Double Italian domination in trees

Double Italian domination in trees W eiping Shang 1 ∗ , Shanshan Zhang 2 ∗ 1 Sc ho ol of Mathematics and Statistics, Zhengzhou Universit y , Zhengzhou 450001, China 2 Sc ho ol of Mathematics and Statistics, Jiangsu Normal Univ ersit y , Xuzhou 221116, China Abstract Let G b e a graph with vertex set V = V ( G ). A double Roman dominating function on a graph G is a function f : V → { 0 , 1 , 2 , 3 } satisfying the conditions that if f ( v ) = 0, then vertex v must hav e at least t w o neighbors in V 2 or one neigh b or in V 3 ; if f ( v ) = 1, then vertex v must hav e at least one neighbor in V 2 ∪ V 3 . The weigh t of a double Roman dominating function f is the sum f ( V ) = P v ∈ V f ( v ), and the double Roman domination n um b er γ dR ( G ) is the minimum w eigh t of a double Roman dominating function on G . A double Italian dominating function on a graph G is a function f : V → { 0 , 1 , 2 , 3 } satis- fying the condition that for every v ertex u ∈ V , if f ( u ) ∈ { 0 , 1 } , then P v ∈ N [ u ] f ( v ) ≥ 3. The double Roman domination num ber γ dI ( G ) is the minimum w eigh t of a double Ital- ian dominating function on G . Mo jdeh and V olkmann [D.A. Mo jdeh and L. V olkmann, Roman 3-domination (double Italian domination), Discrete Appl. Math. 283 (2020), 555–564] pro v ed that γ dI ( T ) = γ dR ( T ) for any tree T . Ho w ev er, we find that there is a minor issue in the pro of. In this pap er, w e first pro ve that γ dI ( T )  = γ dR ( T ). Sub- sequen tly , w e presen t a sharp b ound on the double Italian domination num ber of an y non-trivial tree T , and c haracterize the trees attaining this b ound. Keyw ords domination, double Roman domination n um b er, double Italian domination n um b er 2020 MR Sub ject Classification 05C70 1 In tro duction In this pap er, we only consider simple undirected graphs. F or standard graph-theoretical notions and terminology , w e refer the reader to [ 2 ]. Let G b e a graph with v ertex set V = V ( G ) and edge set E = E ( G ). The or der of G is the n um b er of its vertices. The op en neighborho o d of a vertex v ∈ V is the set N ( v ) = { u ∈ V | uv ∈ E } , and its closed neigh b orhoo d is ∗ e-mail: shangwp@zzu.edu.cn, zhangss@jsnu.edu.cn. 1 N [ v ] = N ( v ) ∪ { v } . V ertices in N ( v ) are called the neighb ors of v . The de gr e e of vertex v ∈ V is d ( v ) = | N ( v ) | . A vertex with degree one is called a le af , and its neighbor is a supp ort vertex . A supp ort vertex with tw o or more leaf neighbors is called a str ong supp ort vertex . An edge adjacen t to a leaf is called a p endant e dge . A graph is trivial if it has only one v ertex, and non-trivial otherwise. A tree is an acyclic connected graph. F or an y S ⊆ V , w e denote by G − S the graph obtained from G b y deleting all the v ertices in S together with all edges incident with the vertices in S , and by G [ S ] the subgraph induced by S . A set S ⊆ V in a graph G is called a dominating set if every v ertex of G is either in S or adjacen t to a v ertex of S . The domination numb er γ ( G ) equals the minim um cardinalit y of a dominating set in G , and a dominating set of G with cardinalit y γ ( G ) is called a γ -set of G . Giv en a graph G and a p ositiv e integer k ≥ 2, assume that f : V → { 0 , 1 , 2 , . . . , k } is a function, and supp ose that ( V 0 , V 1 , V 2 , . . . , V k ) is the ordered partition of V induced b y f , where V i = { v ∈ V : f ( v ) = i } for i ∈ { 0 , 1 , . . . , k } . So w e can write f = ( V 0 , V 1 , V 2 , . . . , V k ). A R oman dominating function on a graph G is a function f : V → { 0 , 1 , 2 } satisfying the condition that ev ery v ertex u for whic h f ( u ) = 0 is adjacent to at least one vertex v for whic h f ( v ) = 2. The weight of a Roman dominating function is the sum f ( V ) = P v ∈ V f ( v ). The minim um w eigh t of a Roman dominating function on G is called the R oman domination numb er of G and is denoted γ R ( G ). A Roman dominating function on G with weigh t γ R ( G ) is called a γ R -function of G . The original study of Roman domination was motiv ated by the defense strategies used to defend the Roman Empire during the reign of Emp eror Constantine the Great. He decreed that for all cities in the Roman Empire, at most tw o legions should b e stationed. F urther, if a location having no legions was attack ed, then it m ust be within the vicinit y of at least one legion b eing stationed, and so that one of the t w o legions could b e sen t to defend the attac k ed city . This part of the history of the Roman Empire ga v e rise to the mathematical concept of Roman domination, originally defined and discussed by Stewart [ 9 ] in 1999, and subsequen tly developed by Co c k ayne et al. [ 5 ] in 2004. Since then, a lot of related v ariations and generalizations hav e b een studied. Beeler et al. [ 3 ] in tro duced the concept of double Roman domination. What they propose is a stronger v ersion of Roman domination that doubles the protection by ensuring that an y attac k can b e defended b y at least tw o legions. A double R oman dominating function on a graph G is a function f : V → { 0 , 1 , 2 , 3 } satisfying the following conditions: (i) If f ( v ) = 0, then v ertex v must ha ve at least tw o neighbors in V 2 or one neigh b or in V 3 . (ii) If f ( v ) = 1, then vertex v m ust ha v e at least one neighbor in V 2 ∪ V 3 . The weigh t of a double Roman dominating function f on G is the sum f ( V ) = P v ∈ V f ( v ), and the minimum w eigh t of f ov er every double Roman dominating function on G is called the 2 double R oman domination numb er of G . W e denote this n um b er with γ dR ( G ), and a double Roman dominating function of G with w eigh t γ dR ( G ) is a γ dR -function of G . Mo jdeh and V olkmann[ 10 ] defined a v ariant of double Roman domination, namely double Italian domination (Roman 3-domination). F ormally , a double Italian dominating function on a graph G is a function f : V → { 0 , 1 , 2 , 3 } satisfying the condition that for ev ery vertex u ∈ V , if f ( u ) ∈ { 0 , 1 } , then P v ∈ N [ u ] f ( v ) ≥ 3. Sp ecifically , it satisfies the follo wing conditions: (i) If f ( v ) = 0, then v ertex v m ust hav e at least three neigh b ors in V 1 , or one neigh b or in V 1 and one neighbor in V 2 , or tw o neighbors in V 2 , or one neighbor in V 3 . (ii) If f ( v ) = 1, then vertex v m ust ha ve at least t w o neigh b ors in V 1 , or one neigh b or in V 2 ∪ V 3 . The w eigh t of a double Italian dominating function f on G is the sum f ( V ) = P v ∈ V f ( v ), and the minimum w eight of f ov er every double Italian dominating function on G is called the double Italian domination numb er of G . W e denote this n um b er with γ dI ( G ), and a double Italian dominating function of G with w eigh t γ dI ( G ) is a γ dI -function of G . Compared with the requiremen ts of a double Roman dominating function, a double Italian dominating function relaxes the restrictions on the domination metho d and is more flexible. It is easy to see that every double Roman dominating function is a double Italian dominating function, so γ dI ( G ) ≤ γ dR ( G ). F or any non-trivial tree T , Mo jdeh and V olkmann[ 10 ] prov ed that γ dI ( T ) = γ dR ( T ) for any tree T . How ever, w e find that there is a minor issue in the pro of. In this pap er, we first pro ve that γ dI ( T )  = γ dR ( T ). Subsequen tly , we present a sharp b ound on the double Italian domination n um b er of any non-trivial tree T , and give a c haracterization of trees with γ dI ( T ) = 2 γ ( T ) + 1. 2 Preliminaries In this section, w e first present some preliminary results, which are used in the proof of our main results. Prop osition 2.1 ([ 3 ]) . In a double R oman dominating function of weight γ dR ( G ) , no vertex ne e ds to b e assigne d the value 1. By Prop osition 2.1 , we can assume that V 1 = ∅ for all double Roman dominating functions under consideration. Prop osition 2.2 ([ 3 ]) . F or any gr aph G , 2 γ ( G ) ≤ γ dR ( G ) ≤ 3 γ ( G ) . Prop osition 2.3 ([ 3 ]) . If T is a non-trivial tr e e, then γ dR ( T ) ≥ 2 γ ( T ) + 1 . 3 F or a positive in teger t , a wounde d spider is obtained from a star K 1 ,t b y sub dividing at most t − 1 edges. Similarly , for an integer t ≥ 2, a he althy spider is obtained from a star K 1 ,t b y sub dividing all of its edges. In a w ounded spider, a vertex of degree t is called the he ad vertex , and the vertex that is distance t wo from the head v ertex is the fo ot vertex . The neigh b or of a foot v ertex is called the sub division vertex . The head and fo ot vertices are well defined except when the wounded spider is the path on t wo or four vertices. F or P 2 , w e will consider both vertices to b e head v ertices, and in the case of P 4 , w e will consider b oth end v ertices as fo ot v ertices and b oth in terior v ertices as head vertices. Prop osition 2.4 ([ 1 , 11 ]) . If T is a non-trivial tr e e, then γ dR ( T ) = 2 γ ( T ) + 1 if and only if T is a wounde d spider. Theorem 2.5 ([ 10 ]) . F or any tr e e T , γ dI ( T ) = γ dR ( T ) . Since γ dI ( G ) ≤ γ dR ( G ), if there exists a γ dI -function on G that is also a double Roman dominating function, then this function is also a γ dR -function and γ dR ( G ) = γ dI ( G ). Before the discussion, we first presen t an imp ortan t lemma, whic h is frequen tly used in the follo wing section. Lemma 2.6. Ther e exist a γ dI -function and a γ dR -function on G such that every le af and its unique neighb or ar e assigne d values 2 and 0 , or 0 and 3 , r esp e ctively, and every str ong supp ort vertex is assigne d a value 3. Pr o of. Let f b e a γ dI -function, v b e a leaf of G , and u b e its unique neighbor. By the definition of f , if f ( v ) = 0, then f ( u ) = 3. If f ( v ) = 1, then f ( u ) = 2. Redefine a function g suc h that g ( v ) = 0, g ( u ) = 3, and g ( w ) = f ( w ) for all other v ertices. Clearly , g is also a double Italian dominating function with the same weigh t. If f ( v ) = 3, then f ( u ) = 0. Similarly , redefine a function g suc h that g ( v ) = 0, g ( u ) = 3, and g ( w ) = f ( w ) for all other v ertices. Clearly , g is also a double Italian dominating function with the same weigh t. If f ( v ) = 2 and f ( u ) = 1, then replace them with 0 and 3 as ab o v e. Hence if f is a γ dI -function, then f ( v ) = 2 and f ( u ) = 0 or f ( v ) = 0 and f ( u ) = 3. By Proposition 2.1 , let f b e a γ dR ( G )-function with no v ertex assigned v alue 1. Let v be a leaf v ertex of G and u b e its unique neigh b or. By the definition of f , if f ( v ) = 0, then f ( u ) = 3 and if f ( v ) = 2, then f ( u ) = 0. If f ( v ) = 3, then f ( u ) = 0. Similarly , redefine a function g suc h that g ( v ) = 0, g ( u ) = 3, and g ( w ) = f ( w ) for all other v ertices. Clearly , g is also a double Italian dominating function with the same weigh t. Hence if f is a γ dR -function, then f ( v ) = 2 and f ( u ) = 0 or f ( v ) = 0 and f ( u ) = 3. Let w b e a strong supp ort vertex of G . Since w has at least tw o leaf neighbors, if any dou- ble Italian dominating function or double Roman dominating function of T assigns a v alue less than 3 to w , then the total w eigh t of assigned to w and its leaf neigh b ors is at least 4. This im- plies that an y γ dI -function or γ dR -function of T will assign a 3 to u and 0 to its leaf neigh b ors. □ 4 3 Main results In this section we presen t a tree for whic h the tw o domination num b ers are not equal. The double star S p,q is the tree with exactly t w o adjacen t non-leaf v ertices (cen tral vertices), one of which is adjacen t to p lea ves and the other to q leav es. Let u 1 and u 2 b e the t w o central v ertices of the double star S 2 , 2 , and T 1 b e the tree obtained b y sub dividing the three p endan t edges of the double star S 2 , 2 , as shown in the follo wing figure. 1 u 1 v 1 w 1 2 v 2 w 2 2 3 u 2 v 3 w 3 2 v 4 2 u 1 v 1 w 1 2 v 2 w 2 2 3 u 2 v 3 w 3 2 v 4 Figure 1: γ dI ( T 1 ) = 10 and γ dR ( T 1 ) = 11 Theorem 3.1. Ther e exists a tr e e T 1 such that γ dI ( T 1 ) < γ dR ( T 1 ) . Pr o of. Recall that u 1 and u 2 are the t wo central v ertices of the double star S 2 , 2 , and T 1 b e the tree obtained from a double star S 2 , 2 b y sub dividing t wo p endan t edges inciden t to u 1 , and one p endan t edge incident to u 2 . Let v 1 and v 2 b e the other tw o neigh b ors of u 1 , v 3 and v 4 b e the other t wo neigh b ors of u 2 , and w i b e the leaf adjacen t to v i , for i = 1 , 2 , 3 in T 1 . Since one p endan t edge inciden t to u 2 is not sub divided, v 4 is also a leaf. By Lemma 2.6 , there exists a γ dI -function f such that either f ( w i ) = 2 and f ( v i ) = 0, or f ( w i ) = 0 and f ( v i ) = 3, for i = 1 , 2 , 3, and either f ( v 4 ) = 2 and f ( u 2 ) = 0, or f ( v 4 ) = 0 and f ( u 2 ) = 3. If f ( w 3 ) = 2 and f ( v 3 ) = 0, then by the definition of a double Italian dominating function, f ( w 3 ) + f ( u 2 ) ≥ 3, whic h implies f ( u 2 ) ≥ 1. F rom the abov e discussion, w e ha v e f ( u 2 ) = 3 and f ( v 4 ) = 0, so f ( u 2 ) + f ( v 3 ) + f ( w 3 ) + f ( v 4 ) = 5. If f ( w 3 ) = 0 and f ( v 3 ) = 3, then f ( u 2 ) + f ( v 3 ) + f ( w 3 ) + f ( v 4 ) ≥ 5. Similarly , w e obtain f ( u 1 ) + f ( v 1 ) + f ( w 1 ) ≥ 3 and f ( v 2 ) + f ( w 2 ) ≥ 2. Hence γ dI ( T 1 ) ≥ 10. On the other hand, let f ( w i ) = 2 for i = 1 , 2 , 3, f ( v i ) = 0 for i = 1 , 2 , 3 , 4, f ( u 1 ) = 1, and f ( u 2 ) = 3. It is straightforw ard to v erify that this is a double Italian dominating function with w eigh t 10. Thus γ dI ( T 1 ) = 10. Similarly , b y Lemma 2.6 , there exists a γ dR ( T 1 )-function g such that either g ( w i ) = 2 and g ( v i ) = 0, or g ( w i ) = 0 and g ( v i ) = 3, for i = 1 , 2 , 3, and either g ( v 4 ) = 2 and g ( u 2 ) = 0, or g ( v 4 ) = 0 and g ( u 2 ) = 3. If g ( w 3 ) = 2 and g ( v 3 ) = 0, then by the definition of a double Roman dominating function, g ( u 2 ) ≥ 2. F rom the abov e discussion, w e hav e g ( u 2 ) = 3 and g ( v 4 ) = 0, so g ( u 2 ) + g ( v 3 ) + 5 g ( w 3 ) + g ( v 4 ) = 5. If g ( w 3 ) = 0 and g ( v 3 ) = 3, then g ( u 2 ) + g ( v 3 ) + g ( w 3 ) + g ( v 4 ) ≥ 5. Similarly , if g ( w i ) = 2 and g ( v i ) = 0, for i = 1 or 2, then b y the definition of a double Roman dominating function, g ( u 1 ) ≥ 2, and so g ( u 1 ) + g ( v 1 ) + g ( w 1 ) + g ( v 2 ) + g ( w 2 ) ≥ 6. If g ( w i ) = 0 and g ( v i ) = 3 for i = 1 , 2, then g ( u 1 ) + g ( v 1 ) + g ( w 1 ) + g ( v 2 ) + g ( w 2 ) ≥ 6. Hence γ dR ( T 1 ) ≥ 11. On the other hand, let g ( w i ) = 2 for i = 1 , 2 , 3, g ( v i ) = 0 for i = 1 , 2 , 3 , 4, g ( u 1 ) = 2, and g ( u 2 ) = 3. It is straightforw ard to verify that this is a double Roman dominating function with w eigh t 11. Thus γ dR ( T 1 ) = 11 and γ dI ( T 1 ) < γ dR ( T 1 ). □ By theorem 3.1 , w e conclude that the statement γ dI ( T ) = γ dR ( T ) for any tree T is incorrect. Let u 1 and u 2 b e the t w o cen tral vertices of the double star S p,q , where p ≥ 2 and q ≥ 2. Let T b e the tree obtained from a double star S p,q b y sub dividing all p endan t edges inciden t to u 1 , and sub dividing at most q p endan t edges inciden t to u 2 . By a similar argumen t, we ha v e γ dI ( T ) < γ dR ( T ). W e now presen t a low er b ound on the double Italian domination num b er of any non-trivial tree T. Theorem 3.2. If T is a non-trivial tr e e, then γ dI ( T ) ≥ 2 γ ( T ) + 1 . Pr o of. Let T b e a non-trivial tree of order n . W e pro ceed by induction on n . Note that if T is the star K 1 ,n − 1 , then γ ( K 1 ,n − 1 ) = 1 and γ dI ( K 1 ,n − 1 ) = 3 = 2 γ ( K 1 ,n − 1 ) + 1. If diam( T ) = 3, then T is a double star S p,q (where 1 ≤ p ≤ q ), and γ ( S p,q ) = 2 and γ dI ( S p,q ) = 5 when p = 1 and γ dI ( S p,q ) = 6 when p ≥ 2. W e ha v e γ dI ( S 1 ,q ) = 2 γ ( S 1 ,q ) + 1 and γ dI ( S p,q ) > 2 γ ( S p,q ) + 1 when p ≥ 2. Hence, w e may assume that the diameter of T is at least 4. This implies that n ≥ 5. Assume that any tree T ′ with order 2 ≤ n ′ < n has γ dI ( T ′ ) ≥ 2 γ ( T ′ ) + 1. W e first sho w that the result holds if T has a strong supp ort vertex. Assume that T has a strong supp ort vertex u adjacen t to a leaf v . Since u is a strong supp ort v ertex, an y γ dI -function of T will assign a 3 to u and 0 to its leaf neighbors. Consider T ′ = T − v . Since u is a supp ort in T ′ , we ha ve γ ( T ′ ) = γ ( T ) and γ dI ( T ′ ) ≤ γ dI ( T ). Moreo v er, T ′ is a non-trivial tree, so w e can apply our inductiv e hypothesis sho wing that γ dI ( T ) ≥ γ dI ( T ′ ) ≥ 2 γ ( T ′ ) + 1 = 2 γ ( T ) + 1. Since the desired result holds for graphs ha ving a strong supp ort v ertex, henceforth w e may assume that every support v ertex is adjacen t to exactly one leaf. Among all lea ves of T , choose r and u to b e tw o leav es suc h that the distance b et w een r and u is the diameter of T . W e root the tree T at v ertex r . Let w b e the unique neighbor of u , and x b e the paren t of w . Note that b y our choice of u , ev ery child of w is a leaf of T . Since T has no strong supp ort v ertices, it follows that d ( w ) = 2. Among all γ dI -functions of T , let f b e one suc h that f ( u ) + f ( w ) is minimized. By lemma 2.6 , w e ha v e either f ( u ) = 2 and f ( w ) = 0 or f ( u ) = 0 and f ( w ) = 3. W e consider the follo wing t w o cases: Case 1. f ( u ) = 2 and f ( w ) = 0. 6 Let T ′ = T − { u, w } , and f ′ b e the restriction of f onto T ′ . Note that f ′ is a double Italian dominating function of T ′ . Then γ dI ( T ′ ) ≤ γ dI ( T ) − 2. Moreov er, an y dominating set of T ′ can b e extended to a dominating set of T by adding w , so γ ( T ) ≤ γ ( T ′ ) + 1. By inductive h yp othesis, we hav e γ dI ( T ) ≥ γ dI ( T ′ ) + 2 ≥ 2 γ ( T ′ ) + 1 + 2 ≥ 2( γ ( T ) − 1) + 3 = 2 γ ( T ) + 1, as desired. Case 2. f ( u ) = 0 and f ( w ) = 3. W e claim that f ( x ) = 0 and d ( x ) = 2. Supp ose that f ( x ) ≥ 1. Let g b e the function suc h that g ( u ) = 2, g ( w ) = 0, g ( x ) = min { f ( x ) + 1 , 3 } , and g ( v ) = f ( v ) for all other v ertices. Note that g is a function with weigh t no more than f , and g ( u ) + g ( w ) < f ( u ) + f ( w ), con tradicting our c hoice of f . Th us, f ( x ) = 0. Supp ose that d ( x ) ≥ 3. As w e ha v e established, ev ery c hild of x is a leaf or a support of a leaf. Since f ( x ) = 0, any leaf neigh b or of x m ust b e assigned 2, and every support v ertex adjacen t to x is assigned 3 under f . If x has one leaf neighbor z , then let g b e the function such that g ( x ) = 3, g ( z ) = g ( w ) = 0, g ( u ) = 2 and g ( v ) = f ( v ) for all other vertices. Note that g is a function with the same weigh t as f , and g ( u ) + g ( w ) < f ( u ) + f ( w ), contradicting our c hoice of f . If x has only t w o or more support v ertices as c hildren, let g b e the function obtained assigning 2 to x , 0 to eac h of the children of x , 2 to their leaf neigh b ors, and g ( v ) = f ( v ) for all other v ertices, then g is a function with w eight no more than f , and g ( u ) + g ( w ) < f ( u ) + f ( w ), con tradicting our choice of f . Thus, d ( x ) = 2. Consider T ′ = T − { u, w , x } . Since diam( T ) ≥ 4, T ′ is a non-trivial tree. F urther, since any γ -set of T ′ can b e extended to a dominating set of T by adding the vertex w , w e ha v e γ ( T ) ≤ γ ( T ′ ) + 1. Let f ′ b e the restriction of f on to T ′ . Recall that f ( w ) = 3 and f ( u ) = f ( x ) = 0, then f ′ is a double Italian dominating function of T ′ . Thus, γ dI ( T ′ ) ≤ γ dI ( T ) − 3. By inductiv e h yp othesis, w e hav e γ dI ( T ) ≥ γ dI ( T ′ ) + 3 ≥ 2 γ ( T ′ ) + 1 + 3 ≥ 2( γ ( T ) − 1) + 4 = 2 γ ( T ) + 2. The result follows. □ W e next c haracterize the class of trees T for whic h γ dI ( T ) = 2 γ ( T ) + 1. Theorem 3.3. If T is a non-trivial tr e e, then γ dI ( T ) = 2 γ ( T ) + 1 if and only if T is a wounde d spider. Pr o of. Let T b e a wounded spider and u b e the head v ertex. Let S = { v : d ( u, v ) = 2 } b e the set of fo ot v ertices. Since ev ery leaf or its supp ort vertex must b e in any dominating set, it is clearly that S ∪ { u } forms a γ -set for T . Also, if V 0 = V − S − { u } , V 2 = S , and V 3 = { u } , then f = ( V 0 , V 2 , V 3 ) is an double Italian dominating function with f ( V ) = 2 γ ( T ) + 1. Therefore, f is a γ dI -function. No w we will prov e that if γ dI ( T ) = 2 γ ( T ) + 1, then T is a w ounded s p ider. Supp ose to the con trary , and let T b e a smallest counterexample, we ha v e γ dI ( T ) = 2 γ ( T ) + 1 and T is not a wounded spider. By the pro of of Theorem 3.2 , if T is the star K 1 ,n − 1 , then γ dI ( K 1 ,n − 1 ) = 2 γ ( K 1 ,n − 1 ) + 1. If diam( T ) = 3, then T is the double star S p,q (where 1 ≤ p ≤ q ). 7 W e hav e γ dI ( S 1 ,q ) = 2 γ ( S 1 ,q ) + 1 and γ dI ( S p,q ) > 2 γ ( S p,q ) + 1 when p ≥ 2. K 1 ,n − 1 and S 1 ,q are w ounded spiders. Hence, we assume that the diameter of T is at least 4. Assume that T has a strong supp ort vertex u adjacent to a leaf v . Since u has at least tw o leaf neigh b ors, an y γ dI -function of T will assign a 3 to u and 0 to its leaf neighbors. Consider T ′ = T − v . Since u is a supp ort in T ′ , we ha v e γ ( T ′ ) = γ ( T ) and γ dI ( T ′ ) ≤ γ dI ( T ). Moreo v er, T ′ is a non-trivial tree, by theorem 3.2 , γ dI ( T ) ≥ γ dI ( T ′ ) ≥ 2 γ ( T ′ ) + 1 = 2 γ ( T ) + 1. Note that γ dR ( T ) = 2 γ ( T ) + 1, whic h implies γ dR ( T ′ ) = 2 γ ( T ′ ) + 1. By the minimality of T , T ′ ia a w ounded spider. If u is the center of T ′ , then T is also a w ounded spider, con tradicting the assumption that T is a counterexample. Hence, u is a sub division vertex of T ′ and the degree of the center of T ′ is at least 3. Note that every v ertex of T is either a leaf or a supp ort v ertex. By lemma 2.6 , there exists a γ dI -function f of T , no v ertex is assigned the v alue 1. Note that f is also a γ dR -function of T . Then γ dI ( T ) = γ dR ( T ). By prop osition 2.3 and 2.4 , we ha v e γ dI ( T ) > 2 γ ( T ) + 1, a con tradiction. Hence, we may assume that every supp ort v ertex is adjacen t to exactly one leaf. Among all lea ves of T , choose r and u to b e tw o leav es suc h that the distance b et w een r and u is the diameter of T . W e root the tree T at v ertex r . Let w b e the unique neighbor of u , and x b e the paren t of w . Note that b y our choice of u , ev ery child of w is a leaf of T . Since T has no strong supp ort v ertices, it follows that d ( w ) = 2. Among all γ dI -functions of T , let f b e one suc h that f ( u ) + f ( w ) is minimized. By lemma 2.6 , w e hav e either f ( u ) = 2 and f ( w ) = 0 or f ( u ) = 0 and f ( w ) = 3. By the case 2 of theorem 3.2 , if f ( u ) = 0 and f ( w ) = 3, then γ dI ( T ) ≥ 2 γ ( T ) + 2, contradicting that γ dI ( T ) = 2 γ ( T ) + 1. Hence, f ( u ) = 2 and f ( w ) = 0. Let T ′ = T − { u, w } , and f ′ b e the restriction of f onto T ′ . Note that f ′ is a double Italian dominating function of T ′ . Then γ dI ( T ′ ) ≤ γ dI ( T ) − 2. Moreov er, an y dominating set of T ′ can b e extended to a dominating set of T b y adding w , so γ ( T ) ≤ γ ( T ′ ) + 1. By theorem 3.2 , we hav e γ dI ( T ) ≥ γ dI ( T ′ ) + 2 ≥ 2 γ ( T ′ ) + 1 + 2 ≥ 2( γ ( T ) − 1) + 3 = 2 γ ( T ) + 1. Note that γ dI ( T ) = 2 γ ( T ) + 1, which implies that γ dI ( T ′ ) = 2 γ ( T ′ ) + 1 and γ ( T ) = γ ( T ′ ) + 1. By the minimality of T , T ′ is a w ounded spider. If x is the head v ertex of T ′ , then T is also a w ounded spider, contradicting the assumption that T is a coun terexample. If x is a subdivision v ertex of T ′ and the degree of head v ertex of T ′ is at least 3, then ev ery v ertex of T is either a leaf or a supp ort vertex. By lemma 2.6 , there exists a γ dI -function f of T , no v ertex is assigned the v alue 1. Note that f is also a γ dR -function of T , and γ dI ( T ) = γ dR ( T ). By prop osition 2.3 and 2.4 , we ha ve γ dI ( T ) > 2 γ ( T ) + 1, a con tradiction. If x is a fo ot v ertex of T ′ , then let S b e the set of fo ot v ertices of T ′ , and y b e the head v ertex of T ′ . Since ev ery leaf or its support v ertex must b e in any dominating set, we ha v e D ′ = S ∪ { y } forms a γ -set for T ′ , where x ∈ D ′ . Let D = D ′ ∪ { w } − { x } . Then D is a γ -set for T . Th us, γ ( T ) = γ ( T ′ ), con tradicting that γ ( T ) = γ ( T ′ ) + 1. If x is a leaf neigh b or of the head vertex y , and y has only one leaf neigh b or. Note that γ ( T ) = γ ( T ′ ), con tradicting that γ ( T ) = γ ( T ′ ) + 1. Hence, x is a leaf neigh b or of y , and y 8 has at least t wo leaf neighbors. By lemma 2.6 , w e hav e either f ( y ) = 0 or f ( y ) = 3 in T . If f ( y ) = 0, then an y leaf neigh bor of y m ust b e assigned 2. Recall f ( u ) = 2 and f ( w ) = 0, implying that f ( x ) = 2. Let g b e the function obtained assigning 3 to y , 0 to each of the leaf neigh b or of y , 1 to x , and g ( v ) = f ( v ) for all other v ertices. Then g is a function with w eight no more than f and g ( y ) = 3. Without loss of generalit y , let f ( y ) = 3. Note that f ( u ) = 2 and f ( w ) = 0, we ha ve f ( x ) = 1. Consider T ′′ = T − { u, w , x } . Since diam( T ) ≥ 4, T ′′ is a non-trivial tree. F urther, since an y γ -set of T ′′ can b e extended to a dominating set of T by adding the vertex w , γ ( T ) ≤ γ ( T ′′ ) + 1. Let f ′′ b e the restriction of f on to T ′′ . Recall f ( y ) = 3, then f ′′ is a double Italian dominating function of T ′′ . Th us, γ dI ( T ′′ ) ≤ γ dI ( T ) − 3. By theorem 3.2 , w e hav e γ dI ( T ) ≥ γ dI ( T ′′ ) + 3 ≥ 2 γ ( T ′′ ) + 1 + 3 ≥ 2( γ ( T ) − 1) + 4 = 2 γ ( T ) + 2, a contradiction. Therefore, T is a wounded spider. The result follows. □ Ac kno wledgeme n ts This w ork is supp orted b y the National Natural Science F oundation of China under grant n um b er 12571381. References [1] H.A. Ahangar, M. Chellali, S.M. Sheikholeslami, On the double Roman domination in graphs, Discrete Appl. 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