{u}:f(x)=1\}$. Since $f(u)=0$, $|L_{u}\cup A|\geq 2$. If now we define $f^{\prime}:V(G)\rightarrow\{0,1\}$ by

[TABLE]

it follows that $S^{\prime}=\{v\in V(T):f^{\prime}(v)\neq 0\}$ is an independent dominating set of $T$. Thus, $i(T)\leq|S^{\prime}|\leq w(f^{\prime})\leq w(f)-1=i_{\{R2\}}(T)-1$. ∎

In what follows, for the sake of completeness, we characterize the familly of all trees for which the lower bound in Theorem 14 holds with equality. To this aim, we begin with the following lemma.

Lemma 15**.**

Let $T$ be a tree and $f=(V_{0},V_{1},V_{2})$ be a $\gamma_{\{R2\}}(T)$-function. If $w(f)=\gamma_{\{R2\}}(T)=\gamma(T)+1$, then $V_{1}\cup V_{2}$ is an independent set.

Proof.

Since $f=(V_{0},V_{1},V_{2})$ be a $\gamma_{\{R2\}}(T)$-function, it follows that $|V_{1}|+2|V_{2}|=\gamma_{\{R2\}}(T)=\gamma(T)+1\leq|V_{1}|+|V_{2}|+1$. Therefore, $|V_{2}|\leq 1$. Let two vertices $v$ and $u$ in $V_{1}\cup V_{2}$ be adjacent. Since $T$ is a tree, no vertex in $V_{0}$ is adjacent to both $u$ and $v$. Let $u\in V_{1}$ and $v\in V_{2}$. Then each vertex $w$ in $V_{0}\cap N(u)$ has another neighbor in $V_{1}$. Therefore, $V_{1}\setminus\{u\}\cup\{v\}$ is a dominating set in $T$ of cardinality $|V_{1}|<|V_{1}|+2|V_{2}|-1=\gamma(T)$, a contradiction. Let $\{u,v\}\subseteq V_{1}$. If $z\in N(u)\cap V_{1}$ and $z\neq v$, then $V_{1}\setminus\{z,v\}\cup V_{2}$ is a dominating set in $T$ of cardinality $|V_{1}|<|V_{1}|+2|V_{2}|-1=\gamma(T)$, a contradiction. Similarly, for $z\in N(v)\cap V_{1}$ and $z\neq u$ we achieve the same contradiction. If every vertex $z\in N(u)\cup N(v)\{u,v\}$ is in $V_{0}$, then there are $z_{1}\in N(u)\cap V_{0}$ and $z_{2}\in N(v)\cap V_{0}$ such that $z_{1}$ has a neighbor $w_{1}$ other than $u$ and $z_{2}$ has a neighbor $w_{2}$ other than $v$ with positive weights. Now if $f(w_{1})=f(w_{2})=1$, then the set $V_{1}\setminus\{u,v,w_{1},w_{2}\}\cup\{z_{1},z_{2}\}\cup V_{2}$ is a dominating set of cardinality $\gamma_{\{R2\}}(T)-2$, a contradiction. If $f(w_{1})=1$ and $f(w_{2})=2$, then the set $V_{1}\setminus\{u,v,w_{1}\}\cup\{z_{1},z_{2}\}\cup V_{2}$ is a dominating set of size $\gamma_{\{R2\}}(T)-2$, which is again a contradiction. Thus $V_{1}\cup V_{2}$ is an independent set. ∎

From Lemma 15, we have the following.

Corollary 16**.**

Let $T$ be a tree and $f=(V_{0},V_{1},V_{2})$ be a $\gamma_{\{R2\}}(T)$-function. If $\gamma_{\{R2\}}(T)=\gamma(T)+1$, then $i_{\{R2\}}(T)=\gamma_{\{R2\}}(T)$ and $i(T)=\gamma(T)$.

Proof.

In Lemma 15, it has been shown that $V_{1}\cup V_{2}$ is independent. If $V_{2}\neq\emptyset$, then $V_{1}\cup V_{2}$ is an independent dominating set and so $i_{\{R2\}}(T)=\gamma_{\{R2\}}(T)$. If $V_{2}=\emptyset$, then there exists a vertex $v\in V_{0}$ for which $v$ has exactly two neighbors in $V_{1}$ like $u,w$. We define the function $g:V\rightarrow\{0,1\}$ by

[TABLE]

It follows that $S=\{v\in V(T):g(v)=1\}$ is a minimum dominating set of $T$ of cardinality $\gamma_{\{R2\}}(T)-1$. Since $S$ is an independent set, it also follows that $|S|=i(T)$. ∎

In [8], Henning and Klostermeyer characterized all trees $T$ of order $n\geq 2$ for which $\gamma_{\{R2\}}(T)=\gamma(T)+1$. To this aim, they introduced two families of trees.

For positive integers $r$ and $s$, let $F_{r,s}$ be the tree obtained from a double star $S_{r,s}$ by subdividing every edge exactly once. For example, $P_{7}=F_{1,1}$. The tree $F_{4,4}$ is shown in Figure 1. Let $\mathfrak{F}$ be the family of all such trees $F_{r,s}$, that is, $\mathfrak{F}=\{F_{r,s}:r,s\geq 1\}$.

Let $\mathcal{T}$ be the family of trees $T_{k,j}$ of order $k\geq 2$ where $k\geq 2j+1$ and $j\geq 0$, obtained from a star by subdividing $j$ edges exactly once. The tree $T_{12,4}$ is shown in Figure 2.

They proved that;

Theorem 17**.**

([8], Theorem 6)* Let $T$ be a non-trivial tree. Then,*

[TABLE]

Now we characterize all trees $T$ of order $n\geq 2$ for which $i_{\{R2\}}(T)=i(T)+1$. We deduce this result from Lemma 15, Corollary 16 and Theorem 17.

Theorem 18**.**

Let $T$ be a non-trivial tree. Then,

[TABLE]

Theorem 8, Proposition 11 and Theorem 14 yield the following for trees.

Corollary 19**.**

For any tree $T$ of order $n\geq 2$, $2i(T)+1\leq i_{dR}(T)\leq 3i(T)$.

Our final result in this section shows that every value in the range of Corollary 19 is realizable for trees, that is, all values between the lower and upper bounds of Corollary 19 are realizable. We first recall that the corona $G\circ K_{1}$ of a graph $G$ is formed from $G$ by adding a new vertex $w$ and edge $vw$ for each vertex $v\in V(G)$.

Theorem 20**.**

An ordered pair $(a,b)$ is realizable as the IDN and IDRDN of some non-trivial tree if and only if $2a+1\leq b\leq 3a$.

Proof.

Let $T$ be a tree with $i(T)=a$ and $i_{dR}(T)=b$. By Corollary 19, $2a+1\leq b\leq 3a$. Next we show that each ordered pair is realizable. For $b=2a+1$, consider the corona of the star $K_{1,t}$, for $t\geq 1$. We assign the value $1$ to all support vertices other than the center and also assign value $1$ to the leaf neighbor of the center. It is straightforward to check that $i(K_{1,t}\circ K_{1})=t+1$. For the IDRDN we assign the value $2$ to all leaves other than the leaf neighbor of the center and the value $3$ to the center. It is easy to see that $i_{dR}(K_{1,t}\circ K_{1})=2t+3=2i(K_{1,t}\circ K_{1})+1$. Assume now that $b\geq 2a+2$. Let $T$ be the tree formed from a subdivided star $K^{*}_{1,a}$ by choosing $b-(2a+2)$ support vertices of $K^{*}_{1,a}$ and adding another vertex as a neighbor of leaf of each of them. Thus, $T$ has $b-2a-2$ vertices other than the center which are neither support vertices nor leaves. Again, it is straightforward to check that $i(T)=a$ (the set of support vertices form an $i(T)$-set). To see that $i_{dR}(T)=b$, note that each of the $b-2a-2$ new support vertices must be assigned a value $3$ under any $i_{dR}(T)$-function. Assigning a value $2$ to the leaf non-adjacent to the new support vertices and the center of $T$ and [math] to the other vertices. It is simple to check that this function is in fact of the minimum weight. Hence, we have $i_{dR}(T)=3(b-2a-2)+2(a-(b-(2a+2)))+2=b$, as desired. This completes the proof. ∎