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Double Roman domination in trees
Information Processing Letters 134 (2018) 31–34
Contents lists available at ScienceDirect
Information Processing Letters www.elsevier.com/locate/ipl
Double Roman domination in trees ✩ Xiujun Zhang a , Zepeng Li b , Huiqin Jiang a , Zehui Shao c,∗ a b c
School of Information Science and Engineering, Chengdu University, Chengdu 610106, China School of Information Science and Engineering, Lanzhou University, Lanzhou 730000, China Research Institute of Intelligence Software, Guangzhou University, Guangzhou 510006, China
a r t i c l e
i n f o
Article history: Received 6 September 2016 Accepted 19 January 2018 Available online 7 February 2018 Communicated by Jinhui Xu Keywords: Double Roman domination number Roman domination Domination number Tree Combinatorial problems
a b s t r a c t A subset S of the vertex set of a graph G is a dominating set if every vertex of G not in S has at least one neighbor in S. The domination number γ (G ) is defined to be the minimum cardinality among all dominating set of G. A Roman dominating function on a graph G is a function f : V (G ) → {0, 1, 2} satisfying the condition that every vertex u for which f (u ) = 0 is adjacent to at least one vertex v for which f ( v ) = 2. The weight of a Roman dominating function f is the value f ( V (G )) = u ∈ V (G ) f (u ). The minimum weight of a Roman dominating function on a graph G is called the Roman domination number γ R (G ) of G. A double Roman dominating function on a graph G is a function f : V (G ) → {0, 1, 2, 3} satisfying the condition that every vertex u for which f (u ) = 0 is adjacent to at least one vertex v for which f ( v ) = 3 or two vertices v 1 and v 2 for which f ( v 1 ) = f ( v 2 ) = 2, and every vertex u for which f (u ) = 1 is adjacent to at least one vertex v for which f ( v ) ≥ 2. The weight of a double Roman dominating function f is the value f ( V (G )) = u ∈ V (G ) f (u ). The minimum weight of a double Roman dominating function on a graph G is called the double Roman domination number γdR (G ) of G. Beeler et al. (2016) [6] showed that 2γ (G ) ≤ γdR (G ) ≤ 3γ (G ) and showed that 2γ ( T ) + 1 ≤ γdR ( T ) ≤ 3γ ( T ) for any nontrivial tree T and posed a problem that if it is possible to construct a polynomial algorithm for computing the value of γdR ( T ) for any tree T . In this paper, we answer this problem by giving a linear time algorithm to compute the value of γdR ( T ) for any tree T . Moreover, we give characterizations of trees with 2γ ( T ) + 1 = γdR ( T ) and γdR ( T ) + 1 = 2γ R ( T ). © 2018 Elsevier B.V. All rights reserved.
1. Introduction In this paper, we shall only consider graphs without multiple edges or loops. For notation and graph theory terminology we general follow [1]. Let G be a graph,
✩ This work was supported by the National Natural Science Foundation of China under the grants 61127005, 61309015, China Postdoctoral Science Foundation under grant 2014M560851, and 973 Program of China 2013CB329600. Corresponding author. E-mail address: [email protected] (Z. Shao).
*
https://doi.org/10.1016/j.ipl.2018.01.004 0020-0190/© 2018 Elsevier B.V. All rights reserved.
v ∈ V (G ), the neighborhood of v in G is denoted by N ( v ). That is to say N ( v ) = {u |uv ∈ E (G ), u ∈ V (G )}. The closed neighborhood N [ v ] of v in G is defined as N [ v ] = { v } ∪ N ( v ). The degree of v in G, denoted by d G ( v ), is the cardinality of its open neighborhood in G. The distance of two vertices u and v in G, denoted by d G (u , v ), is the length of a shortest path between u and v. A vertex of degree one is called a leaf. A graph is trivial if it has a single vertex. Denote by P n the path on n vertices. Let T be a tree. A vertex with exactly one neighbor is called a leaf and its neighbor is a support vertex. A support vertex with two or more leaf neighbors is called a strong support vertex.
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X. Zhang et al. / Information Processing Letters 134 (2018) 31–34
A set S ⊆ V (G ) in a graph G is called a dominating set if N [ S ] = V (G ). The domination number γ (G ) equals the minimum cardinality of a dominating set in G. A Roman dominating function (RDF) on a graph G is a function f : V (G ) → {0, 1, 2} satisfying the condition that every vertex u for which f (u ) = 0 is adjacent to at least one vertex v for which f ( v ) = 2. The weight of a Roman dominating function f is the value f ( V (G )) = u ∈ V (G ) f (u ). The minimum weight of a Roman dominating function on a graph G is called the Roman domination number γ R (G ) of G. Domination and Roman domination and their variations have received considerable attention [2–5]. As a variation of Roman domination, the concept of double Roman domination was proposed. A double Roman dominating function (DRDF) on a graph G is a function f : V (G ) → {0, 1, 2, 3} satisfying the condition that every vertex u for which f (u ) = 0 is adjacent to at least one vertex v for which f ( v ) = 3 or two vertices v 1 and v 2 for which f ( v 1 ) = f ( v 2 ) = 2, and every vertex u for which f (u ) = 1 is adjacent to at least one vertex v for which f ( v ) ≥ 2. The weight ω( f ) of a double Roman dominating function f is the value ω( f ) = u∈ V (G ) f (u ). The minimum weight of a double Roman dominating function on a graph G is called the double Roman domination number of G. We denote by w S ( f ) the weight of a double Roman dominating func tion f in S ⊆ V (G ), i.e. ω S ( f ) = x∈ S f (x). We say that a function f of G is a γdR -function if it is a DRDF and ω( f ) = γdR (G ). For an RDF f of G, let ( V 0 , V 1 , V 2 ) be the ordered partition of V (G ) induce by f such that V i = {x : f (x) = i } for i = 0, 1, 2. Note that there exists a 1-1 correspondence between the function f and the partition ( V 0 , V 1 , V 2 ) of V , we write f = ( V 0 , V 1 , V 2 ). Similarly, for a DRDF f of G, let ( V 0 , V 1 , V 2 , V 3 ) be the ordered partition of V (G ) induce by f such that V i = {x : f (x) = i } for i = 0, 1, 2, 3. Note that there exists a 1-1 correspondence between the function f and the partition ( V 0 , V 1 , V 2 , V 3 ) of V , we write f = ( V 0 , V 1 , V 2 , V 3 ). Lemma 1. [6] For any graph G, there exists a γdR -function of G such that no vertex needs to be assigned the value 1. Beeler et al. [6] showed that 2γ (G ) ≤ γdR (G ) ≤ 3γ (G ) and defined a graph G to be double Roman if γdR (G ) = 3γ (G ). Moreover, they proposed the following problem. Problem 1. Characterize the double Roman graphs. In particular, characterize the double Roman trees. Especially, for any non-trivial tree T , Beeler et al. [6] proved that 2γ ( T ) + 1 ≤ γdR ( T ) ≤ 3γ ( T ) and every value in this range is realizable for trees.
a linear algorithm to compute the double Roman domination number of trees. Moreover, we give a characterization of trees with 2γ ( T ) + 1 = γdR ( T ) and γdR ( T ) + 1 = 2γ R ( T ). 2. A characterization of trees T with γd R ( T ) = 2γ ( T ) + 1 and γd R ( T ) + 1 = 2γ R ( T ) For any integers r ≥ 1 and t ≥ 0, let F r ,t be a tree formed by joining r edges and t vertex-disjoint paths of length 2 as pendent paths to a single vertex u, which is called the center of F r ,t . Note that F r ,t is a (r + t )-star with t edges subdivided once. We say F r ,t is a wounded spider if r ≥ 1 and t ≥ 0 and F r ,t is a healthy spider if r = 0 and t ≥ 2. The center vertex of F r ,t is also called the head vertex and the vertex at distance two from the head vertex is called the foot vertex. The vertex adjacent to the head vertex is called the wounding vertex. Theorem 1. For any integers r ≥ 1 and t ≥ 0, we have γdR ( F r ,t ) = 2γ ( F r ,t ) + 1 = 2t + 3. Proof. Let V ( F r ,t ) = {u , v i , w j : 1 ≤ i ≤ r + t , 1 ≤ j ≤ t } and E ( F r ,t ) = {uv i , v j w j : 1 ≤ i ≤ r + t , 1 ≤ j ≤ t }. Since any dominating set of F r ,t contains at least one vertex of each edge in {uv t +1 , v j w j : 1 ≤ j ≤ t }, we have γ ( F r ,t ) ≥ t + 1. On the other hand, since r ≥ 1 and {u , w j : 1 ≤ j ≤ t } is a dominating set of F r ,t , we have γ ( F r ,t ) ≤ t + 1. So γ ( F r ,t ) = t + 1. Now we prove that γdR ( F r ,t ) = 2t + 3. Let f be a DRDF of F r ,t . Then for any j ∈ {1, 2 . . . , t }, f ( v j ) + f ( w j ) ≥ 2 and equality holds if and only if f ( v j ) = 0 and f ( w j ) = 2. Moreover, f (u ) + f ( v t +1 ) ≥ 2 and equality holds if and only if f (u ) = 0 and f ( v t +1 ) = 2. Since f is a DRDF of F r ,t , we know that f ( v j ) + f ( w j ) = 2 and f (u ) + f ( v t +1 ) = 2 do not appear simultaneously. Hence, ω( f ) = u ∈ V (G ) f (u ) ≥ 2t + 3. On the other hand, let g (u ) = 3, g ( v i ) = 0 and g ( w j ) = 2, where i = 1, 2, . . . , r + t and j = 1, 2, . . . , t. It can be checked that g is a DRDF of F r ,t with weight 2t + 3. Therefore, γdR ( F r ,t ) = 2t + 3 = 2γ ( F r ,t ) + 1. 2 Theorem 2. Let T be a nontrivial tree. Then 2γ ( T ) + 1 if and only if T is a wounded spider.
γdR ( T ) =
Proof. The sufficiency follows from Theorem 1. Now we will prove that if γdR ( T ) = 2γ ( T ) + 1, there exist two integers r ≥ 1 and t ≥ 0 such that T = F r ,t . Suppose that this is not true and let T be a smallest counterexample to the theorem. Then γdR ( T ) = 2γ ( T ) + 1 and T = F r ,t for any integers r ≥ 1 and t ≥ 0. Claim 1. T has no strong support vertex.
Lemma 2. [6] For any non-trivial tree T , 2γ ( T ) + 1 ≤ γdR ( T ) ≤ 3γ ( T ). Beeler et al. [6] also asked that if it is possible to construct a polynomial algorithm to compute the double Roman domination number of trees. In this paper, we give
Proof of Claim 1. Suppose that T has a strong support vertex x. Let y , z be two leaves of T adjacent to x and T = T − y. Since x is a support vertex in T , we have γ ( T ) = γ ( T ) and γdR ( T ) ≥ γdR ( T ). Hence, by Lemma 2, γdR ( T ) ≥ γdR ( T ) ≥ 2γ ( T ) + 1 = 2γ ( T ) + 1. Note that
X. Zhang et al. / Information Processing Letters 134 (2018) 31–34
γdR ( T ) = 2γ ( T ) + 1, so γdR ( T ) = 2γ ( T ) + 1. By the mini-
mality of T , there exist two integers r ≥ 1 and t ≥ 0 such that T = F r ,t . If x is the center of T , then T = F r +1,t , contradicting the assumption that T is a counterexample. Hence, x is a 2-vertex of T and the degree of the center of T is at least 3. Then we have γdR ( T ) > γdR ( T ) = 2t + 3 by Theorem 1. Note that γ ( T ) = γ ( T ) = t , we have γdR ( T ) > 2γ ( T ) + 1, a contradiction. Therefore, Claim 1 is true. 2 By Claim 1, T has a support vertex x with d T (x) = 2. We denote by y the leaf neighbor of x and by z the nonleaf neighbor of x. Let f be a γdR -function of T and consider the following two cases: Case 1. f ( z) ≤ 1. Let T = T −{x, y , z} and f be the restriction of f to T . Then f is a DRDF of T . Since f is a γdR -function of T and f ( z) ≤ 1, we have f (x) + f ( y ) + f ( z) ≥ 3. Hence,
γdR ( T ) ≤ γdR ( T ) − 3
(1)
Since any dominating set of T together with x is a dominating set of T , we have
γ (T ) ≤ γ (T ) + 1
(2)
Note that T = F r ,t for any integers r ≥ 1 and t ≥ 0, we can obtain that T is a forest with at least one nontrivial component. Since γdR ( T 1 ) = 2γ ( T 1 ) for any trivial tree T 1 and γdR ( T 2 ) ≥ 2γ ( T 2 ) + 1 for any nontrivial tree T 2 , we have
γdR ( T ) ≥ 2γ ( T ) + 1
(3)
By Formulae (1), (2) and (3), we have γdR ( T ) ≥ γdR ( T ) + 3 ≥ 2γ ( T ) + 1 + 3 ≥ 2(γ ( T ) − 1) + 4 = 2γ ( T ) + 2, contradicting the assumption that γdR ( T ) = 2γ ( T ) + 1. Case 2. f ( z) ≥ 2. Let T = T − {x, y } and f be the restriction of f to γdR -function of T and y is a leaf of T , we have f (x) + f ( y ) ≥ 2. Hence, γdR ( T ) ≤ γdR ( T ) − 2. Since γ ( T ) ≤ γ ( T ) + 1 and γdR ( T ) ≥ 2γ ( T ) + 1 by Lemma 2, we have γdR ( T ) ≥ γdR ( T ) + 2 ≥ 2γ ( T ) + 1 + 2 ≥ 2(γ ( T ) − 1) + 3 = 2γ ( T ) + 1. Note that γdR ( T ) = 2γ ( T ) + 1, so γdR ( T ) = 2γ ( T ) + 1. By the minimality of T , there exist two integers r ≥ 1 and t ≥ 0 such that T = F r ,t . By the symmetry of T , there are four possible cases of z in T . T . Then f is a DRDF of T . Since f is a
Subcase 2.1. z is a wounding vertex of T .
In this case, we can obtain that γ ( T ) = t + 2 and γdR ( T ) = 2t + 6, a contradiction.
diction. Subcase 2.2. z is the center of T . In this case, we have T = F r ,t +1 , contradicting the assumption that T is a counterexample. Subcase 2.3. z is a 2-vertex of T but not the center.
γ (T ) + 1 =
Subcase 2.4. z is a foot vertex of T . In this case, we can obtain that γ ( T ) = γ ( T ) = t + 1 and γdR ( T ) = 2t + 4, a contradiction. 2 Proposition 1. [6] i) Let G be a graph and f = ( V 0 , V 1 , V 2 ) a γ R (G )-function. Then γdR (G ) ≤ 2| V 1 | + 3| V 2 |. ii) For any graph G, γdR (G ) ≤ 2γ R (G ) with equality if and only if G = K n . Theorem 3. Let T be a nontrivial tree, then T with 1 = 2γ R ( T ) if and only if T is a wounded spider.
γdR ( T ) +
Proof. ⇐: Since T is a wounded spider, we can assume T = F r ,t for some r ≥ 1. It can be verified that γdR ( T ) = 2t + 3 and γ R ( T ) = t + 2. Therefore, γdR ( T ) + 1 = 2γ R ( T ). ⇒: Let f = ( V 0 , V 1 , V 2 ) be a γ R (G )-function with minimum | V 1 |. Then we have | V 1 | is independent. Together with Proposition 1, we have γdR (G ) ≤ 2| V 1 | + 3| V 2 | ≤ 2| V 1 | + 4| V 2 | = 2γ R (G ) = γdR (G ) + 1. Then we have | V 2 | ≤ 1. If | V 2 | = 0, we have | V 0 | = 0 and thus γ R (G ) = n = | V 1 |. Since V 1 is independent, so it is impossible. If | V 2 | = 1, we have γdR (G ) = 2| V 1 | + 3| V 2 | < 2| V 1 | + 4| V 2 | = 2γ R (G ) = γdR (G ) + 1. Let V 2 = { v }, V 0 = N ( v ) and V 1 = V (G ) − V 0 − V 2 . Since V 1 is independent, we have T is a wounded spider or a healthy spider. Note that if T is a healthy spider, we have γdR ( T ) + 2 = 2γ R ( T ), a contradiction. Therefore, T is a wounded spider. 2 3. A linear algorithm for double Roman domination in trees Let u be a specific vertex of G. Note that a minimum weight of a DRDF on G satisfies that f (u ) ∈ {0, 2, 3} by Lemma 1. So it is useful to consider the following three domination problems. 0 γdR (G , u ) = min{ω( f ) : f is a DRDF of G and f (u ) = 0}. 2 γdR (G , u ) = min{ω( f ) : f is a DRDF of G and f (u ) = 2}. 3 γdR (G , u ) = min{ω( f ) : f is a DRDF of G and f (u ) = 3}.
Lemma 3. For any graph G with a specific vertex u, we have 0 3 2 γdR (G , u ) = min{γdR (G , u ), γdR (G , u ), γdR (G , u )}. 00 γdR (G , u ) = min{ω( f ) : f is a DRDF of G − u }. 02 γdR (G , u ) = min{ω( f ) : f is a DRDF of G + u w and
f (u ) = 0, f ( w ) = 2}.
In this case, if r = 1, then it can be checked that
γ ( T ) = γ ( T ) = t + 1 and γdR ( T ) = 2t + 5, contradicting γdR ( T ) = 2γ ( T ) + 1; if r ≥ 2, then we can obtain that γ ( T ) = γ ( T ) + 1 = t + 2 and γdR ( T ) = 2t + 6, a contra-
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00 0 γdR (G , u ) ≤ γdR (G , u ), since a DRDF f of G 00 (G , u ) ≤ with f (u ) = 0 is also a DRDF of G − u, and γdR 02 γdR (G , u ), since a DRDF f of G + u w with f (u ) = 0 and
Note that
f ( w ) = 2 is also a DRDF of G − {u , w }.
Theorem 4. Suppose G and H are graphs with specific vertices u and v, respectively. Let I be the graph with the specific vertex u, which is obtained from the disjoint union of G and H by joining a new edge uv. Then the following statements hold.
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(i) (ii) (iii) (iv) (v)
X. Zhang et al. / Information Processing Letters 134 (2018) 31–34 0 0 0 02 γdR ( I , u ) = min{γdR (G , u ) + γdR ( H , v ), γdR (G , u ) + 00 3 2 γdR ( H , v ) − 2, γdR (G , u ) + γdR ( H , v )}; 02 2 2 2 γdR ( I , u ) = γdR (G , u ) + min{γdR ( H , v ), γdR ( H , v ), 3 γdR ( H , v )}; 3 3 00 2 γdR ( I , u ) = γdR (G , u ) + min{γdR ( H , v ), γdR ( H , v ), 3 γdR ( H , v )}; 00 00 00 γdR ( I , u ) = γdR (G , u ) + γdR ( H ) = γdR (G , u ) + 0 3 2 min{γdR ( H , v ), γdR ( H , v ), γdR ( H , v )}; 02 02 0 00 γdR ( I , u ) = min{γdR (G , u ) + γdR ( H , v ), γdR (G , u ) + 00 3 2 γdR ( H , v ), γdR (G , u ) + γdR ( H , v )}.
Proof. (i) It follows from the fact that f is a DRDF of I with f (u ) = 0 if and only if f = f ∪ f
, where f
is a DRDF of G with f (u ) = 0 and f
is a DRDF of H with f
( v ) = 0, f is a DRDF of G + u w with f (u ) = 0, f ( w ) = 2 and f
is a DRDF of H with f
( v ) = 2, or f is a DRDF of G − u and f
is a DRDF of H with f
( v ) = 3. (ii) It follows from the fact that f is a DRDF of I with f (u ) = 2 if and only if f = f ∪ f
, where f is a DRDF of G with f (u ) = 2, f
is a DRDF of H + v w with f
( v ) = 0, f
( w ) = 2 or a DRDF of H with f
( v ) ∈ {2, 3}. (iii) It follows from the fact that f is a DRDF of I with f (u ) = 3 if and only if f = f ∪ f
, where f is a DRDF of G with f (u ) = 3 and f
is a DRDF of H − v or a DRDF of H with f
( v ) ∈ {2, 3}. (iv) It follows from the fact that f is a DRDF of I − u if and only if f = f ∪ f
, where f is a DRDF of G − u and f
is a DRDF of H . (v) It follows from the fact that f is a DRDF of I + u w with f (u ) = 0, f ( w ) = 2 if and only if f = f ∪ f
, where f is a DRDF of G + u w with f (u ) = 0, f ( w ) = 2 and f
is a DRDF of H with f
( v ) = 0, or f is a DRDF of G − u and f
is a DRDF of H with f
( v ) ∈ {2, 3}. 2 Lemma 3 and Theorem 4 give the following dynamic programming algorithm for the double Roman domination problem in trees.
Algorithm DoubleRomanDomination Input: A tree T with a tree ordering [ v 1 , v 2 , · · · , v n ]. Output: the double Roman domination number γdR ( T ) of T . begin for i = 1 to n do γ 00 ( v i ) ← 0; γ 0 ( v i ) ← ∞; γ 2 ( v i ) ← 2; γ 3 ( v i ) ← 3; γ 02 ( v i ) ← ∞; end for for i = 1 to n − 1 do let v j be the parent of v i ; γ 0 ( v j ) ← min{γ 0 ( v j ) + γ 0 ( v i ), γ 02 ( v j ) + γ 2 ( v i ) − 2, γ 00 ( v j ) + γ 3 ( v i )}; γ 2 ( v j ) ← γ 2 ( v j ) + min{γ 02 ( v i ), γ 2 ( v i ), γ 3 ( v i )}; γ 3 ( v j ) ← γ 3 ( v j ) + min{γ 00 ( v i ), γ 2 ( v i ), γ 3 ( v i )}; γ 00 ( v j ) ← γ 00 ( v j ) + min{γ 0 ( v i ), γ 2 ( v i ), γ 3 ( v i )}; γ 02 ( v j ) ← min{γ 02 ( v j ) + γ 0 ( v i ), γ 00 ( v j ) + γ 2 ( v i ), γ 00 ( v j ) + γ 3 ( v i )}; end for return min{γ 0 ( v n ), γ 2 ( v n ), γ 3 ( v n )}; end. References [1] T.W. Haynes, S.T. Hedetniemi, P.J. Slater, Fundamentals of Domination in Graphs, Marcel Dekker, New York, 1998. [2] M.R. Garey, D.S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness, W. H. Freeman and Co., San Francisco, Calif, 1979. [3] M.A. Henning, A characterization of Roman trees, Discuss. Math., Graph Theory 22 (2002) 325–334. [4] O. Ore, Theory of Graphs, American Mathematical Society, Providence, RI, 1967. [5] Z. Shao, S. Klavžar, Z. Li, P. Wu, J. Xu, On the signed Roman k-domination: complexity and thin torus graphs, Discrete Appl. Math. 233 (2017) 175–186. [6] R.A. Beeler, T.W. Haynes, S.T. Hedetniemi, Double Roman domination, Discrete Appl. Math. 211 (2016) 23–29.
Contents lists available at ScienceDirect
Information Processing Letters www.elsevier.com/locate/ipl
Double Roman domination in trees ✩ Xiujun Zhang a , Zepeng Li b , Huiqin Jiang a , Zehui Shao c,∗ a b c
School of Information Science and Engineering, Chengdu University, Chengdu 610106, China School of Information Science and Engineering, Lanzhou University, Lanzhou 730000, China Research Institute of Intelligence Software, Guangzhou University, Guangzhou 510006, China
a r t i c l e
i n f o
Article history: Received 6 September 2016 Accepted 19 January 2018 Available online 7 February 2018 Communicated by Jinhui Xu Keywords: Double Roman domination number Roman domination Domination number Tree Combinatorial problems
a b s t r a c t A subset S of the vertex set of a graph G is a dominating set if every vertex of G not in S has at least one neighbor in S. The domination number γ (G ) is defined to be the minimum cardinality among all dominating set of G. A Roman dominating function on a graph G is a function f : V (G ) → {0, 1, 2} satisfying the condition that every vertex u for which f (u ) = 0 is adjacent to at least one vertex v for which f ( v ) = 2. The weight of a Roman dominating function f is the value f ( V (G )) = u ∈ V (G ) f (u ). The minimum weight of a Roman dominating function on a graph G is called the Roman domination number γ R (G ) of G. A double Roman dominating function on a graph G is a function f : V (G ) → {0, 1, 2, 3} satisfying the condition that every vertex u for which f (u ) = 0 is adjacent to at least one vertex v for which f ( v ) = 3 or two vertices v 1 and v 2 for which f ( v 1 ) = f ( v 2 ) = 2, and every vertex u for which f (u ) = 1 is adjacent to at least one vertex v for which f ( v ) ≥ 2. The weight of a double Roman dominating function f is the value f ( V (G )) = u ∈ V (G ) f (u ). The minimum weight of a double Roman dominating function on a graph G is called the double Roman domination number γdR (G ) of G. Beeler et al. (2016) [6] showed that 2γ (G ) ≤ γdR (G ) ≤ 3γ (G ) and showed that 2γ ( T ) + 1 ≤ γdR ( T ) ≤ 3γ ( T ) for any nontrivial tree T and posed a problem that if it is possible to construct a polynomial algorithm for computing the value of γdR ( T ) for any tree T . In this paper, we answer this problem by giving a linear time algorithm to compute the value of γdR ( T ) for any tree T . Moreover, we give characterizations of trees with 2γ ( T ) + 1 = γdR ( T ) and γdR ( T ) + 1 = 2γ R ( T ). © 2018 Elsevier B.V. All rights reserved.
1. Introduction In this paper, we shall only consider graphs without multiple edges or loops. For notation and graph theory terminology we general follow [1]. Let G be a graph,
✩ This work was supported by the National Natural Science Foundation of China under the grants 61127005, 61309015, China Postdoctoral Science Foundation under grant 2014M560851, and 973 Program of China 2013CB329600. Corresponding author. E-mail address: [email protected] (Z. Shao).
*
https://doi.org/10.1016/j.ipl.2018.01.004 0020-0190/© 2018 Elsevier B.V. All rights reserved.
v ∈ V (G ), the neighborhood of v in G is denoted by N ( v ). That is to say N ( v ) = {u |uv ∈ E (G ), u ∈ V (G )}. The closed neighborhood N [ v ] of v in G is defined as N [ v ] = { v } ∪ N ( v ). The degree of v in G, denoted by d G ( v ), is the cardinality of its open neighborhood in G. The distance of two vertices u and v in G, denoted by d G (u , v ), is the length of a shortest path between u and v. A vertex of degree one is called a leaf. A graph is trivial if it has a single vertex. Denote by P n the path on n vertices. Let T be a tree. A vertex with exactly one neighbor is called a leaf and its neighbor is a support vertex. A support vertex with two or more leaf neighbors is called a strong support vertex.
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X. Zhang et al. / Information Processing Letters 134 (2018) 31–34
A set S ⊆ V (G ) in a graph G is called a dominating set if N [ S ] = V (G ). The domination number γ (G ) equals the minimum cardinality of a dominating set in G. A Roman dominating function (RDF) on a graph G is a function f : V (G ) → {0, 1, 2} satisfying the condition that every vertex u for which f (u ) = 0 is adjacent to at least one vertex v for which f ( v ) = 2. The weight of a Roman dominating function f is the value f ( V (G )) = u ∈ V (G ) f (u ). The minimum weight of a Roman dominating function on a graph G is called the Roman domination number γ R (G ) of G. Domination and Roman domination and their variations have received considerable attention [2–5]. As a variation of Roman domination, the concept of double Roman domination was proposed. A double Roman dominating function (DRDF) on a graph G is a function f : V (G ) → {0, 1, 2, 3} satisfying the condition that every vertex u for which f (u ) = 0 is adjacent to at least one vertex v for which f ( v ) = 3 or two vertices v 1 and v 2 for which f ( v 1 ) = f ( v 2 ) = 2, and every vertex u for which f (u ) = 1 is adjacent to at least one vertex v for which f ( v ) ≥ 2. The weight ω( f ) of a double Roman dominating function f is the value ω( f ) = u∈ V (G ) f (u ). The minimum weight of a double Roman dominating function on a graph G is called the double Roman domination number of G. We denote by w S ( f ) the weight of a double Roman dominating func tion f in S ⊆ V (G ), i.e. ω S ( f ) = x∈ S f (x). We say that a function f of G is a γdR -function if it is a DRDF and ω( f ) = γdR (G ). For an RDF f of G, let ( V 0 , V 1 , V 2 ) be the ordered partition of V (G ) induce by f such that V i = {x : f (x) = i } for i = 0, 1, 2. Note that there exists a 1-1 correspondence between the function f and the partition ( V 0 , V 1 , V 2 ) of V , we write f = ( V 0 , V 1 , V 2 ). Similarly, for a DRDF f of G, let ( V 0 , V 1 , V 2 , V 3 ) be the ordered partition of V (G ) induce by f such that V i = {x : f (x) = i } for i = 0, 1, 2, 3. Note that there exists a 1-1 correspondence between the function f and the partition ( V 0 , V 1 , V 2 , V 3 ) of V , we write f = ( V 0 , V 1 , V 2 , V 3 ). Lemma 1. [6] For any graph G, there exists a γdR -function of G such that no vertex needs to be assigned the value 1. Beeler et al. [6] showed that 2γ (G ) ≤ γdR (G ) ≤ 3γ (G ) and defined a graph G to be double Roman if γdR (G ) = 3γ (G ). Moreover, they proposed the following problem. Problem 1. Characterize the double Roman graphs. In particular, characterize the double Roman trees. Especially, for any non-trivial tree T , Beeler et al. [6] proved that 2γ ( T ) + 1 ≤ γdR ( T ) ≤ 3γ ( T ) and every value in this range is realizable for trees.
a linear algorithm to compute the double Roman domination number of trees. Moreover, we give a characterization of trees with 2γ ( T ) + 1 = γdR ( T ) and γdR ( T ) + 1 = 2γ R ( T ). 2. A characterization of trees T with γd R ( T ) = 2γ ( T ) + 1 and γd R ( T ) + 1 = 2γ R ( T ) For any integers r ≥ 1 and t ≥ 0, let F r ,t be a tree formed by joining r edges and t vertex-disjoint paths of length 2 as pendent paths to a single vertex u, which is called the center of F r ,t . Note that F r ,t is a (r + t )-star with t edges subdivided once. We say F r ,t is a wounded spider if r ≥ 1 and t ≥ 0 and F r ,t is a healthy spider if r = 0 and t ≥ 2. The center vertex of F r ,t is also called the head vertex and the vertex at distance two from the head vertex is called the foot vertex. The vertex adjacent to the head vertex is called the wounding vertex. Theorem 1. For any integers r ≥ 1 and t ≥ 0, we have γdR ( F r ,t ) = 2γ ( F r ,t ) + 1 = 2t + 3. Proof. Let V ( F r ,t ) = {u , v i , w j : 1 ≤ i ≤ r + t , 1 ≤ j ≤ t } and E ( F r ,t ) = {uv i , v j w j : 1 ≤ i ≤ r + t , 1 ≤ j ≤ t }. Since any dominating set of F r ,t contains at least one vertex of each edge in {uv t +1 , v j w j : 1 ≤ j ≤ t }, we have γ ( F r ,t ) ≥ t + 1. On the other hand, since r ≥ 1 and {u , w j : 1 ≤ j ≤ t } is a dominating set of F r ,t , we have γ ( F r ,t ) ≤ t + 1. So γ ( F r ,t ) = t + 1. Now we prove that γdR ( F r ,t ) = 2t + 3. Let f be a DRDF of F r ,t . Then for any j ∈ {1, 2 . . . , t }, f ( v j ) + f ( w j ) ≥ 2 and equality holds if and only if f ( v j ) = 0 and f ( w j ) = 2. Moreover, f (u ) + f ( v t +1 ) ≥ 2 and equality holds if and only if f (u ) = 0 and f ( v t +1 ) = 2. Since f is a DRDF of F r ,t , we know that f ( v j ) + f ( w j ) = 2 and f (u ) + f ( v t +1 ) = 2 do not appear simultaneously. Hence, ω( f ) = u ∈ V (G ) f (u ) ≥ 2t + 3. On the other hand, let g (u ) = 3, g ( v i ) = 0 and g ( w j ) = 2, where i = 1, 2, . . . , r + t and j = 1, 2, . . . , t. It can be checked that g is a DRDF of F r ,t with weight 2t + 3. Therefore, γdR ( F r ,t ) = 2t + 3 = 2γ ( F r ,t ) + 1. 2 Theorem 2. Let T be a nontrivial tree. Then 2γ ( T ) + 1 if and only if T is a wounded spider.
γdR ( T ) =
Proof. The sufficiency follows from Theorem 1. Now we will prove that if γdR ( T ) = 2γ ( T ) + 1, there exist two integers r ≥ 1 and t ≥ 0 such that T = F r ,t . Suppose that this is not true and let T be a smallest counterexample to the theorem. Then γdR ( T ) = 2γ ( T ) + 1 and T = F r ,t for any integers r ≥ 1 and t ≥ 0. Claim 1. T has no strong support vertex.
Lemma 2. [6] For any non-trivial tree T , 2γ ( T ) + 1 ≤ γdR ( T ) ≤ 3γ ( T ). Beeler et al. [6] also asked that if it is possible to construct a polynomial algorithm to compute the double Roman domination number of trees. In this paper, we give
Proof of Claim 1. Suppose that T has a strong support vertex x. Let y , z be two leaves of T adjacent to x and T = T − y. Since x is a support vertex in T , we have γ ( T ) = γ ( T ) and γdR ( T ) ≥ γdR ( T ). Hence, by Lemma 2, γdR ( T ) ≥ γdR ( T ) ≥ 2γ ( T ) + 1 = 2γ ( T ) + 1. Note that
X. Zhang et al. / Information Processing Letters 134 (2018) 31–34
γdR ( T ) = 2γ ( T ) + 1, so γdR ( T ) = 2γ ( T ) + 1. By the mini-
mality of T , there exist two integers r ≥ 1 and t ≥ 0 such that T = F r ,t . If x is the center of T , then T = F r +1,t , contradicting the assumption that T is a counterexample. Hence, x is a 2-vertex of T and the degree of the center of T is at least 3. Then we have γdR ( T ) > γdR ( T ) = 2t + 3 by Theorem 1. Note that γ ( T ) = γ ( T ) = t , we have γdR ( T ) > 2γ ( T ) + 1, a contradiction. Therefore, Claim 1 is true. 2 By Claim 1, T has a support vertex x with d T (x) = 2. We denote by y the leaf neighbor of x and by z the nonleaf neighbor of x. Let f be a γdR -function of T and consider the following two cases: Case 1. f ( z) ≤ 1. Let T = T −{x, y , z} and f be the restriction of f to T . Then f is a DRDF of T . Since f is a γdR -function of T and f ( z) ≤ 1, we have f (x) + f ( y ) + f ( z) ≥ 3. Hence,
γdR ( T ) ≤ γdR ( T ) − 3
(1)
Since any dominating set of T together with x is a dominating set of T , we have
γ (T ) ≤ γ (T ) + 1
(2)
Note that T = F r ,t for any integers r ≥ 1 and t ≥ 0, we can obtain that T is a forest with at least one nontrivial component. Since γdR ( T 1 ) = 2γ ( T 1 ) for any trivial tree T 1 and γdR ( T 2 ) ≥ 2γ ( T 2 ) + 1 for any nontrivial tree T 2 , we have
γdR ( T ) ≥ 2γ ( T ) + 1
(3)
By Formulae (1), (2) and (3), we have γdR ( T ) ≥ γdR ( T ) + 3 ≥ 2γ ( T ) + 1 + 3 ≥ 2(γ ( T ) − 1) + 4 = 2γ ( T ) + 2, contradicting the assumption that γdR ( T ) = 2γ ( T ) + 1. Case 2. f ( z) ≥ 2. Let T = T − {x, y } and f be the restriction of f to γdR -function of T and y is a leaf of T , we have f (x) + f ( y ) ≥ 2. Hence, γdR ( T ) ≤ γdR ( T ) − 2. Since γ ( T ) ≤ γ ( T ) + 1 and γdR ( T ) ≥ 2γ ( T ) + 1 by Lemma 2, we have γdR ( T ) ≥ γdR ( T ) + 2 ≥ 2γ ( T ) + 1 + 2 ≥ 2(γ ( T ) − 1) + 3 = 2γ ( T ) + 1. Note that γdR ( T ) = 2γ ( T ) + 1, so γdR ( T ) = 2γ ( T ) + 1. By the minimality of T , there exist two integers r ≥ 1 and t ≥ 0 such that T = F r ,t . By the symmetry of T , there are four possible cases of z in T . T . Then f is a DRDF of T . Since f is a
Subcase 2.1. z is a wounding vertex of T .
In this case, we can obtain that γ ( T ) = t + 2 and γdR ( T ) = 2t + 6, a contradiction.
diction. Subcase 2.2. z is the center of T . In this case, we have T = F r ,t +1 , contradicting the assumption that T is a counterexample. Subcase 2.3. z is a 2-vertex of T but not the center.
γ (T ) + 1 =
Subcase 2.4. z is a foot vertex of T . In this case, we can obtain that γ ( T ) = γ ( T ) = t + 1 and γdR ( T ) = 2t + 4, a contradiction. 2 Proposition 1. [6] i) Let G be a graph and f = ( V 0 , V 1 , V 2 ) a γ R (G )-function. Then γdR (G ) ≤ 2| V 1 | + 3| V 2 |. ii) For any graph G, γdR (G ) ≤ 2γ R (G ) with equality if and only if G = K n . Theorem 3. Let T be a nontrivial tree, then T with 1 = 2γ R ( T ) if and only if T is a wounded spider.
γdR ( T ) +
Proof. ⇐: Since T is a wounded spider, we can assume T = F r ,t for some r ≥ 1. It can be verified that γdR ( T ) = 2t + 3 and γ R ( T ) = t + 2. Therefore, γdR ( T ) + 1 = 2γ R ( T ). ⇒: Let f = ( V 0 , V 1 , V 2 ) be a γ R (G )-function with minimum | V 1 |. Then we have | V 1 | is independent. Together with Proposition 1, we have γdR (G ) ≤ 2| V 1 | + 3| V 2 | ≤ 2| V 1 | + 4| V 2 | = 2γ R (G ) = γdR (G ) + 1. Then we have | V 2 | ≤ 1. If | V 2 | = 0, we have | V 0 | = 0 and thus γ R (G ) = n = | V 1 |. Since V 1 is independent, so it is impossible. If | V 2 | = 1, we have γdR (G ) = 2| V 1 | + 3| V 2 | < 2| V 1 | + 4| V 2 | = 2γ R (G ) = γdR (G ) + 1. Let V 2 = { v }, V 0 = N ( v ) and V 1 = V (G ) − V 0 − V 2 . Since V 1 is independent, we have T is a wounded spider or a healthy spider. Note that if T is a healthy spider, we have γdR ( T ) + 2 = 2γ R ( T ), a contradiction. Therefore, T is a wounded spider. 2 3. A linear algorithm for double Roman domination in trees Let u be a specific vertex of G. Note that a minimum weight of a DRDF on G satisfies that f (u ) ∈ {0, 2, 3} by Lemma 1. So it is useful to consider the following three domination problems. 0 γdR (G , u ) = min{ω( f ) : f is a DRDF of G and f (u ) = 0}. 2 γdR (G , u ) = min{ω( f ) : f is a DRDF of G and f (u ) = 2}. 3 γdR (G , u ) = min{ω( f ) : f is a DRDF of G and f (u ) = 3}.
Lemma 3. For any graph G with a specific vertex u, we have 0 3 2 γdR (G , u ) = min{γdR (G , u ), γdR (G , u ), γdR (G , u )}. 00 γdR (G , u ) = min{ω( f ) : f is a DRDF of G − u }. 02 γdR (G , u ) = min{ω( f ) : f is a DRDF of G + u w and
f (u ) = 0, f ( w ) = 2}.
In this case, if r = 1, then it can be checked that
γ ( T ) = γ ( T ) = t + 1 and γdR ( T ) = 2t + 5, contradicting γdR ( T ) = 2γ ( T ) + 1; if r ≥ 2, then we can obtain that γ ( T ) = γ ( T ) + 1 = t + 2 and γdR ( T ) = 2t + 6, a contra-
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00 0 γdR (G , u ) ≤ γdR (G , u ), since a DRDF f of G 00 (G , u ) ≤ with f (u ) = 0 is also a DRDF of G − u, and γdR 02 γdR (G , u ), since a DRDF f of G + u w with f (u ) = 0 and
Note that
f ( w ) = 2 is also a DRDF of G − {u , w }.
Theorem 4. Suppose G and H are graphs with specific vertices u and v, respectively. Let I be the graph with the specific vertex u, which is obtained from the disjoint union of G and H by joining a new edge uv. Then the following statements hold.
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(i) (ii) (iii) (iv) (v)
X. Zhang et al. / Information Processing Letters 134 (2018) 31–34 0 0 0 02 γdR ( I , u ) = min{γdR (G , u ) + γdR ( H , v ), γdR (G , u ) + 00 3 2 γdR ( H , v ) − 2, γdR (G , u ) + γdR ( H , v )}; 02 2 2 2 γdR ( I , u ) = γdR (G , u ) + min{γdR ( H , v ), γdR ( H , v ), 3 γdR ( H , v )}; 3 3 00 2 γdR ( I , u ) = γdR (G , u ) + min{γdR ( H , v ), γdR ( H , v ), 3 γdR ( H , v )}; 00 00 00 γdR ( I , u ) = γdR (G , u ) + γdR ( H ) = γdR (G , u ) + 0 3 2 min{γdR ( H , v ), γdR ( H , v ), γdR ( H , v )}; 02 02 0 00 γdR ( I , u ) = min{γdR (G , u ) + γdR ( H , v ), γdR (G , u ) + 00 3 2 γdR ( H , v ), γdR (G , u ) + γdR ( H , v )}.
Proof. (i) It follows from the fact that f is a DRDF of I with f (u ) = 0 if and only if f = f ∪ f
, where f
is a DRDF of G with f (u ) = 0 and f
is a DRDF of H with f
( v ) = 0, f is a DRDF of G + u w with f (u ) = 0, f ( w ) = 2 and f
is a DRDF of H with f
( v ) = 2, or f is a DRDF of G − u and f
is a DRDF of H with f
( v ) = 3. (ii) It follows from the fact that f is a DRDF of I with f (u ) = 2 if and only if f = f ∪ f
, where f is a DRDF of G with f (u ) = 2, f
is a DRDF of H + v w with f
( v ) = 0, f
( w ) = 2 or a DRDF of H with f
( v ) ∈ {2, 3}. (iii) It follows from the fact that f is a DRDF of I with f (u ) = 3 if and only if f = f ∪ f
, where f is a DRDF of G with f (u ) = 3 and f
is a DRDF of H − v or a DRDF of H with f
( v ) ∈ {2, 3}. (iv) It follows from the fact that f is a DRDF of I − u if and only if f = f ∪ f
, where f is a DRDF of G − u and f
is a DRDF of H . (v) It follows from the fact that f is a DRDF of I + u w with f (u ) = 0, f ( w ) = 2 if and only if f = f ∪ f
, where f is a DRDF of G + u w with f (u ) = 0, f ( w ) = 2 and f
is a DRDF of H with f
( v ) = 0, or f is a DRDF of G − u and f
is a DRDF of H with f
( v ) ∈ {2, 3}. 2 Lemma 3 and Theorem 4 give the following dynamic programming algorithm for the double Roman domination problem in trees.
Algorithm DoubleRomanDomination Input: A tree T with a tree ordering [ v 1 , v 2 , · · · , v n ]. Output: the double Roman domination number γdR ( T ) of T . begin for i = 1 to n do γ 00 ( v i ) ← 0; γ 0 ( v i ) ← ∞; γ 2 ( v i ) ← 2; γ 3 ( v i ) ← 3; γ 02 ( v i ) ← ∞; end for for i = 1 to n − 1 do let v j be the parent of v i ; γ 0 ( v j ) ← min{γ 0 ( v j ) + γ 0 ( v i ), γ 02 ( v j ) + γ 2 ( v i ) − 2, γ 00 ( v j ) + γ 3 ( v i )}; γ 2 ( v j ) ← γ 2 ( v j ) + min{γ 02 ( v i ), γ 2 ( v i ), γ 3 ( v i )}; γ 3 ( v j ) ← γ 3 ( v j ) + min{γ 00 ( v i ), γ 2 ( v i ), γ 3 ( v i )}; γ 00 ( v j ) ← γ 00 ( v j ) + min{γ 0 ( v i ), γ 2 ( v i ), γ 3 ( v i )}; γ 02 ( v j ) ← min{γ 02 ( v j ) + γ 0 ( v i ), γ 00 ( v j ) + γ 2 ( v i ), γ 00 ( v j ) + γ 3 ( v i )}; end for return min{γ 0 ( v n ), γ 2 ( v n ), γ 3 ( v n )}; end. References [1] T.W. Haynes, S.T. Hedetniemi, P.J. Slater, Fundamentals of Domination in Graphs, Marcel Dekker, New York, 1998. [2] M.R. Garey, D.S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness, W. H. Freeman and Co., San Francisco, Calif, 1979. [3] M.A. Henning, A characterization of Roman trees, Discuss. Math., Graph Theory 22 (2002) 325–334. [4] O. Ore, Theory of Graphs, American Mathematical Society, Providence, RI, 1967. [5] Z. Shao, S. Klavžar, Z. Li, P. Wu, J. Xu, On the signed Roman k-domination: complexity and thin torus graphs, Discrete Appl. Math. 233 (2017) 175–186. [6] R.A. Beeler, T.W. Haynes, S.T. Hedetniemi, Double Roman domination, Discrete Appl. Math. 211 (2016) 23–29.