On the Outer-Independent Double Roman Domination of Graphs

1 Introduction

We consider only simple connected graphs G with vertex set $V=V\left(G\right)$ and edge set $E=E\left(G\right)$, where $n=\left|V\right|$ is the order of G. The open neighborhood of a vertex $v\in V$ is the set $N\left(v\right)=\left\{u\right|uv\in E\}$, and the degree of v is ${\mathrm{deg}}_G\left(v\right)=\left|N\left(v\right)\right|$. A leaf is a vertex with degree one and its neighbor is called a stem. A strong stem is a stem adjacent to at least two leaves. The diameter of G, denoted by diam (G), is the maximum value among distances between all pairs of vertices of G.

A set $S\subset V$ is independent if no two vertices in S are adjacent. The independence number α (G) of a graph G is the maximum cardinality among the independent sets of vertices of G. A vertex cover of a graph G is a set of vertices such that each edge of the graph is incident to at least one vertex of the set. A minimum vertex cover is a vertex cover of smallest possible size. The vertex cover number α₀ (G) is the minimum cardinality of a vertex cover of G.

In 2016, Beeler et al., Ref. 1, introduced the concept of double Roman domination and defined a double Roman dominating function (DRDF) on a graph G to be a function $h:V\left(G\right)\to \left\{\mathrm{0,1,2,3}\right\}$ such that each vertex with label 0 is adjacent to a vertex labeled 3 or to at least two vertices labeled 2, and each vertex with label 1 is adjacent to a vertex labeled 2 or 3. The weight of a DRDF f is the value $h\left[V\left(G\right)\right]={\displaystyle {\sum }_{u\in V\left(G\right)}h\left(u\right)}$, and the double Roman domination number γ_dR (G) equals the minimum weight of a DRDF on G. Double Roman domination has been studied by several authors; see, for example, Refs. 2–14. For more details on Roman domination and its variations, we refer the reader to Refs. 15–18.

For double Roman domination, one can think of any vertex representing a location in the Roman Empire and any edge being a road between two locations. A location is said to be protected if at least one army is stationed in it or by sending to it two armies from neighboring location(s) having already more than two armies (according to the decree of Emperor Constantine the Great). A locality without an army is certainly vulnerable, and it will be even more vulnerable if one of its neighbors is without army too. Hence, the best situation for a location with no army is to be surrounded by locations where each with at least one army. This leads us to seek an DRDF $h=\left(V_0,V_1,V_2,V_3\right)$, where $V_0$ is an independent set; that is, h is an OIDRDF.

Regarding this, Abdollahzadeh Ahangar et al., Ref. 19, introduced a new variation of double Roman domination called outer-independent double Roman domination. An outer-independent double Roman dominating function (OIDRD-function) of a graph G is a DRDF h such that the set of vertices assigned a 0 under h is independent. The outer-independent double Roman domination number (OIDRD-number for short) γ_oidR (G) is the minimum weight of an OIDRD-function on G. Clearly, γ_dR (G) ≤ γ_oidR (G) holds for every graph G. Recently, Mojdeh et al., Ref. 20, proved that the decision problem associated with γ_oidR (G) is NP-complete even when restricted to planar graphs with maximum degree at most four. They also characterized the families of all connected graphs with small outer-independent double Roman domination numbers.

In the following, we denote the set $\left\{\mathrm{0,1,2,3}\right\}$ by [3].

The authors of Ref. 19 provided an upper bound for the OIDRD-number of trees in terms of the order and number of stems.

Theorem 1.1. For each tree T on n ≥ 3 vertices,

${\gamma }_{oidR}\left(T\right)\le n+\frac{s\left(T\right)}{2},$

where $s\left(T\right)$ is the number of stems of T.Since the number of stems of any tree does not exceed half the order of the tree, the next result is immediate from Theorem 1.1.

Proposition 1.2. For every tree T on n ≥ 4 vertices, γ_oidR (T) ≤ 5n/4.Moreover, it should be noted that the problem of characterizing the trees T attaining equality in the upper bound of Proposition 1.2 was raised in Ref. 19. This problem will be solved in this article, and additional bounds on the OIDRD-number will be given. In particular, we prove that, for any connected graph G of order n with minimum degree at least two in which the set of vertices with degree at least three is independent, γ_oidR (G) ≤ 4n/3.

2 Trees T of Order n With γ_oidR (T) = 5n/4

With the aim of characterizing the trees T of order n ≥ 3 with γ_oidR (T) = 5n/4, let ${\mathrm{}}_t$ be the family of trees defined as follows. Let $H_i$ be a path P₄ whose vertices are labeled in order $v_1^i,v_2^i,v_3^i$, and $v_4^i$. For any integer $t\ge 1$, let ${\mathrm{}}_t$ be the family of trees T obtained from $H_1,\dots ,H_t$ by adding $t-1$ edges between the stems of $H_i$’s so that the resulting graph is a tree. Beeler et al., Ref. 1, proved that, for every tree $T\in {\mathrm{}}_t$, ${\gamma }_{dR}\left(T\right)=5\left|V\left(T\right)\right|/4$ which implies that ${\gamma }_{oidR}\left(T\right)=5\left|V\left(T\right)\right|/4$.

Lemma 2.1. Let$T\in {\mathrm{}}_t$for some integer t ≥ 1. Then, there is a γ_oidR (T) function f such that, for every leaf v of T, $f\left(v\right)\in \left\{\mathrm{1,2}\right\}$.

Proof. Let T be a tree of ${\mathrm{}}_t$ for some integer t ≥ 1. Then, ${\gamma }_{oidR}\left(T\right)=5\left|V\left(T\right)\right|/4$. Since T is a bipartite graph, let X and Y be the partite sets of T. Let $X^{\prime }$ be the set of stems of T belonging to X, and likewise let $Y^{\prime }$ be defined similarly. Clearly, $\left|X\right|=\left|Y\right|$, $\left|X^{\prime }\right|=\left|Y^{\prime }\right|$, and every leaf of T is either in $X-X^{\prime }$ or $Y-Y^{\prime }$. Define the function f on $V\left(T\right)$ by assigning a 2 to all vertices of Y, a 1 to all vertices in $X-X^{\prime }$, and a 0 to vertices in X. Then, f is an OIDRD-function of T of weight $5\left|V\left(T\right)\right|/4$, and thus f is a γ_oidR (T) function with desired property. □

Theorem 2.2. Let T be a tree on n ≥ 4 vertices. Then, γ_oidR (T) = 5n/4 if and only if$T\in {\mathrm{}}_t$for some integer$t\ge 1$.

Proof. We prove only the necessity. Let T be a tree of n ≥ 4 such that γ_oidR (T) = 5n/4. Clearly, $n=4t$ for some integer t ≥ 1. To prove that $T\in T_t$, we use an induction on t. If t = 1, then $T=P_4$, and clearly, $T\in {\mathrm{}}_1$. Let t ≥ 2 and assume that the result is true for any tree T with γ_oidR (T) = 5n/4, where $n=4t^{\prime }$ and $t^{\prime }<t$. Let T be a tree with γ_oidR (T) = 5n/4 and $n=4t$. We deduce from ${\gamma }_{oidR}\left(T\right)=5n/4\le n+s\left(T\right)/2\le 5n/4$ that $s\left(T\right)=n/2$. Therefore, T is the corona of some tree and so T has no strong stem. Moreover, $diam\left(T\right)\ge 4$ because t ≥ 2. Let $P=v_1{v}_2,\dots ,v_k$ be a diametral path in T and root T at $v_k$. Then, ${\mathrm{deg}}_T\left(v_2\right)=2$, and there is a unique leaf w adjacent to $v_3$. Denote by $T_x$ the subtree induced by a vertex x and its descendants in the rooted tree T. We claim that ${\mathrm{deg}}_T\left(v_3\right)=3$. Suppose, to the contrary, that ${\mathrm{deg}}_T\left(v_3\right)\ge 4$ and let $v_2=w_1,w_2,\dots ,w_s$ be the children of $v_3$ with depth one. Since ${\mathrm{deg}}_T\left(w_i\right)=2$ for each i, let ${w_i}^{\prime }$ be the leaf adjacent to $w_i$ for $i\in \left\{2,\dots ,s\right\}$, and consider the tree $T^{\prime }$ obtained from T by removing every $w_i$ and ${w_i}^{\prime }$ for $i\ge 2$. Observe that the subtree rooted at $v_3$ is a path $P_4$. Let f be a γ_oidR($T^{\prime }$)-function, where, without loss of generality, $f\left(v_3\right)=3$. Then, f can be extended to an OIDRD-function of T by assigning a 0 to $w_2,\dots ,w_s$ and a 2 to $w_2{ }^{\prime },\dots ,w_s{ }^{\prime }$. It follows from Proposition 1.2 that

${\gamma }_{oidR}\left(T\right)\le {\gamma }_{oidR}\left(T^{\prime }\right)+2\left(s-1\right)\le \frac{5\left(n-2s+2\right)}{4}+2s-2<\frac{5n}{4},$

which leads to a contradiction. Hence, ${\mathrm{deg}}_T\left(v_3\right)=3$, and thus $T_{v_3}=P_4$. Now, let $T^{\prime \prime }=T-T_{v_3}$. Clearly, any γ_oidR($T^{\prime \prime }$)-function f can be extended to an OIDRD-function of T by assigning a 3 to $v_3$, 2 to $v_1$, and a 0 to $v_2,w$. Hence,

$\frac{5n}{4}={\gamma }_{oidR}\left(T\right)\le {\gamma }_{oidR}\left(T^{\prime \prime }\right)+5\le \frac{5\left(n-4\right)}{4}+5=\frac{5n}{4}.$

Therefore, by the induction hypothesis on $T^{\prime \prime }$, we have $T^{\prime \prime }\in {\mathrm{}}_{n/4-1}$. By the construction of $T^{\prime \prime }$, we may assume that $v_4$ is a vertex of an induced path $P_4=v_1^1{v}_2^1{v}_3^1{v}_4^1$, where ${\mathrm{deg}}_{T^{\prime \prime }}\left(v_1^1\right)={\mathrm{deg}}_{T^{\prime \prime }}\left(v_4^1\right)=1$, ${\mathrm{deg}}_{T^{\prime \prime }}\left(v_3^1\right)=2$, and ${\mathrm{deg}}_{T^{\prime \prime }}\left(v_2^1\right)\ge 2$. Now, if $n/4-1=1$, then clearly $T\in {\mathrm{}}_2$, and we are done. Hence, we assume that $n/4-1\ge 2$. We claim that $v_4\in \left\{v_2^1,v_3^1\right\}$. Suppose, to the contrary, that $v_4\in \left\{v_1^1,v_4^1\right\}$. By Lemma 2.1, there is a ${\gamma }_{oidR}\left(T^{\prime \prime }\right)$-function f such that $f\left(v_4\right)\in \left\{\mathrm{1,2}\right\}$. If $f\left(v_4\right)=1$, then $f\left(v_5\right)=2$ and the function h defined on $V\left(T\right)$ by $h\left(v_3\right)=3,h\left(v_1\right)=2$, $h\left(w\right)=h\left(v_2\right)=h\left(v_4\right)=0$, and $h\left(x\right)=f\left(x\right)$ for $x\in V\left(T^{\prime \prime }\right)-\left\{v_4\right\}$ is an OIDRD-function of T of weight $\omega \left(h_1\right)+4=5\left(n-4\right)/4+4<5n/4$, a contradiction. If $f\left(v_4\right)=2$ and $f\left(v_5\right)\ge 1$, then as above we can get a contradiction. Hence, assume that $f\left(v_4\right)=2$ and $f\left(v_5\right)=0$. Then, $v_5$ has a neighbor with weight at least two and the function $h^{\prime }$ defined on $V\left(T\right)$ by $h^{\prime }\left(v_3\right)=3,h^{\prime }\left(v_1\right)=2$, $h^{\prime }\left(v_5\right)=1$, $h^{\prime }\left(v_4\right)=h^{\prime }\left(w\right)=h^{\prime }\left(v_2\right)=0$, and $h^{\prime }\left(x\right)=f\left(x\right)$ for $x\in V\left(T^{\prime \prime }\right)-\left\{v_4,v_5\right\}$ is an OIDRD-function of T of weight $\omega \left(h^{\prime }\right)+4=5\left(n-4\right)/4+4<5n/4$, a contradiction. Thus, $v_4\in \left\{v_2^1,v_3^1\right\}$, and thus $T\in {\mathrm{}}_t$ as desired. This completes the proof. □

3 Slightly Improved Bounds for Trees

In this section, we present some sharp bounds on the OIDRD-number. We start with some classes of trees where the upper bound in Proposition 1.2 will be slightly improved.

Proposition 3.1. Let T be a tree of order n ≥ 3, where$n\equiv \mathrm{0,1,2}\left(\mathrm{mod}4\right)$. If T contains a strong stem, then γ_oidR(T) ≤ 5n/4−1.

Proof. Let $s\in V\left(T\right)$ be a strong stem of T and let $L\left(s\right)=\left\{x_1,x_2,\dots ,x_t\right\}$ be the set of leaves adjacent to $s$. Consider the forest $T-s$ and assume that it contains $r\ge 0$ components each of order at least four $T_1,\dots ,T_r$. If $r\ge 1$, then $f_i$ be a γ_oidR(T_i)-function for each $i\in \left\{1,\dots r\right\}$. Clearly, by Proposition 1.2, ${\gamma }_{oidR}\left(T_i\right)\le 5\left|V\left(T_i\right)\right|/4$. Moreover, suppose that $T-s$ has $t_2$ components isomorphic to $K_2$ and $t_3$ components isomorphic to $P_3$. Clearly, all $f_i$’s together can be extended to an OIDRD-function to T by assigning a 3 to s and to all center vertices of the components of order three, a 2 to each leaf at distance two from s belonging to a component of order two, and a 0 to the remaining vertices in the components of order at most three. Observe that if $r=0$, then the total weight assigned to the vertices is at most p, and thus ${\gamma }_{oidR}\left(T\right)\le n<5n/4$. Hence, we can assume that $r\ge 1$. Now, using the fact that $n\equiv \mathrm{0,1,2}\left(\mathrm{mod}4\right)$, we obtain

$\begin{array}{c}{\gamma }_{oidR}\left(T\right)\le 3+2t_2+3t_3+{\displaystyle \sum _{i=1}^r\frac{5\left|V\left(T_i\right)\right|}{4}}\\ =3+2t_2+3t_3+\frac{5\left(n-3t_3-2t_2-1-t\right)}{4}\\ \le 5n/4-1,\end{array}$

as desired. □

A closer look at the proof of Proposition 3.1 shows that it can be used to obtain the next two results too.

Proposition 3.2. Let T be a tree of order n ≥ 3, where$n\equiv 3 \left(\mathrm{mod}4\right)$. If T contains one strong stem s such that$T-s$contains a component isomorphic to$P_2$or$P_3$, then γ_oidR (T) ≤ 5n/4−1.

Proposition 3.3. Let T be a tree of order$n\ge 3$, where$n\equiv 3\left(\mathrm{mod}4\right)$. If T contains a strong stem having at least three leaves, then γ_oidR (T) ≤ 5n/4−1.

Proposition 3.4. Let T be a tree of order n ≥ 3, where$n\equiv 3\left(\mathrm{mod}4\right)$. If T contains more than one strong stem, then γ_oidR (T) ≤ 5n/4−1.

Proof. Let $s_1$ and $s_2$ be two strong stems of T and let $L\left(s_i\right)=\left\{x_1^i,x_2^i,\dots ,x_{t_i}^i\right\}$ be the set of leaves adjacent to $s_i$ for $i\in \left\{\mathrm{1,2}\right\}$. Clearly, $diam\left(T\right)\ge 3$. If each component of $T-\left\{s_1,s_2\right\}$ is of order at most three, then assigning a 3 to $s_1,s_2$ and to the center vertex of each component of order three, a 2 to each leaf at distance two from $s_i$ that belongs to a component of order two in $T-\left\{s_1,s_2\right\}$, and a 0 to the remaining vertices provides an OIDRD-function of T of weight at most $p$. Therefore, ${\gamma }_{oidR}\left(T\right)\le n\le 5n/4-1$. Hence, we may assume that $T-s$ contains at least one component of order at least four. Let $T_1,\dots ,T_r(r\ge 1)$ be such components of $T-\left\{s_1,s_2\right\}$ of order at least four. Let $f_i$ be a γ_oidR (T_i) function for each i. In addition, let $T-\left\{s_1,s_2\right\}$ have $t_2$ components isomorphic to $K_2$ and $t_3$ components isomorphic to $P_3$. Then, all f_i’s together can be extended to an OIDRD-function of T by assigning a 3 to $s_1,s_2$ and to the center vertex of each component of order three, a 2 to each leaf at distance two from s that belongs to a component of order two in $T-\left\{s_1,s_2\right\}$, a 3 to the center of all components of order three, $f_i\left(x\right)$ for each $x\in V\left(T_i\right)$ and $i\in \left\{1,\dots ,r\right\}$, and a 0 to the remaining vertices in the components of order at most three. Using the fact that $n\equiv 3\left(\mathrm{mod}4\right)$, we obtain

$\begin{array}{c}{\gamma }_{oidR}\left(T\right)\le 6+2t_2+3t_3+{\displaystyle \sum _{i=1}^r\frac{5\left|V\left(T_i\right)\right|}{4}}\\ =6+2t_2+3t_3+\frac{5\left(n-3t_3-2t_2-2-t_1-t_2\right)}{4}\\ \le 5n/4-1,\end{array}$

as desired. □

4 Graphs With Minimum Degree Two

We begin by recalling the question, posed in Ref. 19, on whether the $5n/4$ upper bound on the OIDRD-number for trees remains valid for arbitrary graphs. In this section, we restrict our attention to graphs with minimum degree at least two such that the set of vertices of degree at least three is independent set, where we shall show that the OIDRD-number is bounded above by $4n/3$. We will use the following result established in Ref. 19.

Proposition 4.1. For n ≥ 3, ${\gamma }_{oidR}\left(C_n\right)=\{\begin{array}{ccc}n& \text{if}& n\equiv 0\left(\text{mod}2\right)\\ n+1& & \text{otherwise.}\end{array}$

Proposition 4.2. For n ≥ 3, the path$P_n$has an OIDRD-function f that assigns positive weight to the end-vertices of$P_n$and$\omega \left(f\right)=n+1\le 4n/3$.

Proof. Let $P_n=v_1{v}_2\dots v_n$ and define the function f on $V\left(P_n\right)$ as follows. If $n\equiv 1\left(\mathrm{mod}2\right)$, then $f\left(v_{2i+1}\right)=2$ for $0\le i\le n-1/2$ and $f\left(x\right)=0$ otherwise, and if $n\equiv 0\left(\mathrm{mod}2\right)$, then $f\left(v_n\right)=1$, $f\left(v_{2i+1}\right)=2$ for $0\le i\le n-2/2$ and $f\left(x\right)=0$ otherwise. Clearly, $\omega \left(f\right)\le 4n/3$ and f is an OIDRD-function of $P_n$ assigning positive weight to the end-vertices of $P_n$. □

For integers $r\ge 3$ and s ≥ 1, let $C_{r,s}$ be the graph obtained from a cycle $C_r=\left(u_1{u}_2\dots u_r\right)$ and a path $P_s=v_1{v}_2\dots v_s$ by adding the edge $u_1{v}_1$. Applying Propositions 4.1 and 4.2, we derive the next result.

Proposition 4.3. For integers$r\ge 3$and s ≥ 1, the graph$C_{r,s}$has an OIDRD-function f that assigns a positive weight to v_sand ω(f) ≤ 4(r+s)/3.

Proof. Assume first that $r+s\in \left\{\mathrm{4,5}\right\}$. If $r+s=4$, then assigning a 1 to $v_1$, a 2 to $u_1,u_2$, and a 0 to $u_3$ provides an OIDRD-function of $C_{\mathrm{3,1}}$ satisfying the conditions. If $r=4$ and $s=1$, then assigning a 1 to $v_1$, a 2 to $u_1,u_3$, and a 0 to $u_2$ and $u_4$ provides an OIDRD-function of $C_{\mathrm{4,1}}$ with the desired properties. If $r=3$ and $s=2$, then assigning a 2 to $u_1,u_3,v_2$ and a 0 to $u_2$ and $v_1$ provides as above an OIDRD-function of $C_{\mathrm{3,2}}$ with the desired properties. Hence, assume that $r+s\ge 6$, and let f be a γ_oidR (C_r)function such that $f\left(u_1\right)\ge 2$. If $s=1$, then the function g defined on $V\left(C_{r,s}\right)$ by $g\left(v_1\right)=1$ and $g\left(x\right)=f\left(x\right)$ otherwise is an OIDRD-function of $C_{r,s}$ with desired properties. If $s=2$, then the function g defined on $V\left(C_{r,s}\right)$ by $g\left(v_1\right)=0$, $g\left(v_2\right)=2$, and $g\left(x\right)=f\left(x\right)$ otherwise is an OIDRD-function of $C_{r,s}$ with the desired properties. Henceforth, we can assume that $s\ge 3$.

Let g be an OIDRD-function of the path $v_1{v}_2\dots v_s$ of weight $s+1$ assigning positive weights to $v_1,v_s$ (Proposition 4.2). Define h on $C_{r,s}$ by $h\left(u_i\right)=f\left(u_i\right)$ for each i and $h\left(v_j\right)=g\left(v_j\right)$ for each j. Clearly, h is an OIDRD-function of $C_{r,s}$ and we deduce from Proposition 4.1 that ${\gamma }_{oidR}\left(C_{r,s}\right)\le r+s+2\le 4\left(r+s\right)/3$.□

Let $\mathrm{ℱ}$ be the family of all simple graphs obtained from some connected multigraph H without loops with $\delta \left(H\right)\ge 3$ by subdividing each edge of H at least once and at most five times. Clearly, any graph in $\mathrm{ℱ}$ has order at least 5. The next result shows that every graph G in $\mathrm{ℱ}$ of order n satisfies ${\gamma }_{oidR}\left(G\right)\le 4n/3$.

Proposition 4.4. For any graph$G\in \mathrm{ℱ}$of order n, there exists an OIDRD-function f of G such that$\omega \left(f\right)\le 4n/3$and$f\left(x\right)\ge 2$for each vertex x of degree at least three.

Proof. Let $G\in \mathrm{ℱ}$ be a graph of order n. We use an induction on n. If n = 5, then $G=K_{\mathrm{2,3}}$ and the function f that assigns a 2 to the vertices of degree 3 and a 0 to the remaining vertices satisfies the conditions as desired. Let $n\ge 6$ and assume that the result holds for all graphs in $\mathrm{ℱ}$ of order $n^{\prime }$, where $5\le n^{\prime }<n$. Let $G\in \mathrm{ℱ}$ be a graph of order $n\ge 6$. Suppose that $A=\{x\in V\left(G\right)|{\mathrm{deg}}_G\left(x\right)\ge 3\}$ and let $B=V\left(G\right)-A$. In the sequel, we will call an induced path P of G an A-ear path if $V\left(P\right)\subset B$ and P is connected to A by either its unique vertex (when $\left|V\left(P\right)\right|=1$) or its two end-vertices (when $\left|V\left(P\right)\right|\ge 2$). For each $i\in \left\{\mathrm{1,2,3,4,5}\right\}$, let $Q_i$ be the set of all A-ear paths P of G of order i and let $\mathrm{}={{{\displaystyle \cup }}^{\text{}}}_{i=1}^5{Q}_i$. Clearly, $B={{{\displaystyle \cup }}^{\text{}}}_{P\in \mathrm{}}V\left(P\right)$. Moreover, for each A-ear path P, let $X_P=\{u\in A|u$ is adjacent to a vertex of $P\}$. Hence, $A={{{\displaystyle \cup }}^{\text{}}}_{P\in \mathrm{}}{X}_P$. Furthermore, since $G\in \mathrm{ℱ}$, we have $\left|X_P\right|=2$ for each $P\in \mathrm{}$, and therefore, $\left|A\right|\ge 2$.

First, let $Q_3{{\displaystyle \cup }}^{\text{}}{Q}_5\ne \varnothing$. Suppose $P=x_1\dots x_{2k+1}\in Q_{2k+1}(k\in \left\{\mathrm{1,2}\right\})$ and let $X_P=\left\{a_1,a_2\right\}$, where $a_1{x}_1,a_2{x}_{2k+1}\in E\left(G\right)$. Let $G^{\prime }$ be the graph obtained from G by first removing all vertices of the path P except $x_{k+1}$ and then adding edges $a_1{x}_{k+1}$ and $a_2{x}_{k+1}$. Clearly, $G^{\prime }\in \mathrm{ℱ}$ of order less than $n$. By the induction hypothesis on $G^{\prime }$, there exists an OIDRD-function $f=\left(V_0,V_1,V_2,V_3\right)$ of $G^{\prime }$ such that $a_1,a_2\in V_2{{\displaystyle \cup }}^{\text{}}{V}_3$ and $\omega \left(f\right)\le 4\left(n-2k\right)/3$. It follows that $f\left(x_{k+1}\right)=0$. Now, if k = 1, then the function g defined on $V\left(G\right)$ by $g\left(x_2\right)=2$, $g\left(x_1\right)=g\left(x_3\right)=0$, and $g\left(x\right)=f\left(x\right)$ otherwise is an OIDRD-function of G such that $g\left(x\right)\ge 2$ for each $x\in A$ and

$\omega \left(g\right)=\omega \left(f\right)+2\le \frac{4\left(n-2\right)}{3}+2<\frac{4n}{3}.$

If k = 2, then the function g defined on $V\left(G\right)$ by $g\left(x_2\right)=g\left(x_4\right)=2$, $g\left(x_1\right)=g\left(x_3\right)=g\left(x_5\right)=0$, and $g\left(x\right)=f\left(x\right)$ otherwise is an OIDRD-function of G such that $g\left(x\right)\ge 2$ for each $x\in A$ and

$\omega \left(g\right)=\omega \left(f\right)+4\le \frac{4\left(n-4\right)}{3}+4<\frac{4n}{3}.$

From now on, we can assume that $Q_3{{\displaystyle \cup }}^{\text{}}{Q}_5=\varnothing$.

Next, assume that $Q_4\ne \varnothing$ and let $P=x_1{x}_2{x}_3{x}_4\in Q_4$ with $X_P=\left\{u,a\right\}$ and $u{x}_1,a{x}_4\in E\left(G\right)$. Let $G^{\prime }$ be the graph obtained from G by deleting $x_1,x_2,x_3$ and adding the edge $u{x}_4$. Clearly, $G^{\prime }\in \mathrm{ℱ}$ and thus by the induction hypothesis, there exists an OIDRD-function f of $G^{\prime }$ such that $f\left(x\right)\ge 2$ for each vertex $x\in A$ and $\omega \left(f\right)\le 4\left(n-3\right)/3$. Then, the function g defined on $V\left(G\right)$ by $g\left(x_2\right)=g\left(x_3\right)=2,g\left(x_1\right)=0$, and $g\left(x\right)=f\left(x\right)$ otherwise is an OIDRD-function of G such that $g\left(x\right)\ge 2$ for each $x\in A$ and $\omega \left(g\right)=\omega \left(f\right)+4\le 4\left(n-3\right)/3+4=4n/3$.

Considering the above situations, we may assume that $\mathrm{}=Q_1{{\displaystyle \cup }}^{\text{}}{Q}_2$. Note that $n=\left|A\right|+m_1+2m_2$ and $m_1+m_2\ge 3$, where $m_i=\left|Q_i\right|$ for $i\in \left\{\mathrm{1,2}\right\}$.

Assume first that $\left|A\right|=2$ and let $A=\left\{u,v\right\}$. If $Q_2\ne \varnothing$, then let $Q_2=\left\{w_1^j{w}_2^j\right|1\le j\le m_2\}$, where $u{w}_1^j\in E\left(G\right)$ for each j. Moreover, if $Q_1\ne \varnothing$, then let $Q_1=\left\{z_1^l\right|1\le l\le m_1\}$. Define the function g on $V\left(G\right)$ by $g\left(u\right)=3,g\left(v\right)=2,g\left(w_2^j\right)=1$ for each j and $g\left(t\right)=0$ otherwise. Then, g is an OIDRD-function of G such that $g\left(x\right)\ge 2$ for each $x\in A$ and we have $\omega \left(g\right)\le 5+m_2\le 4\left(2+2m_2+m_1\right)/3=4n/3$.

Henceforth, we can assume that $\left|A\right|\ge 3$. We consider the following cases.

Case 1.$Q_2=\varnothing$.

Then, $\mathrm{}=Q_1$ and G is obtained from a loopless multigraph $G^{\prime }$ by subdividing each edge of $G^{\prime }$ once. Since each vertex of $G^{\prime }$ has degree at least three, we have $2\left|E\left(G^{\prime }\right)\right|\ge 3n\left(G^{\prime }\right)$ and so $n\left(G\right)=\left|V\left(G^{\prime }\right)\right|+\left|E\left(G^{\prime }\right)\right|\ge 5n\left(G^{\prime }\right)/2$. Define f on $V\left(G\right)$ by $f\left(x\right)=2$ for $x\in V\left(G^{\prime }\right)$ and $f\left(x\right)=0$ otherwise. Clearly, f is an OIDR-function of G such that $f\left(x\right)\ge 2$ for each $x\in A$ and $\omega \left(f\right)=2n\left(G^{\prime }\right)\le 4n/3$.

Case 2.$Q_2\ne \varnothing$.

Let $u\in V\left(G\right)$ be a vertex with the most neighbors in A-ear paths of $Q_2$. We consider the following subcases.

Subcase 2.1.u is adjacent to at least two A-ear paths of $Q_2$.

Let $P_1=x_1{x}_2,P_2=y_1{y}_2\in Q_2$ be two A-ear paths such that $u{x}_1,u{y}_1\in E\left(G\right)$. Assume that $\left\{a{x}_2,b{y}_2\right\}\subseteq E\left(G\right)$, where $a,b\in A-\left\{u\right\}$. Suppose that $a\ne b$, and let $G^{\prime }$ be the graph obtained from G by removing first $u,x_1,x_2$ and then adding the edge $y_{1}a$ and joining by an edge every vertex x in $N\left(u\right)-\left\{x_1,y_1\right\}$ to either a or b provided a or b is not adjacent to the end-vertex of the A-ear path containing x. Clearly, $G^{\prime }\in \mathrm{ℱ}$, and by the induction hypothesis, there exists an OIDRD-function $f=\left(V_0,V_1,V_2,V_3\right)$ of $G^{\prime }$ such that $A-\left\{u\right\}\subseteq V_2{{\displaystyle \cup }}^{\text{}}{V}_3$ and $\omega \left(f\right)\le 4\left(n-3\right)/3$. Define the function g on $V\left(G\right)$ by $g\left(u\right)=3$, $g\left(x_2\right)=1$, $g\left(x_1\right)=0$, and $g\left(x\right)=f\left(x\right)$ otherwise. Then, g is an OIDR-function of G such that $g\left(x\right)\ge 2$ for each $x\in A$ and $\omega \left(g\right)=\omega \left(f\right)+4\le 4\left(n-3\right)/3+4=4n/3$.

Suppose now that $a=b$ and let $w\in A-\left\{u,a\right\}$. Let $G^{\prime }$ be the graph obtained from G by removing first $u,x_1,x_2$ and then adding the edge $y_{1}w$ and joining every vertex x in $N\left(u\right)-\left\{x_1,y_1\right\}$ to either a or w provided a or w is not adjacent to the end-vertex of the A-ear path containing x. Clearly, $G^{\prime }\in \mathrm{ℱ}$, and thus by the induction hypothesis, there exists an OIDRD-function $f=\left(V_0,V_1,V_2,V_3\right)$ of $G^{\prime }$ such that $A-\left\{u\right\}\subseteq V_2{{\displaystyle \cup }}^{\text{}}{V}_3$ and $\omega \left(f\right)\le 4\left(n-3\right)/3$. Now, the function g defined above satisfies the desired conditions.

Subcase 2.2. All neighbors of u but one belong to A-ear paths of $Q_1$.

By the choice of u, we may assume that each vertex in A is adjacent to at most one A-ear path in $Q_2$. In that case, G is obtained from a multigraph H without loops with $\delta \left(H\right)\ge 3$ by subdividing any edge of H at least once and at most twice so that the set of edges of H subdivided twice is independent (in H). Hence, let $u_1{v}_1,\dots ,u_k{v}_k$ be the edges of H subdivided twice and let $A^{\prime \prime }$ be the set of all vertices in H for which all edges that are incident are subdivided once. Therefore, we have $\left|V\left(H\right)\right|=2k+\left|A^{\prime \prime }\right|$ and $\left|E\left(H\right)\right|=\frac{1}{2}{\displaystyle \sum _{v\in V\left(H\right)}{\mathrm{deg}}_H\left(v\right)}\ge \frac{3}{2}\left|V\left(H\right)\right|=3k+\frac{3}{2}\left|A^{\prime \prime }\right|$ (because $\delta \left(H\right)\ge 3$, k edges of H are subdivided twice and the remaining edges are subdivided once). Hence, the order of G is

$n=\left|V\left(H\right)\right|+\left|E\left(H\right)\right|+k\ge 6k+\frac{5}{2}\left|A^{\prime \prime }\right|.$

Assume that, for each i, the edge $u_i{v}_i$ in H once subdivided twice produces the path $u_i{w}_i{z}_i{v}_i$ in G. One can easily see that the function g defined on $V\left(G\right)$ by $g\left(x\right)=2$ for $x\in V\left(H\right)$, $g\left(w_i\right)=g\left(z_i\right)=1$ for each $1\le i\le k$ and $g\left(x\right)=\varnothing$ otherwise is an OIDRD-function of G such that $g\left(x\right)\ge 2$ for each $x\in A$ and

$\omega \left(g\right)=2\left|V\left(H\right)\right|+2k=6k+2\left|A^{\prime \prime }\right|<\frac{4\left(6k+\frac{5}{2}\left|A^{\prime \prime }\right|\right)}{3}\le \frac{4n}{3}.$

This completes the proof. □

Theorem 4.5. If G is a connected n-vertex graph with$\delta \left(G\right)\ge 2$such that the set of vertices with degree at least three is independent, then

${\gamma }_{oidR}\left(G\right)\le \frac{4n}{3}.$

This bound is sharp for $C_3$.

Proof. We use an induction on the order n. Clearly, $n\ge 3$ since $\delta \left(G\right)\ge 2$. If $n\in \left\{\mathrm{3,4}\right\}$, then $G\in \left\{C_3,C_4\right\}$ and the result is true by Proposition 4.1, establishing the base cases. Let $n\ge 5$, and assume that the result holds for all graphs $G^{\prime }$ of order less than n with minimum degree at least two such that the set of vertices with degree at least three is independent. Let G be a graph of order n such that $\delta \left(G\right)\ge 2$ and the set of vertices with degree at least three is independent. If $\Delta \left(G\right)=2$, then $G=C_n$ and the result follows from Proposition 4.1. Hence, we assume that $\text{Δ}\left(G\right)\ge 3$, and let $A=\left[v\in V\left(G\right)|{\mathrm{deg}}_G\left(v\right)\ge 3\right]$ and $B=V\left(G\right)-A$. Consider the A-ear paths and keep the same notations as defined in the proof of Proposition 4.4. Note that $A={{{\displaystyle \cup }}^{\text{}}}_{P\in Q}{X}_P$, $V\left(G\right)=A{{\displaystyle \cup }}^{\text{}}{{{\displaystyle \cup }}^{\text{}}}_{P\in \mathrm{}}V\left(P\right)$, and $1\le \left|X_P\right|\le 2$ for each $P\in Q$.

Assume first that there exists an A-ear path P such that $\delta \left[G-V\left(P\right)\right]=1$. Since G is simple, this means that $\left|V\left(P\right)\right|\ge 2$ and some vertex of G of degree three is adjacent to the end-vertices of $P$. Thus, $\left|X_P\right|=1$. In that case, let $X_P=\left\{a\right\}$ and $N_G\left(a\right)-V\left(P\right)=\left\{b\right\}$. Clearly, $b\in B$ (since A is independent), and thus there is a unique A-ear path $P^{\prime }$ in which b is an end-vertex of $P^{\prime }$. Let c be the other end-vertex of $P^{\prime }$ (possibly $b=c$). Let $G^{\prime }$ be the graph resulting from the deletion of vertex a and all vertices of P and $P^{\prime }$. Then, $\delta \left(G^{\prime }\right)\ge 2$ and by the induction hypothesis, ${\gamma }_{oidR}\left(G^{\prime }\right)\le 4\left|V\left(G^{\prime }\right)\right|/3$. On the other hand, since $G^{\prime \prime }=G\left[V\left(P\right){{\displaystyle \cup }}^{\text{}}V\left(P^{\prime }\right){{\displaystyle \cup }}^{\text{}}\left(a\right)\right]$ is isomorphic to $C_{\left|V\left(P\right)\right|+1,\left|V\left(P\prime \right)\right|}$, by Proposition 4.3, $G^{\prime \prime }$ has an OIDRD-function g such that $\omega \left(g\right)\le 4n\left(G^{\prime \prime }\right)/3$ and $g\left(c\right)\ge 1$. Now, for any ${\gamma }_{oidR}\left(G^{\prime }\right)$ function, the function h defined on $V\left(G\right)$ by $h\left(x\right)=f\left(x\right)$ for all $x\in V\left(G^{\prime }\right)$ and $h\left(x\right)=g\left(x\right)$ for all $x\in V\left(G^{\prime \prime }\right)$ is an OIDRD-function of $G$. Therefore,

${\gamma }_{oidR}\left(G\right)\le {\gamma }_{oidR}\left(G^{\prime }\right)+{\gamma }_{oidR}\left(G^{\prime \prime }\right)$

$\le \frac{4\left|V\left(G^{\prime }\right)\right|}{3}+\frac{4\left|V\left(P\right){{\displaystyle \cup }}^{\text{}}V\left(P^{\prime }\right){{\displaystyle \cup }}^{\text{}}\left\{a\right\}\right|}{3}=\frac{4n}{3}.$

Next, we can assume that $\delta \left[G-V\left(P\right)\right]\ge 2$ for each A-ear path $P\in \mathrm{}$. It follows that $\left|X_P\right|=2$ for each A-ear path $P\in \mathrm{}$. Assume that $\mathrm{}-\left(Q_1{{\displaystyle \cup }}^{\text{}}{Q}_2{{\displaystyle \cup }}^{\text{}}{Q}_3{{\displaystyle \cup }}^{\text{}}{Q}_4{{\displaystyle \cup }}^{\text{}}{Q}_5\right)\ne \varnothing$, and let $P\in \mathrm{}-\left(Q_1{{\displaystyle \cup }}^{\text{}}{Q}_2{{\displaystyle \cup }}^{\text{}}{Q}_3{{\displaystyle \cup }}^{\text{}}{Q}_4{{\displaystyle \cup }}^{\text{}}{Q}_5\right)$. Note that, by Proposition 4.2, P has an OIDRD-function g such that $\omega \left(g\right)\le 4\left|V\left(P\right)\right|/3$ and g assigns positive weight to the end-vertices of the path P. Now, let $G^{\prime }$ be the graph obtained from G by removing all vertices of P. By the induction hypothesis on $G^{\prime }$, we have ${\gamma }_{oidR}\left(G^{\prime }\right)\le 4\left|V\left(G^{\prime }\right)\right|/3$. Clearly, for every γ_oidR(G′)-function f, the function h defined on $V\left(G\right)$ by $h\left(x\right)=f\left(x\right)$ for all $x\in V\left(G^{\prime }\right)$ and $h\left(x\right)=g\left(x\right)$ for all $x\in V\left(P\right)$ is an OIDRD-function of G, and thus ${\gamma }_{oidR}\left(G\right)\le {\gamma }_{oidR}\left(G^{\prime }\right)+{\gamma }_{oidR}\left(P\right)\le 4n/3$. For the remaining part of the proof, we can assume that $\mathrm{}=Q_1{{\displaystyle \cup }}^{\text{}}{Q}_2{{\displaystyle \cup }}^{\text{}}{Q}_3{{\displaystyle \cup }}^{\text{}}{Q}_4{{\displaystyle \cup }}^{\text{}}{Q}_5$. Therefore, $G\in \mathrm{ℱ}$, and thus the result follows from Proposition 4.4. □

5 Conclusion

In this article, we continued the study of outer-independent double Roman domination number and we characterized the trees T of order $n\ge 3$, for which ${\gamma }_{oidR}\left(T\right)\le 5n/4$, answering a problem posed by Abdollahzadeh Ahangar et al., Ref. 19. Moreover, we showed that, for any connected graph G of order n with minimum degree at least two in which the set of vertices with degree at least three is independent, ${\gamma }_{oidR}\left(G\right)\le 4n/3$. Finding a sharp upper bound for the outer-independent double Roman domination number of connected graph G of order n with minimum degree remains open.

Data Availability Statement

The original contributions presented in the study are included in the article/Supplementary Material. Further inquiries can be directed to the corresponding author.

Author Contributions

All authors listed have made a substantial, direct, and intellectual contribution to the work and approved it for publication.

Funding

This work was supported by the National Key R&D Program of China (No. 2018YFB1005100).

Conflict of Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

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