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Computing Roman domatic number of graphs
Information Processing Letters 116 (2016) 554–559
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Information Processing Letters www.elsevier.com/locate/ipl
Computing Roman domatic number of graphs ✩ Haisheng Tan a , Hongyu Liang b , Rui Wang c,∗ , Jipeng Zhou a a b c
Department of Computer Science, Jinan University, Guangzhou, China Facebook Inc., 7006 126th Ave NE, Kirkland, WA 98033, USA South University of Science and Technology of China, Shenzhen, China
a r t i c l e
i n f o
Article history: Received 30 November 2015 Received in revised form 14 March 2016 Accepted 26 April 2016 Available online 27 April 2016 Communicated by Ł. Kowalik Keywords: Roman domatic number Graph Computational complexity Approximation algorithms
a b s t r a c t A Roman dominating function on a graph G = ( V , E ) is a mapping: V → {0, 1, 2} satisfying that every vertex v ∈ V with f ( v ) = 0 is adjacent to some vertex u ∈ V with f (u ) = 2. A Roman dominating family (of functions) on G is a set { f 1 , f 2 , . . . , f d } of Roman dominating d functions on G with the property that i =1 f i ( v ) ≤ 2 for all v ∈ V . The Roman domatic number of G, introduced by Sheikholeslami and Volkmann in 2010 [1], is the maximum number of functions in a Roman dominating family on G. In this paper, we study the Roman domatic number from both algorithmic complexity and graph theory points of view. We show that it is N P -complete to decide whether the Roman domatic number is at least 3, even if the graph is bipartite. To the best of our knowledge, this is the first computational hardness result concerning this concept. We also present an asymptotically optimal approximation threshold of (log n) for computing the Roman domatic number of a graph. Moreover, we determine the Roman domatic number of some particular classes of graphs, such as fans, wheels and complete bipartite graphs. © 2016 Elsevier B.V. All rights reserved.
1. Introduction All graphs considered in this paper are simple and undirected. We generally follow [2] for standard notation and terminology in graph theory. Fix a graph G = ( V , E ). The order of G is | V |. For each v ∈ V , N G ( v ) = {u | uv ∈ E } is the open neighborhood of v, and N G [ v ] = N G ( v ) ∪ { v } is the closed neighborhood of v. Let d G ( v ) = | N G ( v )| denote the degree of v, and δ(G ) = min v ∈ V {d G ( v )} be the minimum degree of any vertex in V . For an integer k ≥ 1,
✩ This work was supported in part by the Fundamental Research Funds for the Central Universities in China (No. 21614324), the NSFC-Guangdong (No. 2014A030310172), the National Natural Science Foundation of China (NSFC) Grants 61502201 and 61373125, and Sci., Tech. and Innovation Commission of Shenzhen Municipality Grant KQCX2014052215132295. Corresponding author. E-mail address: [email protected] (R. Wang).
*
http://dx.doi.org/10.1016/j.ipl.2016.04.010 0020-0190/© 2016 Elsevier B.V. All rights reserved.
a k-coloring of G is a mapping c : V → {1, 2, . . . , k} such that c (u ) = c ( v ) whenever uv ∈ E. We also say G is k-colorable if G has a k-coloring. For any function f that maps the vertices in V to (a subset of) the reals R, and any S ⊆ V , let f ( S ) = v ∈ S f ( v ); the weight of f is defined to be f ( V ). The theory of domination is an important area in graph theory, and has been developed extensively and steadily (see [3,4] for detailed surveys on previous results in domination theory). A set S ⊆ V is called a dominating set of G if v ∈ S N G [ v ] = V . The domination number of G, denoted by γ (G ), is the minimum size of a dominating set of G. A domatic partition of G is a partition of V into disjoint dominating sets of G. The domatic number of G [5], denoted as d(G ), is the maximum number of dominating sets in a domatic partition of G. A function f : V → {0, 1, 2} is called a Roman dominating function on G if with v ∈ V for which f ( v ) = 0,
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there is u ∈ N G ( v ) such that f (u ) = 2. The Roman domination number of G, denoted by γ R (G ), is the minimum weight of a Roman dominating function on G. The idea of Roman domination is suggested in [6,7] for studying how to securely protect locations under the decree of Emperor Constantine the Great (Emperor of Rome, in the fourth century A.D.). The definition of Roman domination number is made explicitly in [8], where graph-theoretic properties of this parameter are studied. Roman domination and its variants have attracted considerable attention these years, from graph-theoretic, algorithmic, and complexity perspectives (e.g., [9–17] and the references therein). A set { f 1 , f 2 , . . . , f d } of distinct Roman dominating functions on G is called a Roman dominating family (of d functions) on G if i =1 f i ( v ) ≤ 2 for all v ∈ V . The Roman domatic number of G, denoted by d R (G ), is the maximum number of functions contained in a Roman dominating family on G. The concept of Roman domatic number was introduced by Sheikholeslami and Volkmann [1] as analogue to the domatic number of a graph [5], which has been extensively studied from various aspects. In [1], some graph-theoretic properties of the Roman domatic number are explored, some tight bounds are presented, and the exact values of the Roman domatic number are determined for cycles and trees. However, to the best of our knowledge, no complexity results have been reported for this graph parameter, which motivates our study. Our contribution: In this paper, we initiate the investigation on the Roman domatic number from the algorithmic complexity point of view. We show that it is N P -complete to decide whether the Roman domatic number of a given graph is at least three, even if the graph is bipartite. To our knowledge, this is the first computational hardness result concerning this concept. We also notice that this result is as good as possible, since any non-empty graph admits a Roman dominating family consisting of two functions. We then prove an asymptotically optimal threshold of (log n) for approximating the Roman domatic number of a graph. More specifically, there is a polynomial time O (log n)-approximation algorithm for computing the Roman domatic number, but every polynomial time algorithm for computing the Roman domatic number must have an approximation ratio of (log n) under some widely-believed complexity assumption. Lastly, we also determine the exact values of the Roman domatic number in some particular classes of graphs. Paper organization: The rest of this paper is organized as follows. In Section 2, we study the complexity to compute Roman domatic numbers. In Section 3, we derive the Roman domatic numbers for some special classes of graphs. Section 4 concludes this work and points out future directions. 2. Complexity issues of Roman domatic number 2.1. N P -completeness results In this subsection we show the hardness of computing the Roman domatic number of a graph. We will prove that even the following restricted decision problem is N P -complete.
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Fig. 1. The graph G: the number beside a node v ∈ { A , B , C , D } is its color number assigned, i.e., the value of c ( v ).
Fig. 2. The graph H : the number beside a node v is the value of f 1 ( v ).
Bipartite 3-Roman Domatic Number (B3RDM) Instance: A bipartite graph G = ( V , E ). Question: Is d R (G ) ≥ 3? The following theorem is as good as possible in terms that it is trivial to decide whether d R (G ) ≥ 2 for any graph G. Theorem 1. The B3RDM problem is N P -complete. Proof. The B3RDM problem is clearly in N P . We now present a polynomial-time reduction from a classic N P -complete problem, namely the Graph 3-Coloring problem (G3C) [18], to B3RDM, thus establishing the N P -completeness of the latter. An instance of G3C is a graph G, and the goal is to decide whether G is 3-colorable. Let G = ( V , E ) be an input graph to G3C. Construct another graph H = ( V , E ) as follows. Let V = {x v , x v | v ∈ V } ∪ { y e , y e | e ∈ E } ∪ {a1 , a2 , a3 }. Note that there are two vertices x v , x v corresponding to a vertex v ∈ V , and two vertices y e , y e corresponding to an edge e ∈ E. Let E = {x v y e | v ∈ e ∈ E } ∪ {x v ai | v ∈ V ; 1 ≤ i ≤ 3} ∪ {x v x v | v ∈ V } ∪ { y e y e | e ∈ E }. An example is illustrated in Figs. 1 and 2, where the graph G has 4 nodes and 5 edges while the corresponding H totally has 21 nodes and 31 edges. It is easy to verify that H is bipartite and can be constructed in polynomial time. We will show that d R ( H ) ≥ 3 if and only if G is 3-colorable, which will finish the reduction and conclude the N P -completeness of B3RDM. First consider the “if” direction. Assume that G is 3-colorable, and that c : V → {1, 2, 3} is a 3-coloring of G. We may assume without loss of generality that | V | ≥ 3, and that for each i ∈ {1, 2, 3} there exists a v ∈ V such that c ( v ) = i; i.e., all three colors are actually used. We define
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three functions f 1 , f 2 , f 3 on H as follows. Let i ∈ {1, 2, 3}. For each v ∈ V , define
f i (x v ) = f i (x v ) :=
2 0
0 1
if c ( v ) = i ; otherwise,
and
if f i (x v ) = 2; otherwise.
For each e = {u , v } ∈ E, define
f i ( y e ) := f i ( y e ) :=
2 0
if i ∈ / {c (u ), c ( v )}; otherwise,
0 1
if f i ( y e ) = 2; otherwise.
and
Finally, for each j ∈ {1, 2, 3}, define
f i (a j ) :=
2 0
if j = i ; otherwise.
As an example, we illustrate in Fig. 2 the f 1 value of each node when the coloring is given in Fig. 1. The f 2 and f 3 values can be calculated similarly and we do not show them here to save space. Next, we will prove that { f 1 , f 2 , f 3 } is a Roman dominating family on H . Fix i ∈ {1, 2, 3} and consider all vertices u with f i (u ) = 0. There are several cases to be examined. 1. u = x v for some v ∈ V with c ( v ) = i. Then u is adjacent to ai for which f i (ai ) = 2. 2. u = y e for some e = v v ∈ E such that i ∈ {c ( v ), c ( v )}. Without loss of generality assume c ( v ) = i. Then u is adjacent to x v for which f i (x v ) = 2. 3. u = x v for some v ∈ V . Due to our construction, f i (x v ) = 0 if and only if f i (x v ) = 2, and x v is adjacent to x v in H . 4. u = y e for some e ∈ E. Similar to the previous case, u is adjacent to y e for which f i ( y e ) = 2. 5. u = ai . Since there exists v ∈ V with c ( v ) = i, ai is adjacent to x v for which f i (x v ) = 2. Therefore, every vertex u ∈ V with f i (u ) = 0 is adjacent to some v ∈ V for which f i ( v ) = 2, which indicates that f i is a Roman dominating function on H . Furthermore, since c is a 3-coloring of G, it holds that c (u ) = c ( v ) for all 3 e = uv ∈ E. Thus, it is easy to verify that i =1 f i ( v ) ≤ 2 for each v ∈ V , and the three functions are distinct. Hence, { f 1 , f 2 , f 3 } is indeed a Roman dominating family on H , implying that d R ( H ) ≥ 3. Now comes the “only if” part of the reduction. Suppose d R ( H ) ≥ 3 and { f 1 , f 2 , f 3 } is a Roman dominating family on H . For each x v ∈ V , the three values f 1 (x v ), f 2 (x v ), f 3 (x v ) must consist of exactly one “0” and two “1”s. To see this, first observe that at least one of them is “0”. If there are two “0”s, say, f 1 (x v ) = f 2 (x v ) = 0, then f 1 (x v ) = f 2 (x v ) = 2, contradicting the property of a Roman dominating family. Thus, among the three values there is exactly one “0”, and the remaining two must both be “1”. Consequently, the three values f 1 (x v ), f 2 (x v ), f 3 (x v ) must contain exactly one “2” and two “0”s. Analogous claim holds for each y e ∈ V ; i.e., the
three values f 1 ( y e ), f 2 ( y e ), f 3 ( y e ) are composed of exactly one “2” and two “0”s. We define a coloring c : V → {1, 2, 3} as follows. For each v ∈ V , let c ( v ) = i if f i (x v ) = 2. This coloring is well-defined since exactly one of f 1 (x v ), f 2 (x v ), and f 3 (x v ) equals 2. We next show that this coloring is a 3-coloring. Let e = uv be an arbitrary edge of G, and assume that c (u ) = c ( v ) = i. Then f i (xu ) = f i (x v ) = 2. As exactly two of f 1 ( y e ), f 2 ( y e ), and f 3 ( y e ) are 0 due to our previous analysis, there exists j ∈ {1, 2, 3} \ {i } such that f j ( y e ) = 0. Since N H ( y e ) = {xu , x v , y e }, and f j ( y e ) = 0 or 1, we have f j (xu ) = 2 or f j (x v ) = 2. Thus, there is x ∈ {u , v } for which f i (x) = f j (x) = 2, contradicting the fact that { f 1 , f 2 , f 3 } is a Roman dominating family on H . Accordingly, c (u ) = c ( v ) for each uv ∈ E, and thus c is indeed a 3-coloring of G. This completes the “only if” part of the reduction, and hence concludes the whole proof. 2 Corollary 1. It is N P -hard to compute the Roman domatic number of a given graph, even if the graph is bipartite. 2.2. Approximation behavior In this subsection we derive an asymptotically optimal approximation threshold for computing the Roman domatic number of a graph. Define the optimization problem Roman Domatic Number as follows: Given a graph G, a feasible solution is a Roman dominating family F on G, and the goal is to maximize the number of functions in F . Clearly, the optimal solution contains d R (G ) functions. Before proposing our results, we specify some notations related to approximation algorithms. Let be a maximization problem, and A be an algorithm for solving . For each instance I of , let OPT (, I ) and A(, I ) denote the objective values of the optimum solution to I and the solution returned by A on I , respectively. For β ≥ 1, A is called a β -approximation algorithm for if A runs in polyOPT (,I ) nomial time, and A(,I ) ≤ β for every instance I of . We refer the readers to [19] for standard definitions and notations not given here. To compute the Roman domatic number of a graph, we have the following theorem: Theorem 2. There is a (ln n + O (ln ln n))-approximation algorithm for Roman Domatic Number, where n is the order of the input graph. Proof. Let G = ( V , E ) be the input graph with | V | = n. We have d R (G ) ≤ δ(G ) + 2 owing to [1]. If δ(G ) ≤ ln n, we just output a trivial Roman dominating family on G (which consists of an arbitrary Roman dominating function on G). The approximation ratio of this solution is at most δ(G ) + 2 ≤ ln n + 2 ≤ ln n + O (ln ln n). Now consider the case where δ(G ) > ln n. Using Theorem 1 from [20], we can construct in polynomial time a domatic partition of G of size (δ(G ) + 1)(1 − K (ln ln n/ ln n))/ ln n for some constant K > 0. This naturally induces a Roman dominating family on G of the same size (by assigning the chosen vertices with function value 2 and others with 0 in each dominating set of this partition), which we output as our
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solution. We know that for ∈ (0, 1/2), 1/(1 − ) < 1 + 2 . Therefore, for sufficiently large n, the approximation ratio is at most
δ(G ) + 2 (δ(G ) + 1)(1 − K (ln ln n/ ln n))/ ln n 1 1 = 1+ ln n 1 − K (ln ln n/ ln n) δ(G ) + 1 ln ln n 1 ≤ 1 + 2K 1+ ln n =
1 + 2K
≤
ln n
ln ln n
+
ln n
1 + (4K + 1)
ln n
1 ln n
ln ln n ln n
+ 2K
ln ln n
(ln n)2
ln n
ln n
= ln n + O (ln ln n) . Therefore, we can always approximate Roman Domatic Number within a factor of ln n + O (ln ln n) in polynomial time, completing the proof of Theorem 2. 2 We next show that this approximation factor is asymptotically optimal. For a function f , D T I M E ( f (n)) denotes the set of decision problems decidable in time f (n) (by a deterministic Turing machine) on input of size n. For a class of functions C , let D T I M E (C ) = f ∈C D T I M E ( f (n)). Note that P = D T I M E (n O (1) ) in our notation. We have the following theorem.
Theorem 3. For every fixed > 0, there is no ( 12 − ) × ln n-approximation algorithm for Roman Domatic Number, where n is the order of the input graph, unless N P ⊆ D T I M E (n O (log log n) ). Proof of Theorem 3. We need the following lemma, which is proved in [20] for establishing the inapproximability result of the domatic number. Lemma 1 (Combining Propositions 10, 13 and Theorem 11 in [20]). Fix > 0 and assume N P D T I M E (n O (log log n) ). Given a graph G of order n, we cannot distinguish in polynomial time between the following two cases (even when we know G belongs to exactly one of them): 1. the size of any dominating set of G is at least r; 2. the domatic number of G is at least q, where r , q are parameters (easily computable from the input) satisfying that qr ≥ (1 − )n ln n. Fix > 0. Let G be an input graph of order n that satisfies exactly one of the cases in Lemma 1, and r , q be the parameters given by Lemma 1. Assume N P D T I M E (n O (log log n) ), and assume to the contrary that there exists a ( 12 − ) ln n-approximation algorithm for Roman Domatic Number. We apply this hypothetical algorithm to
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find, in polynomial time, a Roman dominating family on G of size d with d R (G ) ≥ d ≥ d R (G )/(( 12 − ) ln n). Now we examine the bounds on d for the two cases in Lemma 1.
• In the first case, γ (G ) ≥ r. According to [1], we have γ R (G ) ≥ γ (G ) and d(G ) ≤ d R (G ) ≤ 2n/γ R (G ). Therefore, d ≤ d R (G ) ≤ 2n/γ R (G ) ≤ 2n/γ (G ) ≤ 2n/r. • In the second case, d(G ) ≥ q, and thus d R (G ) ≥ d(G ) ≥ q. We have d ≥ d R (G )/(( 12 − ) ln n) ≥ q/(( 12 − ) ln n). Since qr ≥ (1 − )n ln n, we have d ≥ q/(( 12 − ) ln n) ≥ r ((11/−2−)n )lnlnnn > 2n/r. Therefore, by comparing d and 2n/r, we can distinguish whether the graph G belongs to case 1 or 2 in Lemma 1, which contradicts the indistinguishability result stated by the lemma. This means that our assumption is invalid, that is, there doesn’t exist any ( 12 − ) ln n-approximation algorithm for Roman Domatic Number, unless N P ⊆ D T I M E (n O (log log n) ). The proof is concluded. 2 3. Roman domatic number in particular graph classes In this section we determine the exact values of the Roman domatic number of some special classes of graphs. We first prove a useful theorem, from which several (old and new) results follow immediately. Theorem 4. Let G be an arbitrary graph. Construct H by adding a new vertex to G and connecting it with all the vertices in G. Then, d R ( H ) = d R (G ) + 1. Proof. If G is empty (i.e., containing no edges), we have d R (G ) = 1, and H is a tree of order at least 2. By Theorem 15 in [1], d R ( H ) = 2, and thus the theorem holds. In the following we will assume that G is non-empty. Denote by V the vertex set of G, and by v 0 be the vertex of H that is not in V (i.e., the vertex that is added to G for obtaining H ). The vertex set of H is thus V ∪ { v 0 }. Let { f 1 , f 2 , . . . , f d } be a Roman dominating family on H with d = d R ( H ). For each i ∈ {1, 2, . . . , d}, define f i : V → {0, 1, 2} as: f i ( v ) = f i ( v ) for all v ∈ V . Let I = {i | f i ( v 0 ) < 2; 1 ≤ i ≤ d}, and consider the family of functions F = { f i | i ∈ I }. For each function f i in F , if f i ( v ) = 0 for some v ∈ V , then there exists u ∈ N G ( v ) such that f i (u ) = f i (u ) = 2, since f i ( v 0 ) < 2 and f i is a Roman dominating function on H . Therefore, every f i in F is a Roman dominating function on G. It is also clear that, for d
each v ∈ V , i∈ I f i ( v ) ≤ i =1 f i ( v ) ≤ 2. We now prove that all these functions are distinct. Choose two arbitrary indices i 1 , i 2 ∈ I such that i 1 = i 2 . If f i 1 ( v 0 ) = f i 2 ( v 0 ), then f i and f i must be different, 1 2 since f i 1 and f i 2 are different. If f i 1 ( v 0 ) = f i 2 ( v 0 ), say, f i 1 ( v 0 ) = 0 and f i 2 ( v 0 ) = 1, then there exists v ∈ V for which f i 1 ( v ) = 2. Hence, f i 2 ( v ) must be 0, implying that f i 1 and f i 2 are distinct. Therefore, F is a Roman dominating family on G of size | I |. By the definition of Roman dominating family, there is at most one i for which
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f i ( v 0 ) = 2. Thus, | I | ≥ d − 1, from which it follows that d R ( H ) = d ≤ | I | + 1 ≤ d R (G ) + 1. We next show that this upper bound is attainable. Let { f 1 , . . . , f d R (G ) } be a (maximum) Roman dominating family on G. Since G is non-empty, d R (G ) ≥ 2 holds due to Observation 2 in [1]. We claim that, for each i ∈ {1, 2, . . . , d R (G )}, there exists v ∈ V such that f i ( v ) = 2. If this is not the case, that is, there exists a function f i for which f i ( v ) < 2 for all v ∈ V , then it must hold that f i ( v ) = 1 for all v ∈ V . Now choose j ∈ {1, 2, . . . , d R (G )} \ {i } (recall that d R (G ) ≥ 2). We have f j ( v ) ≤ 2 − f i ( v ) ≤ 1, and thus f j ( v ) = 1 = f i ( v ) for all v ∈ V , which contradicts with the distinctness requirement in the definition of Roman dominating family. The claim is thus proved. Now define functions f 1 , f 2 , . . . , f d (G )+1 as follows: R
• For each i ∈ {1, 2, . . . , d R (G )}, f i ( v 0 ) = 0 and f i ( v ) = f i ( v ) for all v ∈ V ; • f d R (G )+1 ( v 0 ) = 2 and f d R (G )+1 ( v ) = 0 for all v ∈ V . For each i ∈ {1, 2, . . . , d R (G )}, f i is a Roman dominating function on H since by previous analysis, there exists v ∈ V for which f i ( v ) = f i ( v ) = 2. As v 0 is adjacent to all other vertices in H , f d R (G )+1 is also a Roman dominating function on H . Therefore, it is easy to see that { f 1 , f 2 , . . . , f d R (G )+1 } is a Roman dominating family on H of size d R (G ) + 1. Combining with the inequality d R ( H ) ≤ d R (G ) + 1 obtained before, we have d R ( H ) = d R (G ) + 1, which completes the proof of Theorem 4. 2 Based on the above Theorem 4, we can get the following results.
• Complete graphs Corollary 2 (Observation 1 in [1]). Let K n be the complete graph of order n ≥ 1. Then d R ( K n ) = n. Proof. Start with a single vertex and apply Theorem 4 for n − 1 times. 2
• Fan graphs For n ≥ 3, F n denotes the graph with V ( F n ) = { v 0 , v 1 , v 2 , . . . , v n−1 } and E ( F n ) = { v i v i +1 | 1 ≤ i ≤ n − 2} ∪ { v 0 v i | 1 ≤ i ≤ n − 1}. Corollary 3. Let F n be the fan graph of order n ≥ 3. Then d R ( F n ) = 3. Proof. For any tree T of order n ≥ 2, d R ( T ) = 2 (Theorem 15 in [1]). F n is actually to add a new vertex to a tree of n − 1 nodes and connect it to all the n − 1 vertices. Therefore, according to Theorem 4, we have d R ( F n ) = 3. 2
• Wheel graphs For n ≥ 3, let W n denote the wheel graph with V ( W n ) = { v 0 , v 1 , v 2 , . . . , v n } and E ( W n ) = { v i v i +1 | 1 ≤ i ≤ n − 1} ∪ { v n v 1 } ∪ { v 0 v i | 1 ≤ i ≤ n}.
Corollary 4. For n ≥ 3, let W n be the wheel graph of order n + 1. Then,
d R (W n ) =
4 3
if n ≡ 0 (mod 3); otherwise.
Proof. For n ≥ 3, we denote C n to be the cycle of length n. Then, W n is obtained by adding a new vertex v 0 to the cycle C n and connecting v 0 with all the original cycle vertices. According to Proposition 8 in [1], we have
d R (C n ) =
3 2
if n ≡ 0 (mod 3); otherwise.
Therefore, our corollary is achieved based on Theorem 4. 2
• Complete bipartite graphs For a1 , a2 ≥ 1, denote by K a1 ,a2 the complete bipartite graph with partite sets V 1 and V 2 such that | V 1 | = a1 and | V 2 | = a2 . Theorem 5. For all integers a1 , a2 ≥ 1, d R ( K a1 ,a2 ) = max{2, min{a1 , a2 }}. Proof. Assume without loss of generality that a1 ≤ a2 . If a1 = 1, K a1 ,a2 is a tree of order at least 2, and the theorem follows from Theorem 15 in [1]. Thus, in the following we suppose a ≥ 2, in which case max{2, min{a1 , a2 }} = a1 . Let V 1 = { v 1,1 , v 1,2 , . . . , v 1,a1 } and V 2 = { v 2,1 , v 2,2 , . . . , v 2,a2 }. Assume that { f 1 , f 2 , . . . , f d } is a Roman dominating family on K a1 ,a2 with d = d R ( K a1 ,a2 ). Trivially d ≥ 2. In what follows we examine two cases separately. 1. There exist i ∈ {1, . . . , d} and j ∈ {1, 2} such that f i ( v j ,k ) ≥ 1 for all k ∈ {1, . . . , a j }. Without loss of generality assume i = j = 1. (The proof for this case does not rely on the fact that a1 ≤ a2 , so similar argument goes through for j = 2.) Then, for each 2 ≤ i ≤ d, f i ( v 1,k ) ≤ 1 for all 1 ≤ k ≤ a1 . Consequently, f i ( v 2,k ) = 0 (and thus ≥ 1) for all 1 ≤ k ≤ a2 , since otherwise f i would not be a Roman dominating function on K a1 ,a2 . Therefore, there exists at most two such i ; that is, d ≤ 3. Now assume d = 3. According to the analysis above, it must hold that for each i ∈ {2, 3}, f i ( v 2,k ) = 1 for all 1 ≤ k ≤ a2 . Hence, f i ( v 1,k ) = 0 for all i ∈ { 2, 3} and 1 ≤ k ≤ a1 . This implies, however, that 3 i =1 f i ( v 1,1 ) ≥ 3, contradicting with the property of a Roman dominating family. Thus, we have d ≤ 2. 2. For all i ∈ {1, . . . , d} and j ∈ {1, 2}, there is k ∈ {1, . . . , a j } for which f i ( v j ,k ) = 0. This indicates that, for each i ∈ {1, . . . , d} and j ∈ {1, 2}, there exists k ∈ {1, . . . , a j } such that f i ( v j ,k ) = 2. Since each vertex can have value 2 under at most one function in a Roman dominating family, d ≤ min{a1 , a2 } = a1 holds. Therefore, we have d ≤ max{2, a1 } = a1 . On the other hand, define F = { f 1 , . . . , f a1 }, where for each 1 ≤
H. Tan et al. / Information Processing Letters 116 (2016) 554–559
i ≤ a1 , f i ( v 1,i ) = f i ( v 2,i ) = 2 and f i ( v ) = 0 for all v ∈ ( V 1 ∪ V 2 ) \ { v 1,i , v 2,i }. It is easy to see that F is a Roman dominating family on K a1 ,a2 of size a1 . Thus, d R ( K a1 ,a2 ) = d = a1 , completing the proof of Theorem 5. 2 4. Conclusion and future work In this paper, we gave the first computational hardness result for Roman domatic number, and prove an asymptotically optimal approximation threshold for computing this graph parameter. It is interesting to see whether we can further close the gap between the approximation ratio of (1 + o(1)) ln n and the inapproximability factor of ( 12 − o(1)) ln n. Also, it remains open whether the Roman domatic number can be computed efficiently on some well-studied graph classes such as line graphs and interval graphs. References [1] S. Sheikholeslami, L. Volkmann, The Roman domatic number of a graph, Appl. Math. Lett. 23 (2010) 1295–1300. [2] R. Diestel, Graph Theory, 4th edition, Springer-Verlag, 2010. [3] T. Haynes, S.T. Hedetniemi, P. Slater, Domination in Graphs: Advanced Topics, Marcel Dekker, 1998. [4] T. Haynes, S.T. Hedetniemi, P. Slater, Fundamentals of Domination in Graphs, Marcel Dekker, 1998. [5] E. Cockayne, S. Hedetniemi, Towards a theory of domination in graphs, Networks 7 (1977) 247–261.
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[6] C. Revelle, K. Rosing, Defendens Imperium Romanum: a classical problem in military strategy, Am. Math. Mon. 107 (7) (2000) 585–594. [7] I. Stewart, Defend the Roman Empire!, Sci. Am. 281 (6) (1999) 136–139. [8] E. Cockaynea, P. Dreyer Jr., S. Hedetniemi, S. Hedetniemi, Roman domination in graphs, Discrete Math. 278 (2004) 11–22. [9] H. Ahangar, J. Amjadi, S. Sheikholeslami, L. Volkmann, Y. Zhao, Signed roman edge domination numbers in graphs, J. Comb. Optim. 31 (2016). [10] H. Ahangar, M. Henning, C. Löwenstein, Y. Zhao, V. Samodivkin, Signed roman domination in graphs, J. Comb. Optim. 27 (2014) 241–255. [11] E. Chambers, B. Kinnersley, N. Prince, D. West, Extremal problems for Roman domination, SIAM J. Discrete Math. 23 (2009) 1575–1586. [12] X. Fu, Y. Yang, B. Jiang, Roman domination in regular graphs, Discrete Math. 309 (2009) 1528–1537. [13] A. Hansberg, L. Volkmann, Upper bounds on the k-domination number and the k-Roman domination number, Discrete Appl. Math. 157 (2009) 1634–1639. [14] M. Liedloffa, T. Kloks, J. Liu, S.-L. Peng, Efficient algorithms for Roman domination on some classes of graphs, Discrete Appl. Math. 156 (2008) 3400–3415. [15] C.-H. Liu, G. Chang, Roman domination on strongly chordal graphs, J. Comb. Optim. 26 (2012) 608–619. [16] E. Targhi, N. Rad, L. Volkmann, Unique response Roman domination in graphs, Discrete Appl. Math. 159 (11) (2011) 1110–1117. [17] H.-M. Xing, X. Chen, X.-G. Chen, A note on Roman domination in graphs, Discrete Math. 306 (2006) 3338–3340. [18] M. Garey, D. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness, W.H. Freeman, 1979. [19] V. Vazirani, Approximation Algorithms, Springer, 2004. [20] U. Feige, M. Halldórsson, G. Kortsarz, A. Srinivasan, Approximating the domatic number, SIAM J. Comput. 32 (1) (2002) 172–195.
Contents lists available at ScienceDirect
Information Processing Letters www.elsevier.com/locate/ipl
Computing Roman domatic number of graphs ✩ Haisheng Tan a , Hongyu Liang b , Rui Wang c,∗ , Jipeng Zhou a a b c
Department of Computer Science, Jinan University, Guangzhou, China Facebook Inc., 7006 126th Ave NE, Kirkland, WA 98033, USA South University of Science and Technology of China, Shenzhen, China
a r t i c l e
i n f o
Article history: Received 30 November 2015 Received in revised form 14 March 2016 Accepted 26 April 2016 Available online 27 April 2016 Communicated by Ł. Kowalik Keywords: Roman domatic number Graph Computational complexity Approximation algorithms
a b s t r a c t A Roman dominating function on a graph G = ( V , E ) is a mapping: V → {0, 1, 2} satisfying that every vertex v ∈ V with f ( v ) = 0 is adjacent to some vertex u ∈ V with f (u ) = 2. A Roman dominating family (of functions) on G is a set { f 1 , f 2 , . . . , f d } of Roman dominating d functions on G with the property that i =1 f i ( v ) ≤ 2 for all v ∈ V . The Roman domatic number of G, introduced by Sheikholeslami and Volkmann in 2010 [1], is the maximum number of functions in a Roman dominating family on G. In this paper, we study the Roman domatic number from both algorithmic complexity and graph theory points of view. We show that it is N P -complete to decide whether the Roman domatic number is at least 3, even if the graph is bipartite. To the best of our knowledge, this is the first computational hardness result concerning this concept. We also present an asymptotically optimal approximation threshold of (log n) for computing the Roman domatic number of a graph. Moreover, we determine the Roman domatic number of some particular classes of graphs, such as fans, wheels and complete bipartite graphs. © 2016 Elsevier B.V. All rights reserved.
1. Introduction All graphs considered in this paper are simple and undirected. We generally follow [2] for standard notation and terminology in graph theory. Fix a graph G = ( V , E ). The order of G is | V |. For each v ∈ V , N G ( v ) = {u | uv ∈ E } is the open neighborhood of v, and N G [ v ] = N G ( v ) ∪ { v } is the closed neighborhood of v. Let d G ( v ) = | N G ( v )| denote the degree of v, and δ(G ) = min v ∈ V {d G ( v )} be the minimum degree of any vertex in V . For an integer k ≥ 1,
✩ This work was supported in part by the Fundamental Research Funds for the Central Universities in China (No. 21614324), the NSFC-Guangdong (No. 2014A030310172), the National Natural Science Foundation of China (NSFC) Grants 61502201 and 61373125, and Sci., Tech. and Innovation Commission of Shenzhen Municipality Grant KQCX2014052215132295. Corresponding author. E-mail address: [email protected] (R. Wang).
*
http://dx.doi.org/10.1016/j.ipl.2016.04.010 0020-0190/© 2016 Elsevier B.V. All rights reserved.
a k-coloring of G is a mapping c : V → {1, 2, . . . , k} such that c (u ) = c ( v ) whenever uv ∈ E. We also say G is k-colorable if G has a k-coloring. For any function f that maps the vertices in V to (a subset of) the reals R, and any S ⊆ V , let f ( S ) = v ∈ S f ( v ); the weight of f is defined to be f ( V ). The theory of domination is an important area in graph theory, and has been developed extensively and steadily (see [3,4] for detailed surveys on previous results in domination theory). A set S ⊆ V is called a dominating set of G if v ∈ S N G [ v ] = V . The domination number of G, denoted by γ (G ), is the minimum size of a dominating set of G. A domatic partition of G is a partition of V into disjoint dominating sets of G. The domatic number of G [5], denoted as d(G ), is the maximum number of dominating sets in a domatic partition of G. A function f : V → {0, 1, 2} is called a Roman dominating function on G if with v ∈ V for which f ( v ) = 0,
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there is u ∈ N G ( v ) such that f (u ) = 2. The Roman domination number of G, denoted by γ R (G ), is the minimum weight of a Roman dominating function on G. The idea of Roman domination is suggested in [6,7] for studying how to securely protect locations under the decree of Emperor Constantine the Great (Emperor of Rome, in the fourth century A.D.). The definition of Roman domination number is made explicitly in [8], where graph-theoretic properties of this parameter are studied. Roman domination and its variants have attracted considerable attention these years, from graph-theoretic, algorithmic, and complexity perspectives (e.g., [9–17] and the references therein). A set { f 1 , f 2 , . . . , f d } of distinct Roman dominating functions on G is called a Roman dominating family (of d functions) on G if i =1 f i ( v ) ≤ 2 for all v ∈ V . The Roman domatic number of G, denoted by d R (G ), is the maximum number of functions contained in a Roman dominating family on G. The concept of Roman domatic number was introduced by Sheikholeslami and Volkmann [1] as analogue to the domatic number of a graph [5], which has been extensively studied from various aspects. In [1], some graph-theoretic properties of the Roman domatic number are explored, some tight bounds are presented, and the exact values of the Roman domatic number are determined for cycles and trees. However, to the best of our knowledge, no complexity results have been reported for this graph parameter, which motivates our study. Our contribution: In this paper, we initiate the investigation on the Roman domatic number from the algorithmic complexity point of view. We show that it is N P -complete to decide whether the Roman domatic number of a given graph is at least three, even if the graph is bipartite. To our knowledge, this is the first computational hardness result concerning this concept. We also notice that this result is as good as possible, since any non-empty graph admits a Roman dominating family consisting of two functions. We then prove an asymptotically optimal threshold of (log n) for approximating the Roman domatic number of a graph. More specifically, there is a polynomial time O (log n)-approximation algorithm for computing the Roman domatic number, but every polynomial time algorithm for computing the Roman domatic number must have an approximation ratio of (log n) under some widely-believed complexity assumption. Lastly, we also determine the exact values of the Roman domatic number in some particular classes of graphs. Paper organization: The rest of this paper is organized as follows. In Section 2, we study the complexity to compute Roman domatic numbers. In Section 3, we derive the Roman domatic numbers for some special classes of graphs. Section 4 concludes this work and points out future directions. 2. Complexity issues of Roman domatic number 2.1. N P -completeness results In this subsection we show the hardness of computing the Roman domatic number of a graph. We will prove that even the following restricted decision problem is N P -complete.
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Fig. 1. The graph G: the number beside a node v ∈ { A , B , C , D } is its color number assigned, i.e., the value of c ( v ).
Fig. 2. The graph H : the number beside a node v is the value of f 1 ( v ).
Bipartite 3-Roman Domatic Number (B3RDM) Instance: A bipartite graph G = ( V , E ). Question: Is d R (G ) ≥ 3? The following theorem is as good as possible in terms that it is trivial to decide whether d R (G ) ≥ 2 for any graph G. Theorem 1. The B3RDM problem is N P -complete. Proof. The B3RDM problem is clearly in N P . We now present a polynomial-time reduction from a classic N P -complete problem, namely the Graph 3-Coloring problem (G3C) [18], to B3RDM, thus establishing the N P -completeness of the latter. An instance of G3C is a graph G, and the goal is to decide whether G is 3-colorable. Let G = ( V , E ) be an input graph to G3C. Construct another graph H = ( V , E ) as follows. Let V = {x v , x v | v ∈ V } ∪ { y e , y e | e ∈ E } ∪ {a1 , a2 , a3 }. Note that there are two vertices x v , x v corresponding to a vertex v ∈ V , and two vertices y e , y e corresponding to an edge e ∈ E. Let E = {x v y e | v ∈ e ∈ E } ∪ {x v ai | v ∈ V ; 1 ≤ i ≤ 3} ∪ {x v x v | v ∈ V } ∪ { y e y e | e ∈ E }. An example is illustrated in Figs. 1 and 2, where the graph G has 4 nodes and 5 edges while the corresponding H totally has 21 nodes and 31 edges. It is easy to verify that H is bipartite and can be constructed in polynomial time. We will show that d R ( H ) ≥ 3 if and only if G is 3-colorable, which will finish the reduction and conclude the N P -completeness of B3RDM. First consider the “if” direction. Assume that G is 3-colorable, and that c : V → {1, 2, 3} is a 3-coloring of G. We may assume without loss of generality that | V | ≥ 3, and that for each i ∈ {1, 2, 3} there exists a v ∈ V such that c ( v ) = i; i.e., all three colors are actually used. We define
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three functions f 1 , f 2 , f 3 on H as follows. Let i ∈ {1, 2, 3}. For each v ∈ V , define
f i (x v ) = f i (x v ) :=
2 0
0 1
if c ( v ) = i ; otherwise,
and
if f i (x v ) = 2; otherwise.
For each e = {u , v } ∈ E, define
f i ( y e ) := f i ( y e ) :=
2 0
if i ∈ / {c (u ), c ( v )}; otherwise,
0 1
if f i ( y e ) = 2; otherwise.
and
Finally, for each j ∈ {1, 2, 3}, define
f i (a j ) :=
2 0
if j = i ; otherwise.
As an example, we illustrate in Fig. 2 the f 1 value of each node when the coloring is given in Fig. 1. The f 2 and f 3 values can be calculated similarly and we do not show them here to save space. Next, we will prove that { f 1 , f 2 , f 3 } is a Roman dominating family on H . Fix i ∈ {1, 2, 3} and consider all vertices u with f i (u ) = 0. There are several cases to be examined. 1. u = x v for some v ∈ V with c ( v ) = i. Then u is adjacent to ai for which f i (ai ) = 2. 2. u = y e for some e = v v ∈ E such that i ∈ {c ( v ), c ( v )}. Without loss of generality assume c ( v ) = i. Then u is adjacent to x v for which f i (x v ) = 2. 3. u = x v for some v ∈ V . Due to our construction, f i (x v ) = 0 if and only if f i (x v ) = 2, and x v is adjacent to x v in H . 4. u = y e for some e ∈ E. Similar to the previous case, u is adjacent to y e for which f i ( y e ) = 2. 5. u = ai . Since there exists v ∈ V with c ( v ) = i, ai is adjacent to x v for which f i (x v ) = 2. Therefore, every vertex u ∈ V with f i (u ) = 0 is adjacent to some v ∈ V for which f i ( v ) = 2, which indicates that f i is a Roman dominating function on H . Furthermore, since c is a 3-coloring of G, it holds that c (u ) = c ( v ) for all 3 e = uv ∈ E. Thus, it is easy to verify that i =1 f i ( v ) ≤ 2 for each v ∈ V , and the three functions are distinct. Hence, { f 1 , f 2 , f 3 } is indeed a Roman dominating family on H , implying that d R ( H ) ≥ 3. Now comes the “only if” part of the reduction. Suppose d R ( H ) ≥ 3 and { f 1 , f 2 , f 3 } is a Roman dominating family on H . For each x v ∈ V , the three values f 1 (x v ), f 2 (x v ), f 3 (x v ) must consist of exactly one “0” and two “1”s. To see this, first observe that at least one of them is “0”. If there are two “0”s, say, f 1 (x v ) = f 2 (x v ) = 0, then f 1 (x v ) = f 2 (x v ) = 2, contradicting the property of a Roman dominating family. Thus, among the three values there is exactly one “0”, and the remaining two must both be “1”. Consequently, the three values f 1 (x v ), f 2 (x v ), f 3 (x v ) must contain exactly one “2” and two “0”s. Analogous claim holds for each y e ∈ V ; i.e., the
three values f 1 ( y e ), f 2 ( y e ), f 3 ( y e ) are composed of exactly one “2” and two “0”s. We define a coloring c : V → {1, 2, 3} as follows. For each v ∈ V , let c ( v ) = i if f i (x v ) = 2. This coloring is well-defined since exactly one of f 1 (x v ), f 2 (x v ), and f 3 (x v ) equals 2. We next show that this coloring is a 3-coloring. Let e = uv be an arbitrary edge of G, and assume that c (u ) = c ( v ) = i. Then f i (xu ) = f i (x v ) = 2. As exactly two of f 1 ( y e ), f 2 ( y e ), and f 3 ( y e ) are 0 due to our previous analysis, there exists j ∈ {1, 2, 3} \ {i } such that f j ( y e ) = 0. Since N H ( y e ) = {xu , x v , y e }, and f j ( y e ) = 0 or 1, we have f j (xu ) = 2 or f j (x v ) = 2. Thus, there is x ∈ {u , v } for which f i (x) = f j (x) = 2, contradicting the fact that { f 1 , f 2 , f 3 } is a Roman dominating family on H . Accordingly, c (u ) = c ( v ) for each uv ∈ E, and thus c is indeed a 3-coloring of G. This completes the “only if” part of the reduction, and hence concludes the whole proof. 2 Corollary 1. It is N P -hard to compute the Roman domatic number of a given graph, even if the graph is bipartite. 2.2. Approximation behavior In this subsection we derive an asymptotically optimal approximation threshold for computing the Roman domatic number of a graph. Define the optimization problem Roman Domatic Number as follows: Given a graph G, a feasible solution is a Roman dominating family F on G, and the goal is to maximize the number of functions in F . Clearly, the optimal solution contains d R (G ) functions. Before proposing our results, we specify some notations related to approximation algorithms. Let be a maximization problem, and A be an algorithm for solving . For each instance I of , let OPT (, I ) and A(, I ) denote the objective values of the optimum solution to I and the solution returned by A on I , respectively. For β ≥ 1, A is called a β -approximation algorithm for if A runs in polyOPT (,I ) nomial time, and A(,I ) ≤ β for every instance I of . We refer the readers to [19] for standard definitions and notations not given here. To compute the Roman domatic number of a graph, we have the following theorem: Theorem 2. There is a (ln n + O (ln ln n))-approximation algorithm for Roman Domatic Number, where n is the order of the input graph. Proof. Let G = ( V , E ) be the input graph with | V | = n. We have d R (G ) ≤ δ(G ) + 2 owing to [1]. If δ(G ) ≤ ln n, we just output a trivial Roman dominating family on G (which consists of an arbitrary Roman dominating function on G). The approximation ratio of this solution is at most δ(G ) + 2 ≤ ln n + 2 ≤ ln n + O (ln ln n). Now consider the case where δ(G ) > ln n. Using Theorem 1 from [20], we can construct in polynomial time a domatic partition of G of size (δ(G ) + 1)(1 − K (ln ln n/ ln n))/ ln n for some constant K > 0. This naturally induces a Roman dominating family on G of the same size (by assigning the chosen vertices with function value 2 and others with 0 in each dominating set of this partition), which we output as our
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solution. We know that for ∈ (0, 1/2), 1/(1 − ) < 1 + 2 . Therefore, for sufficiently large n, the approximation ratio is at most
δ(G ) + 2 (δ(G ) + 1)(1 − K (ln ln n/ ln n))/ ln n 1 1 = 1+ ln n 1 − K (ln ln n/ ln n) δ(G ) + 1 ln ln n 1 ≤ 1 + 2K 1+ ln n =
1 + 2K
≤
ln n
ln ln n
+
ln n
1 + (4K + 1)
ln n
1 ln n
ln ln n ln n
+ 2K
ln ln n
(ln n)2
ln n
ln n
= ln n + O (ln ln n) . Therefore, we can always approximate Roman Domatic Number within a factor of ln n + O (ln ln n) in polynomial time, completing the proof of Theorem 2. 2 We next show that this approximation factor is asymptotically optimal. For a function f , D T I M E ( f (n)) denotes the set of decision problems decidable in time f (n) (by a deterministic Turing machine) on input of size n. For a class of functions C , let D T I M E (C ) = f ∈C D T I M E ( f (n)). Note that P = D T I M E (n O (1) ) in our notation. We have the following theorem.
Theorem 3. For every fixed > 0, there is no ( 12 − ) × ln n-approximation algorithm for Roman Domatic Number, where n is the order of the input graph, unless N P ⊆ D T I M E (n O (log log n) ). Proof of Theorem 3. We need the following lemma, which is proved in [20] for establishing the inapproximability result of the domatic number. Lemma 1 (Combining Propositions 10, 13 and Theorem 11 in [20]). Fix > 0 and assume N P D T I M E (n O (log log n) ). Given a graph G of order n, we cannot distinguish in polynomial time between the following two cases (even when we know G belongs to exactly one of them): 1. the size of any dominating set of G is at least r; 2. the domatic number of G is at least q, where r , q are parameters (easily computable from the input) satisfying that qr ≥ (1 − )n ln n. Fix > 0. Let G be an input graph of order n that satisfies exactly one of the cases in Lemma 1, and r , q be the parameters given by Lemma 1. Assume N P D T I M E (n O (log log n) ), and assume to the contrary that there exists a ( 12 − ) ln n-approximation algorithm for Roman Domatic Number. We apply this hypothetical algorithm to
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find, in polynomial time, a Roman dominating family on G of size d with d R (G ) ≥ d ≥ d R (G )/(( 12 − ) ln n). Now we examine the bounds on d for the two cases in Lemma 1.
• In the first case, γ (G ) ≥ r. According to [1], we have γ R (G ) ≥ γ (G ) and d(G ) ≤ d R (G ) ≤ 2n/γ R (G ). Therefore, d ≤ d R (G ) ≤ 2n/γ R (G ) ≤ 2n/γ (G ) ≤ 2n/r. • In the second case, d(G ) ≥ q, and thus d R (G ) ≥ d(G ) ≥ q. We have d ≥ d R (G )/(( 12 − ) ln n) ≥ q/(( 12 − ) ln n). Since qr ≥ (1 − )n ln n, we have d ≥ q/(( 12 − ) ln n) ≥ r ((11/−2−)n )lnlnnn > 2n/r. Therefore, by comparing d and 2n/r, we can distinguish whether the graph G belongs to case 1 or 2 in Lemma 1, which contradicts the indistinguishability result stated by the lemma. This means that our assumption is invalid, that is, there doesn’t exist any ( 12 − ) ln n-approximation algorithm for Roman Domatic Number, unless N P ⊆ D T I M E (n O (log log n) ). The proof is concluded. 2 3. Roman domatic number in particular graph classes In this section we determine the exact values of the Roman domatic number of some special classes of graphs. We first prove a useful theorem, from which several (old and new) results follow immediately. Theorem 4. Let G be an arbitrary graph. Construct H by adding a new vertex to G and connecting it with all the vertices in G. Then, d R ( H ) = d R (G ) + 1. Proof. If G is empty (i.e., containing no edges), we have d R (G ) = 1, and H is a tree of order at least 2. By Theorem 15 in [1], d R ( H ) = 2, and thus the theorem holds. In the following we will assume that G is non-empty. Denote by V the vertex set of G, and by v 0 be the vertex of H that is not in V (i.e., the vertex that is added to G for obtaining H ). The vertex set of H is thus V ∪ { v 0 }. Let { f 1 , f 2 , . . . , f d } be a Roman dominating family on H with d = d R ( H ). For each i ∈ {1, 2, . . . , d}, define f i : V → {0, 1, 2} as: f i ( v ) = f i ( v ) for all v ∈ V . Let I = {i | f i ( v 0 ) < 2; 1 ≤ i ≤ d}, and consider the family of functions F = { f i | i ∈ I }. For each function f i in F , if f i ( v ) = 0 for some v ∈ V , then there exists u ∈ N G ( v ) such that f i (u ) = f i (u ) = 2, since f i ( v 0 ) < 2 and f i is a Roman dominating function on H . Therefore, every f i in F is a Roman dominating function on G. It is also clear that, for d
each v ∈ V , i∈ I f i ( v ) ≤ i =1 f i ( v ) ≤ 2. We now prove that all these functions are distinct. Choose two arbitrary indices i 1 , i 2 ∈ I such that i 1 = i 2 . If f i 1 ( v 0 ) = f i 2 ( v 0 ), then f i and f i must be different, 1 2 since f i 1 and f i 2 are different. If f i 1 ( v 0 ) = f i 2 ( v 0 ), say, f i 1 ( v 0 ) = 0 and f i 2 ( v 0 ) = 1, then there exists v ∈ V for which f i 1 ( v ) = 2. Hence, f i 2 ( v ) must be 0, implying that f i 1 and f i 2 are distinct. Therefore, F is a Roman dominating family on G of size | I |. By the definition of Roman dominating family, there is at most one i for which
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H. Tan et al. / Information Processing Letters 116 (2016) 554–559
f i ( v 0 ) = 2. Thus, | I | ≥ d − 1, from which it follows that d R ( H ) = d ≤ | I | + 1 ≤ d R (G ) + 1. We next show that this upper bound is attainable. Let { f 1 , . . . , f d R (G ) } be a (maximum) Roman dominating family on G. Since G is non-empty, d R (G ) ≥ 2 holds due to Observation 2 in [1]. We claim that, for each i ∈ {1, 2, . . . , d R (G )}, there exists v ∈ V such that f i ( v ) = 2. If this is not the case, that is, there exists a function f i for which f i ( v ) < 2 for all v ∈ V , then it must hold that f i ( v ) = 1 for all v ∈ V . Now choose j ∈ {1, 2, . . . , d R (G )} \ {i } (recall that d R (G ) ≥ 2). We have f j ( v ) ≤ 2 − f i ( v ) ≤ 1, and thus f j ( v ) = 1 = f i ( v ) for all v ∈ V , which contradicts with the distinctness requirement in the definition of Roman dominating family. The claim is thus proved. Now define functions f 1 , f 2 , . . . , f d (G )+1 as follows: R
• For each i ∈ {1, 2, . . . , d R (G )}, f i ( v 0 ) = 0 and f i ( v ) = f i ( v ) for all v ∈ V ; • f d R (G )+1 ( v 0 ) = 2 and f d R (G )+1 ( v ) = 0 for all v ∈ V . For each i ∈ {1, 2, . . . , d R (G )}, f i is a Roman dominating function on H since by previous analysis, there exists v ∈ V for which f i ( v ) = f i ( v ) = 2. As v 0 is adjacent to all other vertices in H , f d R (G )+1 is also a Roman dominating function on H . Therefore, it is easy to see that { f 1 , f 2 , . . . , f d R (G )+1 } is a Roman dominating family on H of size d R (G ) + 1. Combining with the inequality d R ( H ) ≤ d R (G ) + 1 obtained before, we have d R ( H ) = d R (G ) + 1, which completes the proof of Theorem 4. 2 Based on the above Theorem 4, we can get the following results.
• Complete graphs Corollary 2 (Observation 1 in [1]). Let K n be the complete graph of order n ≥ 1. Then d R ( K n ) = n. Proof. Start with a single vertex and apply Theorem 4 for n − 1 times. 2
• Fan graphs For n ≥ 3, F n denotes the graph with V ( F n ) = { v 0 , v 1 , v 2 , . . . , v n−1 } and E ( F n ) = { v i v i +1 | 1 ≤ i ≤ n − 2} ∪ { v 0 v i | 1 ≤ i ≤ n − 1}. Corollary 3. Let F n be the fan graph of order n ≥ 3. Then d R ( F n ) = 3. Proof. For any tree T of order n ≥ 2, d R ( T ) = 2 (Theorem 15 in [1]). F n is actually to add a new vertex to a tree of n − 1 nodes and connect it to all the n − 1 vertices. Therefore, according to Theorem 4, we have d R ( F n ) = 3. 2
• Wheel graphs For n ≥ 3, let W n denote the wheel graph with V ( W n ) = { v 0 , v 1 , v 2 , . . . , v n } and E ( W n ) = { v i v i +1 | 1 ≤ i ≤ n − 1} ∪ { v n v 1 } ∪ { v 0 v i | 1 ≤ i ≤ n}.
Corollary 4. For n ≥ 3, let W n be the wheel graph of order n + 1. Then,
d R (W n ) =
4 3
if n ≡ 0 (mod 3); otherwise.
Proof. For n ≥ 3, we denote C n to be the cycle of length n. Then, W n is obtained by adding a new vertex v 0 to the cycle C n and connecting v 0 with all the original cycle vertices. According to Proposition 8 in [1], we have
d R (C n ) =
3 2
if n ≡ 0 (mod 3); otherwise.
Therefore, our corollary is achieved based on Theorem 4. 2
• Complete bipartite graphs For a1 , a2 ≥ 1, denote by K a1 ,a2 the complete bipartite graph with partite sets V 1 and V 2 such that | V 1 | = a1 and | V 2 | = a2 . Theorem 5. For all integers a1 , a2 ≥ 1, d R ( K a1 ,a2 ) = max{2, min{a1 , a2 }}. Proof. Assume without loss of generality that a1 ≤ a2 . If a1 = 1, K a1 ,a2 is a tree of order at least 2, and the theorem follows from Theorem 15 in [1]. Thus, in the following we suppose a ≥ 2, in which case max{2, min{a1 , a2 }} = a1 . Let V 1 = { v 1,1 , v 1,2 , . . . , v 1,a1 } and V 2 = { v 2,1 , v 2,2 , . . . , v 2,a2 }. Assume that { f 1 , f 2 , . . . , f d } is a Roman dominating family on K a1 ,a2 with d = d R ( K a1 ,a2 ). Trivially d ≥ 2. In what follows we examine two cases separately. 1. There exist i ∈ {1, . . . , d} and j ∈ {1, 2} such that f i ( v j ,k ) ≥ 1 for all k ∈ {1, . . . , a j }. Without loss of generality assume i = j = 1. (The proof for this case does not rely on the fact that a1 ≤ a2 , so similar argument goes through for j = 2.) Then, for each 2 ≤ i ≤ d, f i ( v 1,k ) ≤ 1 for all 1 ≤ k ≤ a1 . Consequently, f i ( v 2,k ) = 0 (and thus ≥ 1) for all 1 ≤ k ≤ a2 , since otherwise f i would not be a Roman dominating function on K a1 ,a2 . Therefore, there exists at most two such i ; that is, d ≤ 3. Now assume d = 3. According to the analysis above, it must hold that for each i ∈ {2, 3}, f i ( v 2,k ) = 1 for all 1 ≤ k ≤ a2 . Hence, f i ( v 1,k ) = 0 for all i ∈ { 2, 3} and 1 ≤ k ≤ a1 . This implies, however, that 3 i =1 f i ( v 1,1 ) ≥ 3, contradicting with the property of a Roman dominating family. Thus, we have d ≤ 2. 2. For all i ∈ {1, . . . , d} and j ∈ {1, 2}, there is k ∈ {1, . . . , a j } for which f i ( v j ,k ) = 0. This indicates that, for each i ∈ {1, . . . , d} and j ∈ {1, 2}, there exists k ∈ {1, . . . , a j } such that f i ( v j ,k ) = 2. Since each vertex can have value 2 under at most one function in a Roman dominating family, d ≤ min{a1 , a2 } = a1 holds. Therefore, we have d ≤ max{2, a1 } = a1 . On the other hand, define F = { f 1 , . . . , f a1 }, where for each 1 ≤
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i ≤ a1 , f i ( v 1,i ) = f i ( v 2,i ) = 2 and f i ( v ) = 0 for all v ∈ ( V 1 ∪ V 2 ) \ { v 1,i , v 2,i }. It is easy to see that F is a Roman dominating family on K a1 ,a2 of size a1 . Thus, d R ( K a1 ,a2 ) = d = a1 , completing the proof of Theorem 5. 2 4. Conclusion and future work In this paper, we gave the first computational hardness result for Roman domatic number, and prove an asymptotically optimal approximation threshold for computing this graph parameter. It is interesting to see whether we can further close the gap between the approximation ratio of (1 + o(1)) ln n and the inapproximability factor of ( 12 − o(1)) ln n. Also, it remains open whether the Roman domatic number can be computed efficiently on some well-studied graph classes such as line graphs and interval graphs. References [1] S. Sheikholeslami, L. Volkmann, The Roman domatic number of a graph, Appl. Math. Lett. 23 (2010) 1295–1300. [2] R. Diestel, Graph Theory, 4th edition, Springer-Verlag, 2010. [3] T. Haynes, S.T. Hedetniemi, P. Slater, Domination in Graphs: Advanced Topics, Marcel Dekker, 1998. [4] T. Haynes, S.T. Hedetniemi, P. Slater, Fundamentals of Domination in Graphs, Marcel Dekker, 1998. [5] E. Cockayne, S. Hedetniemi, Towards a theory of domination in graphs, Networks 7 (1977) 247–261.
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