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ORIGINAL RESEARCH article

Front. Appl. Math. Stat., 05 February 2021
Sec. Optimization
This article is part of the Research Topic 2021 Editor’s Pick: Applied Mathematics and Statistics View all 13 articles

On the Outer-Independent Double Roman Domination of Graphs

Yongsheng RaoYongsheng Rao1Saeed Kosari
Saeed Kosari1*Seyed Mahmoud SheikholeslamiSeyed Mahmoud Sheikholeslami2M. ChellaliM. Chellali3Mahla KheibariMahla Kheibari2
  • 1Institute of Computing Science and Technology, Guangzhou University, Guangzhou, China
  • 2Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, Iran
  • 3LAMDA-RO Laboratory, Department of Mathematics, University of Blida, Blida, Algeria

An outer-independent double Roman dominating function (OIDRDF) of a graph G is a function h:V(G){0,1,2,3} such that i) every vertex v with f(v)=0 is adjacent to at least one vertex with label 3 or to at least two vertices with label 2, ii) every vertex v with f(v)=1 is adjacent to at least one vertex with label greater than 1, and iii) all vertices labeled by 0 are an independent set. The weight of an OIDRDF is the sum of its function values over all vertices. The outer-independent double Roman domination number γoidR (G) is the minimum weight of an OIDRDF on G. It has been shown that for any tree T of order n ≥ 3, γoidR (T) ≤ 5n/4 and the problem of characterizing those trees attaining equality was raised. In this article, we solve this problem and we give additional bounds on the outer-independent double Roman domination number. In particular, we show 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, γoidR (T) ≤ 4n/3.

1 Introduction

We consider only simple connected graphs G with vertex set V=V(G) and edge set E=E(G), where n=|V| is the order of G. The open neighborhood of a vertex vV is the set N(v)={u|uvE}, and the degree of v is degG(v)=|N(v)|. 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 SV 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(G){0,1,2,3} 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[V(G)]=uV(G)h(u), and the double Roman domination number γdR (G) equals the minimum weight of a DRDF on G. Double Roman domination has been studied by several authors; see, for example, Refs. 214. For more details on Roman domination and its variations, we refer the reader to Refs. 1518.

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=(V0,V1,V2,V3), where V0 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) γoidR (G) is the minimum weight of an OIDRD-function on G. Clearly, γdR (G) ≤ γoidR (G) holds for every graph G. Recently, Mojdeh et al., Ref. 20, proved that the decision problem associated with γoidR (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 {0,1,2,3} 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,

γoidR(T)n+s(T)2,

where s(T) 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, γoidR (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, γoidR (G) ≤ 4n/3.

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

With the aim of characterizing the trees T of order n ≥ 3 with γoidR (T) = 5n/4, let t be the family of trees defined as follows. Let Hi be a path P4 whose vertices are labeled in order v1i,v2i,v3i, and v4i. For any integer t1, let t be the family of trees T obtained from H1,,Ht by adding t1 edges between the stems of Hi’s so that the resulting graph is a tree. Beeler et al., Ref. 1, proved that, for every tree Tt, γdR(T)=5|V(T)|/4 which implies that γoidR(T)=5|V(T)|/4.

Lemma 2.1. LetTtfor some integer t ≥ 1. Then, there is a γoidR (T) function f such that, for every leaf v of T, f(v){1,2}.

Proof. Let T be a tree of t for some integer t ≥ 1. Then, γoidR(T)=5|V(T)|/4. Since T is a bipartite graph, let X and Y be the partite sets of T. Let X be the set of stems of T belonging to X, and likewise let Y be defined similarly. Clearly, |X|=|Y|, |X|=|Y|, and every leaf of T is either in XX or YY. Define the function f on V(T) by assigning a 2 to all vertices of Y, a 1 to all vertices in XX, and a 0 to vertices in X. Then, f is an OIDRD-function of T of weight 5|V(T)|/4, and thus f is a γoidR (T) function with desired property. □

Theorem 2.2. Let T be a tree on n ≥ 4 vertices. Then, γoidR (T) = 5n/4 if and only ifTtfor some integert1.

Proof. We prove only the necessity. Let T be a tree of n ≥ 4 such that γoidR (T) = 5n/4. Clearly, n=4t for some integer t ≥ 1. To prove that TTt, we use an induction on t. If t = 1, then T=P4, and clearly, T1. Let t ≥ 2 and assume that the result is true for any tree T with γoidR (T) = 5n/4, where n=4t and t<t. Let T be a tree with γoidR (T) = 5n/4 and n=4t. We deduce from γoidR(T)=5n/4n+s(T)/25n/4 that s(T)=n/2. Therefore, T is the corona of some tree and so T has no strong stem. Moreover, diam(T)4 because t ≥ 2. Let P=v1v2,,vk be a diametral path in T and root T at vk. Then, degT(v2)=2, and there is a unique leaf w adjacent to v3. Denote by Tx the subtree induced by a vertex x and its descendants in the rooted tree T. We claim that degT(v3)=3. Suppose, to the contrary, that degT(v3)4 and let v2=w1,w2,,ws be the children of v3 with depth one. Since degT(wi)=2 for each i, let wi be the leaf adjacent to wi for i{2,,s}, and consider the tree T obtained from T by removing every wi and wi for i2. Observe that the subtree rooted at v3 is a path P4. Let f be a γoidR(T)-function, where, without loss of generality, f(v3)=3. Then, f can be extended to an OIDRD-function of T by assigning a 0 to w2,,ws and a 2 to w2 ,,ws . It follows from Proposition 1.2 that

γoidR(T)γoidR(T)+2(s1)5(n2s+2)4+2s2<5n4,

which leads to a contradiction. Hence, degT(v3)=3, and thus Tv3=P4. Now, let T=TTv3. Clearly, any γoidR(T)-function f can be extended to an OIDRD-function of T by assigning a 3 to v3, 2 to v1, and a 0 to v2,w. Hence,

5n4=γoidR(T)γoidR(T)+55(n4)4+5=5n4.

Therefore, by the induction hypothesis on T, we have Tn/41. By the construction of T, we may assume that v4 is a vertex of an induced path P4=v11v21v31v41, where degT(v11)=degT(v41)=1, degT(v31)=2, and degT(v21)2. Now, if n/41=1, then clearly T2, and we are done. Hence, we assume that n/412. We claim that v4{v21,v31}. Suppose, to the contrary, that v4{v11,v41}. By Lemma 2.1, there is a γoidR(T)-function f such that f(v4){1,2}. If f(v4)=1, then f(v5)=2 and the function h defined on V(T) by h(v3)=3,h(v1)=2, h(w)=h(v2)=h(v4)=0, and h(x)=f(x) for xV(T){v4} is an OIDRD-function of T of weight ω(h1)+4=5(n4)/4+4<5n/4, a contradiction. If f(v4)=2 and f(v5)1, then as above we can get a contradiction. Hence, assume that f(v4)=2 and f(v5)=0. Then, v5 has a neighbor with weight at least two and the function h defined on V(T) by h(v3)=3,h(v1)=2, h(v5)=1, h(v4)=h(w)=h(v2)=0, and h(x)=f(x) for xV(T){v4,v5} is an OIDRD-function of T of weight ω(h)+4=5(n4)/4+4<5n/4, a contradiction. Thus, v4{v21,v31}, and thus Tt 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, wheren0,1,2(mod4). If T contains a strong stem, then γoidR(T) ≤ 5n/4−1.

Proof. Let sV(T) be a strong stem of T and let L(s)={x1,x2,,xt} be the set of leaves adjacent to s. Consider the forest Ts and assume that it contains r0 components each of order at least four T1,,Tr. If r1, then fi be a γoidR(Ti)-function for each i{1,r}. Clearly, by Proposition 1.2, γoidR(Ti)5|V(Ti)|/4. Moreover, suppose that Ts has t2 components isomorphic to K2 and t3 components isomorphic to P3. Clearly, all fi’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 γoidR(T)n<5n/4. Hence, we can assume that r1. Now, using the fact that n0,1,2(mod4), we obtain

γoidR(T)3+2t2+3t3+i=1r5|V(Ti)|4=3+2t2+3t3+5(n3t32t21t)45n/41,

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, wheren3 (mod4). If T contains one strong stem s such thatTscontains a component isomorphic toP2orP3, then γoidR (T) ≤ 5n/4−1.

Proposition 3.3. Let T be a tree of ordern3, wheren3(mod4). If T contains a strong stem having at least three leaves, then γoidR (T) ≤ 5n/4−1.

Proposition 3.4. Let T be a tree of order n ≥ 3, wheren3(mod4). If T contains more than one strong stem, then γoidR (T) ≤ 5n/4−1.

Proof. Let s1 and s2 be two strong stems of T and let L(si)={x1i,x2i,,xtii} be the set of leaves adjacent to si for i{1,2}. Clearly, diam(T)3. If each component of T{s1,s2} is of order at most three, then assigning a 3 to s1,s2 and to the center vertex of each component of order three, a 2 to each leaf at distance two from si that belongs to a component of order two in T{s1,s2}, and a 0 to the remaining vertices provides an OIDRD-function of T of weight at most p. Therefore, γoidR(T)n5n/41. Hence, we may assume that Ts contains at least one component of order at least four. Let T1,,Tr(r1) be such components of T{s1,s2} of order at least four. Let fi be a γoidR (Ti) function for each i. In addition, let T{s1,s2} have t2 components isomorphic to K2 and t3 components isomorphic to P3. Then, all fi’s together can be extended to an OIDRD-function of T by assigning a 3 to s1,s2 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{s1,s2}, a 3 to the center of all components of order three, fi(x) for each xV(Ti) and i{1,,r}, and a 0 to the remaining vertices in the components of order at most three. Using the fact that n3(mod4), we obtain

γoidR(T)6+2t2+3t3+i=1r5|V(Ti)|4=6+2t2+3t3+5(n3t32t22t1t2)45n/41,

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, γoidR(Cn)={nifn0(mod2)n+1otherwise.

Proposition 4.2. For n ≥ 3, the pathPnhas an OIDRD-function f that assigns positive weight to the end-vertices ofPnandω(f)=n+14n/3.

Proof. Let Pn=v1v2vn and define the function f on V(Pn) as follows. If n1(mod2), then f(v2i+1)=2 for 0in1/2 and f(x)=0 otherwise, and if n0(mod2), then f(vn)=1, f(v2i+1)=2 for 0in2/2 and f(x)=0 otherwise. Clearly, ω(f)4n/3 and f is an OIDRD-function of Pn assigning positive weight to the end-vertices of Pn. □

For integers r3 and s ≥ 1, let Cr,s be the graph obtained from a cycle Cr=(u1u2ur) and a path Ps=v1v2vs by adding the edge u1v1. Applying Propositions 4.1 and 4.2, we derive the next result.

Proposition 4.3. For integersr3and s ≥ 1, the graphCr,shas an OIDRD-function f that assigns a positive weight to vsand ω(f) ≤ 4(r+s)/3.

Proof. Assume first that r+s{4,5}. If r+s=4, then assigning a 1 to v1, a 2 to u1,u2, and a 0 to u3 provides an OIDRD-function of C3,1 satisfying the conditions. If r=4 and s=1, then assigning a 1 to v1, a 2 to u1,u3, and a 0 to u2 and u4 provides an OIDRD-function of C4,1 with the desired properties. If r=3 and s=2, then assigning a 2 to u1,u3,v2 and a 0 to u2 and v1 provides as above an OIDRD-function of C3,2 with the desired properties. Hence, assume that r+s6, and let f be a γoidR (Cr)function such that f(u1)2. If s=1, then the function g defined on V(Cr,s) by g(v1)=1 and g(x)=f(x) otherwise is an OIDRD-function of Cr,s with desired properties. If s=2, then the function g defined on V(Cr,s) by g(v1)=0, g(v2)=2, and g(x)=f(x) otherwise is an OIDRD-function of Cr,s with the desired properties. Henceforth, we can assume that s3.

Let g be an OIDRD-function of the path v1v2vs of weight s+1 assigning positive weights to v1,vs (Proposition 4.2). Define h on Cr,s by h(ui)=f(ui) for each i and h(vj)=g(vj) for each j. Clearly, h is an OIDRD-function of Cr,s and we deduce from Proposition 4.1 that γoidR(Cr,s)r+s+24(r+s)/3.□

Let be the family of all simple graphs obtained from some connected multigraph H without loops with δ(H)3 by subdividing each edge of H at least once and at most five times. Clearly, any graph in has order at least 5. The next result shows that every graph G in of order n satisfies γoidR(G)4n/3.

Proposition 4.4. For any graphGof order n, there exists an OIDRD-function f of G such thatω(f)4n/3andf(x)2for each vertex x of degree at least three.

Proof. Let G be a graph of order n. We use an induction on n. If n = 5, then G=K2,3 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 n6 and assume that the result holds for all graphs in of order n, where 5n<n. Let G be a graph of order n6. Suppose that A={xV(G)|degG(x)3} and let B=V(G)A. In the sequel, we will call an induced path P of G an A-ear path if V(P)B and P is connected to A by either its unique vertex (when |V(P)|=1) or its two end-vertices (when |V(P)|2). For each i{1,2,3,4,5}, let Qi be the set of all A-ear paths P of G of order i and let =i=15Qi. Clearly, B=PV(P). Moreover, for each A-ear path P, let XP={uA|u is adjacent to a vertex of P}. Hence, A=PXP. Furthermore, since G, we have |XP|=2 for each P, and therefore, |A|2.

First, let Q3Q5. Suppose P=x1x2k+1Q2k+1(k{1,2}) and let XP={a1,a2}, where a1x1,a2x2k+1E(G). Let G be the graph obtained from G by first removing all vertices of the path P except xk+1 and then adding edges a1xk+1 and a2xk+1. Clearly, G of order less than n. By the induction hypothesis on G, there exists an OIDRD-function f=(V0,V1,V2,V3) of G such that a1,a2V2V3 and ω(f)4(n2k)/3. It follows that f(xk+1)=0. Now, if k = 1, then the function g defined on V(G) by g(x2)=2, g(x1)=g(x3)=0, and g(x)=f(x) otherwise is an OIDRD-function of G such that g(x)2 for each xA and

ω(g)=ω(f)+24(n2)3+2<4n3.

If k = 2, then the function g defined on V(G) by g(x2)=g(x4)=2, g(x1)=g(x3)=g(x5)=0, and g(x)=f(x) otherwise is an OIDRD-function of G such that g(x)2 for each xA and

ω(g)=ω(f)+44(n4)3+4<4n3.

From now on, we can assume that Q3Q5=.

Next, assume that Q4 and let P=x1x2x3x4Q4 with XP={u,a} and ux1,ax4E(G). Let G be the graph obtained from G by deleting x1,x2,x3 and adding the edge ux4. Clearly, G and thus by the induction hypothesis, there exists an OIDRD-function f of G such that f(x)2 for each vertex xA and ω(f)4(n3)/3. Then, the function g defined on V(G) by g(x2)=g(x3)=2,g(x1)=0, and g(x)=f(x) otherwise is an OIDRD-function of G such that g(x)2 for each xA and ω(g)=ω(f)+44(n3)/3+4=4n/3.

Considering the above situations, we may assume that =Q1Q2. Note that n=|A|+m1+2m2 and m1+m23, where mi=|Qi| for i{1,2}.

Assume first that |A|=2 and let A={u,v}. If Q2, then let Q2={w1jw2j|1jm2}, where uw1jE(G) for each j. Moreover, if Q1, then let Q1={z1l|1lm1}. Define the function g on V(G) by g(u)=3,g(v)=2,g(w2j)=1 for each j and g(t)=0 otherwise. Then, g is an OIDRD-function of G such that g(x)2 for each xA and we have ω(g)5+m24(2+2m2+m1)/3=4n/3.

Henceforth, we can assume that |A|3. We consider the following cases.

Case 1.Q2=.

Then, =Q1 and G is obtained from a loopless multigraph G by subdividing each edge of G once. Since each vertex of G has degree at least three, we have 2|E(G)|3n(G) and so n(G)=|V(G)|+|E(G)|5n(G)/2. Define f on V(G) by f(x)=2 for xV(G) and f(x)=0 otherwise. Clearly, f is an OIDR-function of G such that f(x)2 for each xA and ω(f)=2n(G)4n/3.

Case 2.Q2.

Let uV(G) be a vertex with the most neighbors in A-ear paths of Q2. We consider the following subcases.

Subcase 2.1.u is adjacent to at least two A-ear paths of Q2.

Let P1=x1x2,P2=y1y2Q2 be two A-ear paths such that ux1,uy1E(G). Assume that {ax2,by2}E(G), where a,bA{u}. Suppose that ab, and let G be the graph obtained from G by removing first u,x1,x2 and then adding the edge y1a and joining by an edge every vertex x in N(u){x1,y1} 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, and by the induction hypothesis, there exists an OIDRD-function f=(V0,V1,V2,V3) of G such that A{u}V2V3 and ω(f)4(n3)/3. Define the function g on V(G) by g(u)=3, g(x2)=1, g(x1)=0, and g(x)=f(x) otherwise. Then, g is an OIDR-function of G such that g(x)2 for each xA and ω(g)=ω(f)+44(n3)/3+4=4n/3.

Suppose now that a=b and let wA{u,a}. Let G be the graph obtained from G by removing first u,x1,x2 and then adding the edge y1w and joining every vertex x in N(u){x1,y1} 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, and thus by the induction hypothesis, there exists an OIDRD-function f=(V0,V1,V2,V3) of G such that A{u}V2V3 and ω(f)4(n3)/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 Q1.

By the choice of u, we may assume that each vertex in A is adjacent to at most one A-ear path in Q2. In that case, G is obtained from a multigraph H without loops with δ(H)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 u1v1,,ukvk be the edges of H subdivided twice and let A be the set of all vertices in H for which all edges that are incident are subdivided once. Therefore, we have |V(H)|=2k+|A| and |E(H)|=12vV(H)degH(v)32|V(H)|=3k+32|A| (because δ(H)3, k edges of H are subdivided twice and the remaining edges are subdivided once). Hence, the order of G is

n=|V(H)|+|E(H)|+k6k+52|A|.

Assume that, for each i, the edge uivi in H once subdivided twice produces the path uiwizivi in G. One can easily see that the function g defined on V(G) by g(x)=2 for xV(H), g(wi)=g(zi)=1 for each 1ik and g(x)= otherwise is an OIDRD-function of G such that g(x)2 for each xA and

ω(g)=2|V(H)|+2k=6k+2|A|<4(6k+52|A|)34n3.

This completes the proof. □

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

γoidR(G)4n3.

This bound is sharp for C3.

Proof. We use an induction on the order n. Clearly, n3 since δ(G)2. If n{3,4}, then G{C3,C4} and the result is true by Proposition 4.1, establishing the base cases. Let n5, and assume that the result holds for all graphs G 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 δ(G)2 and the set of vertices with degree at least three is independent. If Δ(G)=2, then G=Cn and the result follows from Proposition 4.1. Hence, we assume that Δ(G)3, and let A=[vV(G)|degG(v)3] and B=V(G)A. Consider the A-ear paths and keep the same notations as defined in the proof of Proposition 4.4. Note that A=PQXP, V(G)=APV(P), and 1|XP|2 for each PQ.

Assume first that there exists an A-ear path P such that δ[GV(P)]=1. Since G is simple, this means that |V(P)|2 and some vertex of G of degree three is adjacent to the end-vertices of P. Thus, |XP|=1. In that case, let XP={a} and NG(a)V(P)={b}. Clearly, bB (since A is independent), and thus there is a unique A-ear path P in which b is an end-vertex of P. Let c be the other end-vertex of P (possibly b=c). Let G be the graph resulting from the deletion of vertex a and all vertices of P and P. Then, δ(G)2 and by the induction hypothesis, γoidR(G)4|V(G)|/3. On the other hand, since G=G[V(P)V(P)(a)] is isomorphic to C|V(P)|+1,|V(P)|, by Proposition 4.3, G has an OIDRD-function g such that ω(g)4n(G)/3 and g(c)1. Now, for any γoidR(G) function, the function h defined on V(G) by h(x)=f(x) for all xV(G) and h(x)=g(x) for all xV(G) is an OIDRD-function of G. Therefore,

γoidR(G)γoidR(G)+γoidR(G)
4|V(G)|3+4|V(P)V(P){a}|3=4n3.

Next, we can assume that δ[GV(P)]2 for each A-ear path P. It follows that |XP|=2 for each A-ear path P. Assume that (Q1Q2Q3Q4Q5), and let P(Q1Q2Q3Q4Q5). Note that, by Proposition 4.2, P has an OIDRD-function g such that ω(g)4|V(P)|/3 and g assigns positive weight to the end-vertices of the path P. Now, let G be the graph obtained from G by removing all vertices of P. By the induction hypothesis on G, we have γoidR(G)4|V(G)|/3. Clearly, for every γoidR(G′)-function f, the function h defined on V(G) by h(x)=f(x) for all xV(G) and h(x)=g(x) for all xV(P) is an OIDRD-function of G, and thus γoidR(G)γoidR(G)+γoidR(P)4n/3. For the remaining part of the proof, we can assume that =Q1Q2Q3Q4Q5. Therefore, G, 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 n3, for which γoidR(T)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, γoidR(G)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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Keywords: outer independence double Roman domination, outer-independent double Roman dominating function, independent set, double Roman domination, Roman domination, tree

Citation: Rao Y, Kosari S, Sheikholeslami SM, Chellali M and Kheibari M (2021) On the Outer-Independent Double Roman Domination of Graphs. Front. Appl. Math. Stat. 6:559132. doi: 10.3389/fams.2020.559132

Received: 09 November 2020; Accepted: 04 December 2020;
Published: 05 February 2021.

Edited by:

Yong Chen, Hangzhou Dianzi University, China

Reviewed by:

Zepeng Li, Lanzhou University, China
Sarfraz Ahmad, COMSATS University Islamabad, Pakistan

Copyright © 2021 Rao, Kosari, Sheikholeslami, Chellali and Kheibari. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Saeed Kosari, saeedkosari38@yahoo.com

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