Algorithmic aspects of k-tuple total domination in ... - Semantic Scholar
Information Processing Letters 112 (2012) 816–822
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Algorithmic aspects of k-tuple total domination in graphs D. Pradhan 1 Department of Computer Science and Automation, Indian Institute of Science, Bangalore 560012, India
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Article history: Received 17 September 2011 Received in revised form 25 May 2012 Accepted 17 July 2012 Available online 20 July 2012 Communicated by R. Uehara Keywords: Graph algorithms Domination Total domination k-Tuple total domination NP-complete APX-complete
a b s t r a c t For a fixed positive integer k, a k-tuple total dominating set of a graph G = ( V , E ) is a subset T D k of V such that every vertex in V is adjacent to at least k vertices of T D k . In minimum k-tuple total dominating set problem (Min k-Tuple Total Dom Set), it is required to find a k-tuple total dominating set of minimum cardinality and Decide Min k-Tuple Total Dom Set is the decision version of Min k-Tuple Total Dom Set problem. In this paper, we show that Decide Min k-Tuple Total Dom Set is NP-complete for split graphs, doubly chordal graphs and bipartite graphs. For chordal bipartite graphs, we show that Min k-Tuple Total Dom Set can be solved in polynomial time. We also propose some hardness results and approximation algorithms for Min k-Tuple Total Dom Set problem. © 2012 Elsevier B.V. All rights reserved.
1. Introduction For a graph G = ( V , E ), the sets N G ( v ) = {u ∈ V | uv ∈ E } and N G [ v ] = N G ( v ) ∪ { v } denote the neighborhood and the closed neighborhood of a vertex v, respectively. A set D of vertices of a graph G is a dominating set of G if every vertex in V \ D is adjacent to a vertex in D. Equivalently, a subset D of V is a dominating set of G if | N G [ v ] ∩ D | 1 for every v ∈ V . The domination number of a graph G, denoted by γ (G ), is the minimum cardinality of a dominating set of G. The concept of domination and its variations have many applications and have been widely studied in literature (see [11,12]). Among the variations of domination, total domination is one of those. A set D of vertices of a graph G is a total dominating set of G if every vertex in V is adjacent to at least one vertex of D. Equivalently, a subset D of V is a total dominating set of G if | N G ( v ) ∩ D | 1 for every v ∈ V . The total domination number of a graph G, denoted by γt (G ), is the minimum cardinality of a total dominating set of G. As a variation of domination, the concept of
E-mail address: [email protected]. The author was supported by National Board for Higher Mathematics (NBHM) and Dr. D.S. Kothari Postdoctoral Fellowship (DSKPDF), India. 1
0020-0190/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ipl.2012.07.010
k-tuple total domination has been introduced by Henning and Kazemi [13]. For a fixed positive integer k, a k-tuple total dominating set of a graph G is a subset T D k of V such that every vertex in V is adjacent to at least k vertices of T D k . Equivalently, a set T D k ⊆ V is a k-tuple total dominating set of G if | N G ( v ) ∩ T D k | k for every v ∈ V . So k-tuple total domination is the generalization of the usual total domination. The case when k = 2 is called double total domination [13] where some upper bounds are obtained for the minimum cardinality of a double total dominating set. The k-tuple total domination number of a graph G, denoted by γ×k,t (G ), is the minimum cardinality of a k-tuple total dominating set of G. Upper bounds and lower bounds for γ×k,t (G ) are also proposed in [13,14] in terms of different graph parameters. In this paper, we study on k-tuple total domination from algorithmic point of view. As per our knowledge, no algorithmic result has been obtained for k-tuple total domination in graphs. We first present some NP-hardness and polynomial solvable cases of Min k-Tuple Total Dom Set problem on different graph classes. We then extend these studies by investing the approximation hardness of Min k-Tuple Total Dom Set problem. We also propose a constant approximation algorithm for Min k-Tuple Total Dom Set problem. For the general concept of algorithms and
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approximation algorithms, one can see [2,7]. We derive the following main results in this paper: 1. Decide Min k-Tuple Total Dom Set is NP-complete for split graphs, doubly chordal graphs and bipartite graphs. 2. Min k-Tuple Total Dom Set is polynomially solvable for chordal bipartite graphs. 3. Min k-Tuple Total Dom Set cannot be approximated within (1 − ε ) ln | V | for any ε > 0, unless NP ⊆ DTIME(| V | O (log log | V |) ). 4. Min k-Tuple Total Dom Set is APX-complete for graphs of maximum degree k + 2. In particular, this is true for bipartite graphs of maximum degree k + 2. 2. Preliminaries For a graph G = ( V , E ), the degree of a vertex v is | N G ( v )| and is denoted by d G ( v ). If d G ( v ) = 1, then v is called a pendant vertex of G. If d G ( v ) = 0, then v is called an isolated vertex of G. Let (G ) and δ(G ) denote the maximum degree and minimum degree of a graph G, respectively. For S ⊆ V , let G [ S ] denote the subgraph induced by S. If G [C ], C ⊆ V , is a complete subgraph of G, then C is called a clique of G. A set S ⊆ V is an independent set if G [ S ] has no edge. A graph is said to be a chordal graph if every cycle of length at least four has a chord i.e., an edge joining two non-consecutive vertices of the cycle. A chordal graph G = ( V , E ) is a split graph if V can be partitioned into two sets K and S such that K is a clique and S is an independent set. A vertex v ∈ V (G ) is a simplicial vertex of G if N G [ v ] is a clique of G. An ordering α = ( v 1 , v 2 , . . . , v n ) is a perfect elimination ordering (PEO) of G if v i is a simplicial vertex of G i = G [{ v i , v i +1 , . . . , v n }] ∀i , 1 i n. It is characterized that a graph G is chordal if and only if it has a PEO [10]. A vertex u ∈ N G [ v ] is a maximum neighbor of v in G if N G [ w ] ⊆ N G [u ] for all w ∈ N G [ v ]. An ordering σ = ( v 1 , v 2 , . . . , v n ) of V (G ) is called a maximum neighborhood ordering (MNO) of G if v i has a maximum neighbor in G i = G [{ v i , v i +1 , . . . , v n }] for each i, 1 i n. Graphs that admit maximum neighborhood orderings are called as dually chordal graphs [4]. A vertex v is called doubly simplicial in G if it is simplicial and has a maximum neighbor in G. An ordering σ = ( v 1 , v 2 , . . . , v n ) of V (G ) is called a doubly perfect elimination ordering (dpeo) of G if v i is a doubly simplicial vertex in G i = G [{ v i , v i +1 , . . . , v n }] for each i, 1 i n. A graph is doubly chordal if it admits a doubly perfect elimination ordering (dpeo) [4]. Note that every doubly chordal graph is a dually chordal graph. 3. Complexity of Decide Min k-Tuple Total Dom Set problem in graphs In this section, we show that Decide Min k-Tuple Total Dom Set is NP-complete for split graphs, doubly chordal and bipartite graphs. Hence Decide Min k-Tuple Total Dom Set is NP-complete for chordal graphs and dually chordal graphs. A set S ⊆ V of a graph G = ( V , E ) is called a vertex cover of G if for every edge uv ∈ E, either u ∈ S or v ∈ S.
Fig. 1. The constructed split graph G from a graph G.
In minimum vertex cover problem (Min Vertex Cover), it is required to find a vertex cover of minimum cardinality and let Decide Min Vertex Cover be the decision version of Min Vertex Cover problem. Decide Min Vertex Cover is known to be NP-complete for general graphs [15]. We show that Decide Min k-Tuple Total Dom Set is NPcomplete for split graphs by providing a polynomial time transformation from the Decide Min Vertex Cover problem. Theorem 3.1. For a fixed integer k, Decide Min k-Tuple Total Dom Set is NP-complete for split graphs. Proof. Given a graph G = ( V , E ), we construct a split graph G = ( V , E ) with vertex set V = V ∪ S ∪ E ∪ {x} where S = {s1 , s2 , . . . , sk } and x ∈ / V ∪ S ∪ E and with edge set E = {uv | u = v and u , v ∈ V ∪ S } ∪ { ve | v ∈ V , e ∈ E and v ∈ e } ∪ {si e | e ∈ E and 1 i k − 1} ∪ {si x | 1 i k}. Notice that G is a split graph as V ∪ S is a clique and E ∪ {x} is an independent set. Since | V | = | V | + | E | + k + 1, the construction of G can be done in polynomial time. The construction of G is illustrated in Fig. 1 for k = 3. Next, we show that G has a vertex cover of cardinality α if and only if G has a k-tuple total dominating set of cardinality α + k. Let C be a vertex cover of G. Then it is clear that T D k = C ∪ {s1 , s2 , . . . , sk } is a k-tuple total dominating set of G with | T D k | = α + k. On the other hand, suppose that D is a k-tuple total dominating set of G with cardinality α + k. Then it is clear that s1 , s2 , . . . , sk ∈ D since N G (x) = {s1 , s2 , . . . , sk }. Now let C = D \ {s1 , s2 , . . . , sk }. We claim that C is a vertex cover of G. Let ab = e i ∈ E. Since N G (e i ) ⊂ S ∪ {a, b} and | N G (e i ) ∩ S | = k − 1, D contains at least one of a and b. So either a ∈ C or b ∈ C . Hence C is a vertex cover of G with |C | = | D | − k = α . Therefore, Decide Min k-Tuple Total Dom Set is NPcomplete for split graphs. 2 We show that Decide Min k-Tuple Total Dom Set is NPcomplete even for bipartite graphs by giving a polynomial time transformation from the Decide Min Vertex Cover problem.
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Dom Set be the decision version of the problem. Decide Min Total Dom Set is NP-complete for chordal graphs [17]. Now, we show that Decide Min k-Tuple Total Dom Set is NP-complete for another important subclass of chordal graphs known as doubly chordal graphs. Given an instance of Decide Min (k − 1)-Tuple Total Dom Set problem, we construct a doubly chordal graph for the instance of Decide Min k-Tuple Total Dom Set problem. Let G = ( V , E ) be a chordal graph and x ∈ / V . Construct a graph G = ( V , E ) where V = V ∪ {x} and E = E ∪ {xv | v ∈ V }. If σ = ( v 1 , v 2 , . . . , v n ) is a PEO of G, then σ = ( v 1 , v 2 , . . . , v n , x) is a dpeo of G . So G is a doubly chordal graph. Claim. γ×k,t (G ) = γ×(k−1),t (G ) + 1. Fig. 2. The constructed bipartite graph G from a graph G.
Theorem 3.2. For any fixed positive integer k, Decide Min kTuple Total Dom Set is NP-complete for bipartite graphs. Proof. Let S = {s1 , s2 , . . . , sk }, S = {s1 , s2 , . . . , sk }, T = {t 1 , t 2 , . . . , tk }, T = {t 1 , t 2 , . . . , tk }, and T = {t 1 , t 2 , . . . , tk }. Given a graph G = ( V , E ), we construct a bipartite graph G = ( V , E ) with vertex set V = S ∪ T ∪ V ∪ E ∪ T ∪ S ∪ T and edge set E is described as follows:
• {si t j | si ∈ S , t j ∈ T } ∪ {si t j | si ∈ S , t j ∈ T } ∪ {si t j | si ∈ S , t j ∈ T }; • {ut j | u ∈ V , t j ∈ T } ∪ {et j | e ∈ E , t j ∈ T \ {tk }}; • { ve | v ∈ V , e ∈ E and v ∈ e }. Notice that G is a bipartite graph with partite sets V ∪ S ∪ T and E ∪ S ∪ T ∪ T . Since | V | = | V | + | E | + 5k, the construction of G can be done in polynomial time. The construction of G is illustrated in Fig. 2 for k = 3. Next, we show that G has a vertex cover of cardinality α if and only if G has a k-tuple total dominating set of cardinality α + 4k. If C is a vertex cover of G, then C ∪ S ∪ S ∪ T ∪ T is a k-tuple total dominating set of G . Hence G has a ktuple total dominating set of cardinality α + 4k. On the other hand, let D be a k-tuple total dominating set of G of cardinality α + 4k. Then it is clear that ( S ∪ S ∪ T ) ⊂ D. The vertices of set E ∪ T that belongs to D can be interchanged by the vertices of T . So without loss of generality, we assume that ( S ∪ T ) ⊂ D. Since | N G (e ) ∩ T | = k − 1 for each e ∈ E, C = D \ ( S ∪ S ∪ T ∪ T ) = ∅. Clearly |C | = | D | − | S ∪ S ∪ T ∪ T | = α . We now claim that C is a vertex cover of G. Suppose that e ∗ = ab ∈ E. Since | D ∩ N G (e ∗ )| k and e ∗ is adjacent to the vertices of ( T \ {tk }) ∪ {a, b}, D contains either a or b. This implies that a ∈ C or b ∈ C . Therefore, Decide Min k-Tuple Total Dom Set is NPcomplete for bipartite graphs. 2 In minimum total dominating set (Min Total Dom Set) problem, it is required to find a total dominating set of a graph of minimum cardinality and let Decide Min Total
Proof of the Claim. If D is a (k − 1)-tuple total dominating set of G, then D ∪ {x} is a k-tuple total dominating set of G . So, γ×k,t (G ) γ×(k−1),t (G ) + 1. On the other hand, assume that D is a minimum ktuple total dominating set of G . If x ∈ D , then it is clear that D \ {x} is a (k − 1)-tuple total dominating set of G. If x∈ / D , then D = ( D \ { v }) ∪ {x} for some v ∈ D , is also a minimum k-tuple total dominating set of G and D \ {x} is a (k − 1)-tuple total dominating set of G. So, γ×(k−1),t (G ) γ×k,t (G ) − 1 implying the equality. 2 Since Decide Min Total Dom Set is NP-complete for chordal graphs, by induction on k, we can prove that for k > 1, Decide Min k-Tuple Total Dom Set is NP-complete for doubly chordal graphs. Hence we have the following theorem Theorem 3.3. For any fixed integer k > 1, Decide Min k-Tuple Total Dom Set is NP-complete for doubly chordal graphs. By Theorems 3.1 and 3.3, the following corollary arises. Corollary 3.4. For any fixed integer k > 0, Decide Min k-Tuple Total Dom Set is NP-complete for chordal graphs. 4. Polynomial solvable case A graph is called a chordal bipartite graph if it is bipartite and every cycle of length at least six has a chord i.e., an edge joining two non-consecutive vertices of the cycle. Note that chordal bipartite graphs are not necessarily chordal. For any integer n 3, an n-sun is a graph with 2n vertices, say x1 , x2 , . . . , xn and y 1 , y 2 , . . . , yn such that {x1 , x2 , . . . , xn } is a clique and { y 1 , y 2 , . . . , yn } is an independent set and y i is adjacent to xi and xi −1 for 2 i n and y 1 is adjacent to x1 and xn . A strongly chordal graph is a chordal graph containing no n-sun (n 3) as an induced subgraph [5,8]. In this section, we show that Min k-Tuple Total Dom Set can be solved in polynomial time for chordal bipartite graphs. A polynomial time algorithm has been proposed in [18] for computing a minimum k-tuple dominating set in a strongly chordal graph. It is also mentioned in [18] that a minimum k-tuple total dominating set can be computed
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by the same algorithm by replacing the closed neighbors in most of the statements with the concerned neighbors. So it follows that Min k-Tuple Total Dom Set can be solved in polynomial time for strongly chordal graphs. Let G = ( X , Y , E ) be a chordal bipartite graph and C X (G )(C Y (G )) denote the split graph obtained from G by adding edges between every pair of vertices in X (Y ). Then C X (G ) is a strongly chordal graph which follows from the following theorem appears in [3]. Theorem 4.1. (See [3].) Let G = ( X , Y , E ) is a bipartite graph. Then G is a chordal bipartite graph if and only if C X (G ) is a strongly chordal graph. Let G = ( X , Y , E ) be a bipartite graph and k > 0 be an X ( Y ) be the minimum cardinality subset of X integer. Let X | k (| N G (x) ∩ Y | k) for every (Y ) such that | N G ( y ) ∩ y ∈ Y (x ∈ X ). The proof of the following lemma is easy and hence is omitted. Lemma 4.2. X ∪ Y is a minimum k-tuple total dominating set of a bipartite graph G = ( X , Y , E ). The next theorem shows that the Min k-Tuple Total Dom Set problem can be solved in polynomial time for chordal bipartite graphs. Theorem 4.3. Min k-Tuple Total Dom Set is solvable in polynomial time for chordal bipartite graphs. Proof. Suppose that G = ( X , Y , E ) is a chordal bipartite graph. Note that δ(G ) k; otherwise there is no k-tuple total dominating set of G. By Lemma 4.2, we require to X and Y of G in polynomial time. We next show compute X can be computed in polynomial time and comthat how Y is similar. Add edges to G such that the resulting puting graph will be C X (G ). By Theorem 4.1, C X (G ) is a strongly chordal graph. Consider a minimum k-tuple dominating set X is also a kD X of C X (G ). It is clear that | D X | k. Since tuple dominating set of C X (G ), | D X | | X |. Without loss of generality, we assume that D X contains the vertices of X only; otherwise we can replace by an adjacent vertex of the vertex y ∈ Y if y belongs to D X . Note that such a neighbor of y exists; otherwise D X \ { y } is a smaller kX |, then tuple dominating set of C X (G ). Again if | D X | < | we get a contradiction to the choice of X . So our problem is a subproblem of minimum k-tuple dominating set problem for strongly chordal graphs. Since the minimum k-tuple dominating set problem is solvable in polynomial time [18], the theorem follows. 2 5. Hardness and approximation In this section, we present some hardness and approximation results for the Min k-Tuple Total Dom Set problem. First we give an approximation preserving reduction from Min Total Dom Set to obtain a stronger inapproximability result for Min k-Tuple Total Dom Set in general graphs. For this we need the following theorem proved in [6,9].
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Theorem 5.1. (See [6,9].) Min Total Dom Set cannot be approximated within (1 − ε ) ln | V | for any ε > 0, unless NP ⊆ DTIME(| V | O (log log | V |) ). The same results hold for bipartite graphs and chordal (split) graphs. Theorem 5.2. Min k-Tuple Total Dom Set cannot be approximated within (1 − ε ) ln | V | for any ε > 0, unless NP ⊆ DTIME(| V | O (log log | V |) ). Proof. Let G = ( V , E ) be an instance of Min Total Dom Set with n vertices and m edges. Given G = ( V , E ), we construct a graph G = ( V , E ), an instance of Min k-Tuple Total Dom Set, in polynomial time as follows:
• Add k + 1 new vertices v 1 , v 2 , . . . , v k+1 . • Add all possible the edges v i v j , i = j so that { v 1 , v 2 , . . . , v k+1 } forms a clique. • For each v ∈ V , add the edges v v i , 1 i k − 1. Clearly, it can be seen that G can be constructed in polynomial time as | V | = n + k + 1. Next, we prove that γ×k,t (G ) = γt (G ) + k + 1. Let D be a minimum total dominating set of G. Then D k = D ∪ { v 1 , v 2 , . . . , v k+1 } is a k-tuple total dominating set of G . So γ×k,t (G ) | D k | = | D | + k + 1 = γt (G ) + k + 1. On the other hand, suppose that D k∗ is a k-tuple total dom-
inating set of G . Then it is clear that v 1 , v 2 , . . . , v k+1 ∈ D k∗ as d G ( v k ) = d G ( v k+1 ) = k. We prove that D ∗ = D k∗ \
{ v 1 , v 2 , . . . , v k+1 } is a total dominating set of G. Let v ∈ V . Since | N G ( v ) ∩ D k∗ | k and | N G ( v ) ∩ { v 1 , v 2 , . . . , v k+1 }| = k − 1, there exists a vertex u ∈ V such that u ∈ D ∗ . This implies that | N G ( v ) ∩ D ∗ | 1 for each v ∈ V and hence D ∗ is a total dominating set of G. So γt (G ) γ×k,t (G ) − (k + 1) which implies the equality. Now assume that Min k-Tuple Total Dom Set can be approximated within ratio αk 1 by using an algorithm A k . Let l > 0 be an integer. Consider the following algorithm: Algorithm 1: A k,l 1 2 3 4 5 6 7 8
Input: A graph G = ( V , E ); Output: A minimum total dominating set D of G; if (there exists a minimum total dominating set D of G of cardinality < l) then Return D; else Construct G ; Compute a k-tuple total dominating set D k of G using algorithm A k ; D = D k \ S, where S = { v 1 , v 2 , . . . , v k+1 } end Return D;
Step 1 of the algorithm can be completed in polynomial time as l is a constant. Again since A k is a polynomial time algorithm, algorithm A k,l is polynomial. Note that if D is computed in Step 1, then D is optimal. So we analyze the case where D is constructed in next lines. Let D k∗ be an optimal k-tuple total dominating set in G and D ∗ be an optimal total dominating set in G. It is clear that | D ∗ | l. Let D be the total dominating set computed by algorithm A k,l . Then | D | = | D k | − (k + 1) | D k | αk | D k∗ | = αk (1 + k| D+∗1| )| D ∗ | αk (1 + k+l 1 )| D ∗ |. Hence,
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algorithm A k,l approximates Min Total Dom Set within the ratio αk (1 + k+l 1 ). Assume that there exists some fixed ε > 0 such that Min k-Tuple Total Dom Set can be approximated within ratio α = (1 − ε ) ln | V | using algorithm A k . Let l be a positive integer such that k+l 1 < ε2 . Then algorithm A k,l approximates Min Total Dom Set within ratio
ε)(1 + ε2 ) ln | V | = (1 − ε ) ln | V | for
ε
αk (1 + k+l 1 ) (1 −
2 = ε2 + ε2 as ln | V | =
ln(| V | + k + 1) ≈ ln | V | for sufficiently large value of | V |. Therefore, it follows by Theorem 5.1 that Min k-Tuple Total Dom Set cannot be approximated within ratio (1 − ε ) ln | V | for any ε > 0, unless NP ⊆ DTIME(| V | O (log log | V |) ). 2 Theorem 5.3. Min k-Tuple Total Dom Set for bipartite graphs cannot be approximated within (1 − ε ) ln | V | for any ε > 0, unless NP ⊆ DTIME(| V | O (log log | V |) ). Proof. Let G = ( V , E ) be an instance of Min Total Dom Set with n vertices and m edges. Given a bipartite graph G = ( X , Y , E ), we construct a bipartite graph G = ( X , Y , E ), an instance of Min k-Tuple Total Dom Set, in polynomial time as follows:
• Add 2k new vertices {x1 , x2 , . . . , xk } and { y 1 , y 2 , . . . , yk }. Let X 1 = {x1 , x2 , . . . , xk } and Y 1 = { y 1 , y 2 , . . . , yk }. • Add all possible the edges xi y j so that G [ X 1 ∪ Y 1 ] becomes a complete bipartite graph.
Similar to the proof of Theorem 5.2, we can choose the positive integer l such that 2k < ε2 , where ε > 0 and can l prove easily that Min k-Tuple Total Dom Set for bipartite graphs cannot be approximated within ratio (1 − ε ) ln | V | for any ε > 0, unless NP ⊆ DTIME(| V | O (log log | V |) ). 2 Note that if G is a chordal graph, then the constructed graph G as in Theorem 5.2 is also a chordal graph. So we immediately have the following corollary. Corollary 5.4. Min k-Tuple Total Dom Set for chordal graphs cannot be approximated within (1 − ε ) ln | V | for any ε > 0, unless NP ⊆ DTIME(| V | O (log log | V |) ). Next we show that Min k-Tuple Total Dom Set problem is in APX. For this, we need to define some terminologies which are introduced in [16]. Let X be any nonempty set and F be a family of subsets of X . Let k > 0 be an integer.
• An element x ∈ X is k-covered in a set C ⊆ F of subsets of X if x is in at least k sets of C . • A k-cover of ( X , F ) is a subset C of F such that x is k-covered in C for each x ∈ X . • A k-cover of ( X , F ) of minimum cardinality is called a minimum k-cover. Algorithm 2: GEN-SET-COVER
• For each x ∈ X , add the edges xy j , 1 j k − 1. • For each y ∈ Y , add the edges yxi , 1 i k − 1. Notice that G = ( X , Y , E ) is a bipartite graph with X = X ∪ X 1 and Y = Y ∪ Y 1 . Clearly, it can be seen that G can be constructed in polynomial time as | V | = n + 2k. Next, we show that γ×k,t (G ) = γt (G ) + 2k. Let D be a minimum total dominating set of G. Then D k = D ∪ {xi , y i | 1 i k} is a k-tuple total dominating set of G . So γ×k,t (G ) | D k | = | D | + 2k = γt (G ) + 2k. On the other hand, suppose that D k∗ is a k-tuple total dominating set of G . Then it is clear that xi , y i ∈ D k∗ for 1 i k
as d G (xk ) = d G ( yk ) = k. We prove that D ∗ = D k∗ \ {xi , y i | 1 i k} is a total dominating set of G. Let x ∈ X . Since | N G (x) ∩ D k∗ | k and | N G (x) ∩ { y 1 , y 2 , . . . , yk }| = k − 1, there exists a vertex y ∈ Y such that y ∈ D ∗ . This implies that | N G (x) ∩ D ∗ | 1 for each x ∈ X . Similar arguments can be made for each y ∈ Y . Hence D ∗ is a total dominating set of G. Therefore, γt (G ) γ×k,t (G ) − 2k implying the equality. Now assume that Min k-Tuple Total Dom Set can be approximated within ratio αk 1 by using an algorithm A k . Let l > 0 be an integer. Modifying the algorithm A k,l by making the set S = {xi , y i | 1 i k}. Let D k∗ be an optimal k-tuple total dominating set in
G and D ∗ be an optimal total dominating set in G. It is clear that | D ∗ | l. Let D be the total dominating set computed by algorithm A k,l . Then | D | = | D k | − 2k | D k | αk | D k∗ | = αk (1 + | D2k∗ | )| D ∗ | αk (1 + 2kl )| D ∗ |. Hence, algorithm A k,l approximates Min Total Dom Set within the ra). tio αk (1 + 2k l
1 2 3 4 5 6 7 8 9 10
Input: An integer k > 0, a set X and a family F of subsets of X such that F is a k-cover of ( X , F ); Output: A minimum k-cover C of ( X , F ); C = ∅; i = 0; while ( X \ ( S 1 ∪ S 2 ∪ · · · ∪ S i ) = ∅) do i + +; Choose S ∈ F \ C such that | S \ ( S 1 ∪ S 2 ∪ · · · ∪ S i −1 )| is maximized; S i = S; S i = set of elements of X , k-covered in C ∪ { S i } but not in C ; C = C ∪ { S i }; end Output (C );
Theorem 5.5. (See [16].) Let X be a set, F be a family of subsets of X and k > 0 be an integer. Let S M be one set of F with maximum cardinality. Then (i) Algorithm GEN-SET-COVER outputs a k-cover C of ( X , F ) in polynomial time. (ii) The cardinality of the output of Algorithm GEN-SET-COVER with input ( X , F ) is at most ln(| S M |) + 1 times the cardinality of an optimal k-cover set of ( X , F ). Theorem 5.6. Min k-Tuple Total Dom Set problem in any graph G = ( V , E ) with maximum degree (G ) can be approximated with an approximation ratio of ln((G )) + 1. Proof. Clearly, G contains a k-tuple total dominating set if and only if δ(G ) k. If this is the case, then let X = V and F = { N G ( v ) | v ∈ V }. Suppose the algorithm GEN-SETCOVER on ( X , F ) outputs C . Finally, D = { v ∈ V | N G ( v ) ∈ C } is a k-tuple total dominating set of G. In this case, | S M | = (G ). So by Theorem 5.5, the theorem follows. 2
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5.1. k-Tuple total domination in bounded degree graphs By Theorem 5.6, it is clear that if the degree of the graph is bounded by a constant, the approximation ratio is constant. Hence Min k-Tuple Total Dom Set for bounded degree graphs is in APX. Next we prove that Min k-Tuple Total Dom Set is APX-complete even if the degree of the graph is bounded by k + 2. Hereafter, we denote Min kTuple Total Dom Set-(B) as the Min k-Tuple Total Dom Set restricted to graphs (bipartite graphs) of degree at most . Similarly, we denote Min Total Dom Set-(B) as the Min Total Dom Set restricted to graphs (bipartite graphs) of degree at most . To show that Min k-Tuple Total Dom Set-k + 2 is APXcomplete for k 2, we require the following theorem that we are going to prove. Theorem 5.7. Min Total Dom Set-3 is APX-complete. Proof. It is well known that the Min Total Dom Set-3 is in APX. Since Min Vertex Cover-3 is APX-complete [1,19], it is enough to establish an L-reduction [19] from Min Vertex Cover-3. Given a graph G = ( V , E ) of degree at most 3, an instance of Min Vertex Cover-3, we construct a graph G = ( V , E ) as an instance of Min Total Dom Set-3 in polynomial time as follows. For each edge e = uv, we subdivide the edge uv by introducing a new vertex e such that ue , ve ∈ E . Then we introduce a pendant vertex e such that e e ∈ E . Note that G is also a bipartite graph. Next, we prove that γt (G ) = β(G ) + m, where β(G ) is the cardinality of a minimum vertex cover of G. If S is a vertex cover of G, then S ∪ {e | e ∈ E (G )} is a total dominating set of G . So we have γt (G ) β(G ) + m. On the other hand, let D be a minimum total dominating set of G . Suppose that e = uv ∈ E (G ). To dominate the vertex e , either e ∈ D or e ∈ D. If e ∈ D, then e ∈ D. If u ∈ D or v ∈ D, then D \ {e } is a smaller total dominating set of G . This is a contradiction. So assume that u , v ∈ / D. Then ( D \ {e }) ∪ {e } is also a minimum total dominating set of G . So without loss of generality, we assume that / D and e ∈ D for each e ∈ E (G ). Again we have seen e ∈ that for each edge uv ∈ E (G ), either u or v is contained in D. So it is clear that D \ {e | e ∈ E (G )} is a vertex cover of G and hence β(G ) γt (G ) − m. This implies the equality. Since (G ) = 3, we have m 3n and β(G ) n4 . Now 2
γt (G ) β(G ) + m β(G ) + 3n β(G ) + 6β(G ) = 7β(G ). 2 Since e belongs to any total dominating set D of G for each e ∈ E (G ), D \ {e | e ∈ E (G )} is always a vertex cover of G. So |C | − |C ∗ | | D | − m − (| D ∗ | − m) = | D | − | D ∗ |. From these two inequalities, it is clear that the above reduction is an L-reduction with a = 7 and b = 1. Therefore, Min Total Dom Set-3 is APX-complete. 2
Due to the reduction used in Theorem 5.7, it can be seen easily that G is also a bipartite graph and hence the following corollary directly follows from Theorem 5.7. Corollary 5.8. Min Total Dom Set-B3 is APX-complete.
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In the following theorem, we show that Min k-Tuple Total Dom Set is APX-complete for graphs of degree at most k + 2 for any k 2. Theorem 5.9. Min k-Tuple Total Dom Set-(k + 2) is APXcomplete for any k 2. Proof. Clearly, by Theorem 5.6, Min k-Tuple Total Dom Set-(k + 2) is in APX. Since by Theorem 5.7, Min Total Dom Set-3 is APX-complete, it is enough to establish an L-reduction from Min Total Dom Set-3 to Min k-Tuple Total Dom Set-(k + 2). Given a graph G = ( V , E ), an instance of Min Total Dom Set-3, we construct a graph G = ( V , E ) of degree at most k + 2 as follows. For each vertex v ∈ V , we take a complete graph G ( v ) on k + 1 vertices, say v 1 , v 2 , . . . , v k+1 and connect v to v i , 1 i k − 1. Note that (G ) = k + 2. Next, we prove that γ×k,t (G ) = γt (G ) + (k + 1)n. If D is a total dominating set of G, then D ∪ { v 1 , v 2 , . . . , v k+1 | v ∈ V } is a k-tuple total dominating set of G . So γ×k,t (G ) γt (G ) + (k + 1)n. On the other hand, suppose that D k is a k-tuple total dominating set of G . Then it is clear that { v 1 , v 2 , . . . , v k+1 | v ∈ V } ⊆ D k . Since v is adjacent to v i , 1 i k − 1, we have | N G ( v )∩ D k | 1. Let D = D k \ { v 1 , v 2 , . . . , v k+1 | v ∈ V }. It is clear that D is a total dominating set of G and hence γt (G ) γ×k,t (G )−(k + 1)n. This implies the equality and the claim follows. Since (G ) = 3 and γt (G ) γ (G ) n4 , we have γ×k,t (G ) γt (G ) + (k + 1)n γt (G ) + 4(k + 1)γt (G ) = (4k + 5)γt (G ). Again γt (G ) − | D | = γ×k,t (G ) − (k + 1)n − | D k | + (k + 1)n γ×k,t (G ) − | D k |. From these two inequalities, it is clear that the above reduction is an L-reduction with a = 4k + 5 and b = 1. Therefore, Min k-Tuple Total Dom Set-(k + 2) is APX-complete. 2 Theorem 5.9 can be improvised by showing that Min k-Tuple Total Dom Set-B(k + 2) is APX-complete. We prove this in the following theorem. Theorem 5.10. Min k-Tuple Total Dom Set-B(k + 2) is APXcomplete for any k 2. Proof. By Corollary 5.8, Min Total Dom Set-B3 is APXcomplete. So it is enough to establish an L-reduction from Min Total Dom Set-B3 to Min k-Tuple Total Dom SetB(k + 2). Given a bipartite graph G = ( X , Y , E ), an instance of Min Total Dom Set-B3, we construct a bipartite graph G = ( X , Y , E ) of degree at most k + 2, an instance of Min kTuple Total Dom Set-B(k + 2) as follows. For each vertex v ∈ X ∪ Y , we take a complete bipartite graph G ( v ) = K k,k on 2k vertices with bipartite sets X v and Y v , where X v = {x1v , x2v , . . . , xkv } and Y v = { y 1v , y 2v , . . . , ykv }. Connect v to xiv , 1 i k − 1 if v ∈ Y ; otherwise connect v to y iv , 1 i k − 1. Note that (G ) = k + 2. We can proved similarly as proved in Theorem 5.9 that γ×k,t (G ) = γt (G ) + 2kn. Again we can easily proved that the above reduction is an L-reduction with a = 8k + 1 and b = 1. Therefore, Min k-Tuple Total Dom Set-B(k + 2) is APX-complete. 2
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6. Conclusion In this paper, we have shown that Decide Min k-Tuple Total Dom Set problem is NP-complete for split graphs, doubly chordal graphs and bipartite graphs. We have then proved that Min k-Tuple Total Dom Set problem can be solved in polynomial time for chordal bipartite graphs. Apart from these, we have presented inapproximability results of Min k-Tuple Total Dom Set problem for general graphs and bipartite graphs. We have also shown that for k 2, Min k-Tuple Total Dom Set problem is APXcomplete for graphs (bipartite graphs) with maximum degree k + 2. Acknowledgements The author would like to thank the anonymous referees for their helpful comments leading to improvements in the presentation of the paper. References [1] P. Alimonti, V. Kann, Hardness of approximating problems on cubic graphs, in: Proc. of 3rd Italian Conference on Algorithms and Complexity, Rome, in: Lecture Notes in Comput. Sci., vol. 1203, 1997, pp. 288–298. [2] G. Ausiello, P. Crescenzi, G. Gambosi, V. Kann, A. MarchettiSpaccamela, M. Protasi, Complexity and Approximation: Combinatorial Optimization Problems and Their Approximability Properties, Springer-Verlag, Berlin, 1999. [3] A. Brandstädt, Classes of bipartite graphs related to chordal graphs, Discrete Appl. Math. 32 (1) (1991) 51–60. [4] A. Brandstädt, F.F. Dragan, V. Chepoi, V. Voloshin, Dually chordal graphs, SIAM J. Discrete Math. 11 (3) (1998) 437–455.
[5] G.J. Chang, G. Nemhauser, The k-domination and k-stability problems on sun-free chordal graphs, SIAM J. Algebraic Discrete Methods 6 (4) (1985) 721–730. [6] M. Chlebík, J. Chlebíková, Approximation hardness of dominating set problems in bounded degree graphs, Inform. and Comput. 206 (11) (2008) 1264–1275. [7] T.H. Cormen, C.E. Leiserson, R.L. Rivest, C. Stein, Introduction to Algorithms, Prentice Hall, India, 2002. [8] M. Farber, Characterizations of strongly chordal graphs, Discrete Math. 43 (2–3) (1983) 173–189. [9] U. Feige, A threshold of ln n for approximating set cover, J. ACM 45 (4) (1998) 634–652. [10] D.R. Fulkerson, O.A. Gross, Incidence matrices and interval graphs, Pacific J. Math. 15 (1965) 835–855. [11] T.W. Haynes, S.T. Hedetniemi, P.J. Slater (Eds.), Domination in Graphs: Advanced Topics, Monogr. Textb. Pure Appl. Math., vol. 209, Marcel Dekker Inc., New York, 1998. [12] T.W. Haynes, S.T. Hedetniemi, P.J. Slater, Fundamentals of Domination in Graphs, Monogr. Textb. Pure Appl. Math., vol. 208, Marcel Dekker Inc., New York, 1998. [13] M.A. Henning, A.P. Kazemi, k-tuple total domination in graphs, Discrete Appl. Math. 158 (2010) 1006–1011. [14] M.A. Henning, A.P. Kazemi, k-tuple total domination in cross products of graphs, J. Comb. Optim., http://dx.doi.org/10.1007/s10878011-9389-z. [15] R.M. Karp, Reducibility among combinatorial problems, in: Complexity of Computer Computations, Proc. Sympos., IBM Thomas J. Watson Res. Center, Yorktown Heights, New York, 1972, Plenum Press, New York, 1972, pp. 85–103. [16] R. Klasing, C. Laforest, Hardness results and approximation algorithms of k-tuple domination in graphs, Inform. Process. Lett. 89 (2004) 75–83. [17] R. Laskar, J. Pfaff, S.M. Hedetniemi, S.T. Hedetniemi, On the algorithmic complexity of total domination, SIAM J. Alg. Disc. Math. 5 (3) (1984) 420–425. [18] C.S. Liao, G.J. Chang, k-tuple domination in graphs, Inform. Process. Lett. 87 (2003) 45–50. [19] C.H. Papadimitriou, M. Yannakakis, Optimization, approximation and complexity classes, J. Comput. System Sci. 43 (1991) 425–440.
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Information Processing Letters www.elsevier.com/locate/ipl
Algorithmic aspects of k-tuple total domination in graphs D. Pradhan 1 Department of Computer Science and Automation, Indian Institute of Science, Bangalore 560012, India
a r t i c l e
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Article history: Received 17 September 2011 Received in revised form 25 May 2012 Accepted 17 July 2012 Available online 20 July 2012 Communicated by R. Uehara Keywords: Graph algorithms Domination Total domination k-Tuple total domination NP-complete APX-complete
a b s t r a c t For a fixed positive integer k, a k-tuple total dominating set of a graph G = ( V , E ) is a subset T D k of V such that every vertex in V is adjacent to at least k vertices of T D k . In minimum k-tuple total dominating set problem (Min k-Tuple Total Dom Set), it is required to find a k-tuple total dominating set of minimum cardinality and Decide Min k-Tuple Total Dom Set is the decision version of Min k-Tuple Total Dom Set problem. In this paper, we show that Decide Min k-Tuple Total Dom Set is NP-complete for split graphs, doubly chordal graphs and bipartite graphs. For chordal bipartite graphs, we show that Min k-Tuple Total Dom Set can be solved in polynomial time. We also propose some hardness results and approximation algorithms for Min k-Tuple Total Dom Set problem. © 2012 Elsevier B.V. All rights reserved.
1. Introduction For a graph G = ( V , E ), the sets N G ( v ) = {u ∈ V | uv ∈ E } and N G [ v ] = N G ( v ) ∪ { v } denote the neighborhood and the closed neighborhood of a vertex v, respectively. A set D of vertices of a graph G is a dominating set of G if every vertex in V \ D is adjacent to a vertex in D. Equivalently, a subset D of V is a dominating set of G if | N G [ v ] ∩ D | 1 for every v ∈ V . The domination number of a graph G, denoted by γ (G ), is the minimum cardinality of a dominating set of G. The concept of domination and its variations have many applications and have been widely studied in literature (see [11,12]). Among the variations of domination, total domination is one of those. A set D of vertices of a graph G is a total dominating set of G if every vertex in V is adjacent to at least one vertex of D. Equivalently, a subset D of V is a total dominating set of G if | N G ( v ) ∩ D | 1 for every v ∈ V . The total domination number of a graph G, denoted by γt (G ), is the minimum cardinality of a total dominating set of G. As a variation of domination, the concept of
E-mail address: [email protected]. The author was supported by National Board for Higher Mathematics (NBHM) and Dr. D.S. Kothari Postdoctoral Fellowship (DSKPDF), India. 1
0020-0190/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ipl.2012.07.010
k-tuple total domination has been introduced by Henning and Kazemi [13]. For a fixed positive integer k, a k-tuple total dominating set of a graph G is a subset T D k of V such that every vertex in V is adjacent to at least k vertices of T D k . Equivalently, a set T D k ⊆ V is a k-tuple total dominating set of G if | N G ( v ) ∩ T D k | k for every v ∈ V . So k-tuple total domination is the generalization of the usual total domination. The case when k = 2 is called double total domination [13] where some upper bounds are obtained for the minimum cardinality of a double total dominating set. The k-tuple total domination number of a graph G, denoted by γ×k,t (G ), is the minimum cardinality of a k-tuple total dominating set of G. Upper bounds and lower bounds for γ×k,t (G ) are also proposed in [13,14] in terms of different graph parameters. In this paper, we study on k-tuple total domination from algorithmic point of view. As per our knowledge, no algorithmic result has been obtained for k-tuple total domination in graphs. We first present some NP-hardness and polynomial solvable cases of Min k-Tuple Total Dom Set problem on different graph classes. We then extend these studies by investing the approximation hardness of Min k-Tuple Total Dom Set problem. We also propose a constant approximation algorithm for Min k-Tuple Total Dom Set problem. For the general concept of algorithms and
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approximation algorithms, one can see [2,7]. We derive the following main results in this paper: 1. Decide Min k-Tuple Total Dom Set is NP-complete for split graphs, doubly chordal graphs and bipartite graphs. 2. Min k-Tuple Total Dom Set is polynomially solvable for chordal bipartite graphs. 3. Min k-Tuple Total Dom Set cannot be approximated within (1 − ε ) ln | V | for any ε > 0, unless NP ⊆ DTIME(| V | O (log log | V |) ). 4. Min k-Tuple Total Dom Set is APX-complete for graphs of maximum degree k + 2. In particular, this is true for bipartite graphs of maximum degree k + 2. 2. Preliminaries For a graph G = ( V , E ), the degree of a vertex v is | N G ( v )| and is denoted by d G ( v ). If d G ( v ) = 1, then v is called a pendant vertex of G. If d G ( v ) = 0, then v is called an isolated vertex of G. Let (G ) and δ(G ) denote the maximum degree and minimum degree of a graph G, respectively. For S ⊆ V , let G [ S ] denote the subgraph induced by S. If G [C ], C ⊆ V , is a complete subgraph of G, then C is called a clique of G. A set S ⊆ V is an independent set if G [ S ] has no edge. A graph is said to be a chordal graph if every cycle of length at least four has a chord i.e., an edge joining two non-consecutive vertices of the cycle. A chordal graph G = ( V , E ) is a split graph if V can be partitioned into two sets K and S such that K is a clique and S is an independent set. A vertex v ∈ V (G ) is a simplicial vertex of G if N G [ v ] is a clique of G. An ordering α = ( v 1 , v 2 , . . . , v n ) is a perfect elimination ordering (PEO) of G if v i is a simplicial vertex of G i = G [{ v i , v i +1 , . . . , v n }] ∀i , 1 i n. It is characterized that a graph G is chordal if and only if it has a PEO [10]. A vertex u ∈ N G [ v ] is a maximum neighbor of v in G if N G [ w ] ⊆ N G [u ] for all w ∈ N G [ v ]. An ordering σ = ( v 1 , v 2 , . . . , v n ) of V (G ) is called a maximum neighborhood ordering (MNO) of G if v i has a maximum neighbor in G i = G [{ v i , v i +1 , . . . , v n }] for each i, 1 i n. Graphs that admit maximum neighborhood orderings are called as dually chordal graphs [4]. A vertex v is called doubly simplicial in G if it is simplicial and has a maximum neighbor in G. An ordering σ = ( v 1 , v 2 , . . . , v n ) of V (G ) is called a doubly perfect elimination ordering (dpeo) of G if v i is a doubly simplicial vertex in G i = G [{ v i , v i +1 , . . . , v n }] for each i, 1 i n. A graph is doubly chordal if it admits a doubly perfect elimination ordering (dpeo) [4]. Note that every doubly chordal graph is a dually chordal graph. 3. Complexity of Decide Min k-Tuple Total Dom Set problem in graphs In this section, we show that Decide Min k-Tuple Total Dom Set is NP-complete for split graphs, doubly chordal and bipartite graphs. Hence Decide Min k-Tuple Total Dom Set is NP-complete for chordal graphs and dually chordal graphs. A set S ⊆ V of a graph G = ( V , E ) is called a vertex cover of G if for every edge uv ∈ E, either u ∈ S or v ∈ S.
Fig. 1. The constructed split graph G from a graph G.
In minimum vertex cover problem (Min Vertex Cover), it is required to find a vertex cover of minimum cardinality and let Decide Min Vertex Cover be the decision version of Min Vertex Cover problem. Decide Min Vertex Cover is known to be NP-complete for general graphs [15]. We show that Decide Min k-Tuple Total Dom Set is NPcomplete for split graphs by providing a polynomial time transformation from the Decide Min Vertex Cover problem. Theorem 3.1. For a fixed integer k, Decide Min k-Tuple Total Dom Set is NP-complete for split graphs. Proof. Given a graph G = ( V , E ), we construct a split graph G = ( V , E ) with vertex set V = V ∪ S ∪ E ∪ {x} where S = {s1 , s2 , . . . , sk } and x ∈ / V ∪ S ∪ E and with edge set E = {uv | u = v and u , v ∈ V ∪ S } ∪ { ve | v ∈ V , e ∈ E and v ∈ e } ∪ {si e | e ∈ E and 1 i k − 1} ∪ {si x | 1 i k}. Notice that G is a split graph as V ∪ S is a clique and E ∪ {x} is an independent set. Since | V | = | V | + | E | + k + 1, the construction of G can be done in polynomial time. The construction of G is illustrated in Fig. 1 for k = 3. Next, we show that G has a vertex cover of cardinality α if and only if G has a k-tuple total dominating set of cardinality α + k. Let C be a vertex cover of G. Then it is clear that T D k = C ∪ {s1 , s2 , . . . , sk } is a k-tuple total dominating set of G with | T D k | = α + k. On the other hand, suppose that D is a k-tuple total dominating set of G with cardinality α + k. Then it is clear that s1 , s2 , . . . , sk ∈ D since N G (x) = {s1 , s2 , . . . , sk }. Now let C = D \ {s1 , s2 , . . . , sk }. We claim that C is a vertex cover of G. Let ab = e i ∈ E. Since N G (e i ) ⊂ S ∪ {a, b} and | N G (e i ) ∩ S | = k − 1, D contains at least one of a and b. So either a ∈ C or b ∈ C . Hence C is a vertex cover of G with |C | = | D | − k = α . Therefore, Decide Min k-Tuple Total Dom Set is NPcomplete for split graphs. 2 We show that Decide Min k-Tuple Total Dom Set is NPcomplete even for bipartite graphs by giving a polynomial time transformation from the Decide Min Vertex Cover problem.
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Dom Set be the decision version of the problem. Decide Min Total Dom Set is NP-complete for chordal graphs [17]. Now, we show that Decide Min k-Tuple Total Dom Set is NP-complete for another important subclass of chordal graphs known as doubly chordal graphs. Given an instance of Decide Min (k − 1)-Tuple Total Dom Set problem, we construct a doubly chordal graph for the instance of Decide Min k-Tuple Total Dom Set problem. Let G = ( V , E ) be a chordal graph and x ∈ / V . Construct a graph G = ( V , E ) where V = V ∪ {x} and E = E ∪ {xv | v ∈ V }. If σ = ( v 1 , v 2 , . . . , v n ) is a PEO of G, then σ = ( v 1 , v 2 , . . . , v n , x) is a dpeo of G . So G is a doubly chordal graph. Claim. γ×k,t (G ) = γ×(k−1),t (G ) + 1. Fig. 2. The constructed bipartite graph G from a graph G.
Theorem 3.2. For any fixed positive integer k, Decide Min kTuple Total Dom Set is NP-complete for bipartite graphs. Proof. Let S = {s1 , s2 , . . . , sk }, S = {s1 , s2 , . . . , sk }, T = {t 1 , t 2 , . . . , tk }, T = {t 1 , t 2 , . . . , tk }, and T = {t 1 , t 2 , . . . , tk }. Given a graph G = ( V , E ), we construct a bipartite graph G = ( V , E ) with vertex set V = S ∪ T ∪ V ∪ E ∪ T ∪ S ∪ T and edge set E is described as follows:
• {si t j | si ∈ S , t j ∈ T } ∪ {si t j | si ∈ S , t j ∈ T } ∪ {si t j | si ∈ S , t j ∈ T }; • {ut j | u ∈ V , t j ∈ T } ∪ {et j | e ∈ E , t j ∈ T \ {tk }}; • { ve | v ∈ V , e ∈ E and v ∈ e }. Notice that G is a bipartite graph with partite sets V ∪ S ∪ T and E ∪ S ∪ T ∪ T . Since | V | = | V | + | E | + 5k, the construction of G can be done in polynomial time. The construction of G is illustrated in Fig. 2 for k = 3. Next, we show that G has a vertex cover of cardinality α if and only if G has a k-tuple total dominating set of cardinality α + 4k. If C is a vertex cover of G, then C ∪ S ∪ S ∪ T ∪ T is a k-tuple total dominating set of G . Hence G has a ktuple total dominating set of cardinality α + 4k. On the other hand, let D be a k-tuple total dominating set of G of cardinality α + 4k. Then it is clear that ( S ∪ S ∪ T ) ⊂ D. The vertices of set E ∪ T that belongs to D can be interchanged by the vertices of T . So without loss of generality, we assume that ( S ∪ T ) ⊂ D. Since | N G (e ) ∩ T | = k − 1 for each e ∈ E, C = D \ ( S ∪ S ∪ T ∪ T ) = ∅. Clearly |C | = | D | − | S ∪ S ∪ T ∪ T | = α . We now claim that C is a vertex cover of G. Suppose that e ∗ = ab ∈ E. Since | D ∩ N G (e ∗ )| k and e ∗ is adjacent to the vertices of ( T \ {tk }) ∪ {a, b}, D contains either a or b. This implies that a ∈ C or b ∈ C . Therefore, Decide Min k-Tuple Total Dom Set is NPcomplete for bipartite graphs. 2 In minimum total dominating set (Min Total Dom Set) problem, it is required to find a total dominating set of a graph of minimum cardinality and let Decide Min Total
Proof of the Claim. If D is a (k − 1)-tuple total dominating set of G, then D ∪ {x} is a k-tuple total dominating set of G . So, γ×k,t (G ) γ×(k−1),t (G ) + 1. On the other hand, assume that D is a minimum ktuple total dominating set of G . If x ∈ D , then it is clear that D \ {x} is a (k − 1)-tuple total dominating set of G. If x∈ / D , then D = ( D \ { v }) ∪ {x} for some v ∈ D , is also a minimum k-tuple total dominating set of G and D \ {x} is a (k − 1)-tuple total dominating set of G. So, γ×(k−1),t (G ) γ×k,t (G ) − 1 implying the equality. 2 Since Decide Min Total Dom Set is NP-complete for chordal graphs, by induction on k, we can prove that for k > 1, Decide Min k-Tuple Total Dom Set is NP-complete for doubly chordal graphs. Hence we have the following theorem Theorem 3.3. For any fixed integer k > 1, Decide Min k-Tuple Total Dom Set is NP-complete for doubly chordal graphs. By Theorems 3.1 and 3.3, the following corollary arises. Corollary 3.4. For any fixed integer k > 0, Decide Min k-Tuple Total Dom Set is NP-complete for chordal graphs. 4. Polynomial solvable case A graph is called a chordal bipartite graph if it is bipartite and every cycle of length at least six has a chord i.e., an edge joining two non-consecutive vertices of the cycle. Note that chordal bipartite graphs are not necessarily chordal. For any integer n 3, an n-sun is a graph with 2n vertices, say x1 , x2 , . . . , xn and y 1 , y 2 , . . . , yn such that {x1 , x2 , . . . , xn } is a clique and { y 1 , y 2 , . . . , yn } is an independent set and y i is adjacent to xi and xi −1 for 2 i n and y 1 is adjacent to x1 and xn . A strongly chordal graph is a chordal graph containing no n-sun (n 3) as an induced subgraph [5,8]. In this section, we show that Min k-Tuple Total Dom Set can be solved in polynomial time for chordal bipartite graphs. A polynomial time algorithm has been proposed in [18] for computing a minimum k-tuple dominating set in a strongly chordal graph. It is also mentioned in [18] that a minimum k-tuple total dominating set can be computed
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by the same algorithm by replacing the closed neighbors in most of the statements with the concerned neighbors. So it follows that Min k-Tuple Total Dom Set can be solved in polynomial time for strongly chordal graphs. Let G = ( X , Y , E ) be a chordal bipartite graph and C X (G )(C Y (G )) denote the split graph obtained from G by adding edges between every pair of vertices in X (Y ). Then C X (G ) is a strongly chordal graph which follows from the following theorem appears in [3]. Theorem 4.1. (See [3].) Let G = ( X , Y , E ) is a bipartite graph. Then G is a chordal bipartite graph if and only if C X (G ) is a strongly chordal graph. Let G = ( X , Y , E ) be a bipartite graph and k > 0 be an X ( Y ) be the minimum cardinality subset of X integer. Let X | k (| N G (x) ∩ Y | k) for every (Y ) such that | N G ( y ) ∩ y ∈ Y (x ∈ X ). The proof of the following lemma is easy and hence is omitted. Lemma 4.2. X ∪ Y is a minimum k-tuple total dominating set of a bipartite graph G = ( X , Y , E ). The next theorem shows that the Min k-Tuple Total Dom Set problem can be solved in polynomial time for chordal bipartite graphs. Theorem 4.3. Min k-Tuple Total Dom Set is solvable in polynomial time for chordal bipartite graphs. Proof. Suppose that G = ( X , Y , E ) is a chordal bipartite graph. Note that δ(G ) k; otherwise there is no k-tuple total dominating set of G. By Lemma 4.2, we require to X and Y of G in polynomial time. We next show compute X can be computed in polynomial time and comthat how Y is similar. Add edges to G such that the resulting puting graph will be C X (G ). By Theorem 4.1, C X (G ) is a strongly chordal graph. Consider a minimum k-tuple dominating set X is also a kD X of C X (G ). It is clear that | D X | k. Since tuple dominating set of C X (G ), | D X | | X |. Without loss of generality, we assume that D X contains the vertices of X only; otherwise we can replace by an adjacent vertex of the vertex y ∈ Y if y belongs to D X . Note that such a neighbor of y exists; otherwise D X \ { y } is a smaller kX |, then tuple dominating set of C X (G ). Again if | D X | < | we get a contradiction to the choice of X . So our problem is a subproblem of minimum k-tuple dominating set problem for strongly chordal graphs. Since the minimum k-tuple dominating set problem is solvable in polynomial time [18], the theorem follows. 2 5. Hardness and approximation In this section, we present some hardness and approximation results for the Min k-Tuple Total Dom Set problem. First we give an approximation preserving reduction from Min Total Dom Set to obtain a stronger inapproximability result for Min k-Tuple Total Dom Set in general graphs. For this we need the following theorem proved in [6,9].
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Theorem 5.1. (See [6,9].) Min Total Dom Set cannot be approximated within (1 − ε ) ln | V | for any ε > 0, unless NP ⊆ DTIME(| V | O (log log | V |) ). The same results hold for bipartite graphs and chordal (split) graphs. Theorem 5.2. Min k-Tuple Total Dom Set cannot be approximated within (1 − ε ) ln | V | for any ε > 0, unless NP ⊆ DTIME(| V | O (log log | V |) ). Proof. Let G = ( V , E ) be an instance of Min Total Dom Set with n vertices and m edges. Given G = ( V , E ), we construct a graph G = ( V , E ), an instance of Min k-Tuple Total Dom Set, in polynomial time as follows:
• Add k + 1 new vertices v 1 , v 2 , . . . , v k+1 . • Add all possible the edges v i v j , i = j so that { v 1 , v 2 , . . . , v k+1 } forms a clique. • For each v ∈ V , add the edges v v i , 1 i k − 1. Clearly, it can be seen that G can be constructed in polynomial time as | V | = n + k + 1. Next, we prove that γ×k,t (G ) = γt (G ) + k + 1. Let D be a minimum total dominating set of G. Then D k = D ∪ { v 1 , v 2 , . . . , v k+1 } is a k-tuple total dominating set of G . So γ×k,t (G ) | D k | = | D | + k + 1 = γt (G ) + k + 1. On the other hand, suppose that D k∗ is a k-tuple total dom-
inating set of G . Then it is clear that v 1 , v 2 , . . . , v k+1 ∈ D k∗ as d G ( v k ) = d G ( v k+1 ) = k. We prove that D ∗ = D k∗ \
{ v 1 , v 2 , . . . , v k+1 } is a total dominating set of G. Let v ∈ V . Since | N G ( v ) ∩ D k∗ | k and | N G ( v ) ∩ { v 1 , v 2 , . . . , v k+1 }| = k − 1, there exists a vertex u ∈ V such that u ∈ D ∗ . This implies that | N G ( v ) ∩ D ∗ | 1 for each v ∈ V and hence D ∗ is a total dominating set of G. So γt (G ) γ×k,t (G ) − (k + 1) which implies the equality. Now assume that Min k-Tuple Total Dom Set can be approximated within ratio αk 1 by using an algorithm A k . Let l > 0 be an integer. Consider the following algorithm: Algorithm 1: A k,l 1 2 3 4 5 6 7 8
Input: A graph G = ( V , E ); Output: A minimum total dominating set D of G; if (there exists a minimum total dominating set D of G of cardinality < l) then Return D; else Construct G ; Compute a k-tuple total dominating set D k of G using algorithm A k ; D = D k \ S, where S = { v 1 , v 2 , . . . , v k+1 } end Return D;
Step 1 of the algorithm can be completed in polynomial time as l is a constant. Again since A k is a polynomial time algorithm, algorithm A k,l is polynomial. Note that if D is computed in Step 1, then D is optimal. So we analyze the case where D is constructed in next lines. Let D k∗ be an optimal k-tuple total dominating set in G and D ∗ be an optimal total dominating set in G. It is clear that | D ∗ | l. Let D be the total dominating set computed by algorithm A k,l . Then | D | = | D k | − (k + 1) | D k | αk | D k∗ | = αk (1 + k| D+∗1| )| D ∗ | αk (1 + k+l 1 )| D ∗ |. Hence,
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algorithm A k,l approximates Min Total Dom Set within the ratio αk (1 + k+l 1 ). Assume that there exists some fixed ε > 0 such that Min k-Tuple Total Dom Set can be approximated within ratio α = (1 − ε ) ln | V | using algorithm A k . Let l be a positive integer such that k+l 1 < ε2 . Then algorithm A k,l approximates Min Total Dom Set within ratio
ε)(1 + ε2 ) ln | V | = (1 − ε ) ln | V | for
ε
αk (1 + k+l 1 ) (1 −
2 = ε2 + ε2 as ln | V | =
ln(| V | + k + 1) ≈ ln | V | for sufficiently large value of | V |. Therefore, it follows by Theorem 5.1 that Min k-Tuple Total Dom Set cannot be approximated within ratio (1 − ε ) ln | V | for any ε > 0, unless NP ⊆ DTIME(| V | O (log log | V |) ). 2 Theorem 5.3. Min k-Tuple Total Dom Set for bipartite graphs cannot be approximated within (1 − ε ) ln | V | for any ε > 0, unless NP ⊆ DTIME(| V | O (log log | V |) ). Proof. Let G = ( V , E ) be an instance of Min Total Dom Set with n vertices and m edges. Given a bipartite graph G = ( X , Y , E ), we construct a bipartite graph G = ( X , Y , E ), an instance of Min k-Tuple Total Dom Set, in polynomial time as follows:
• Add 2k new vertices {x1 , x2 , . . . , xk } and { y 1 , y 2 , . . . , yk }. Let X 1 = {x1 , x2 , . . . , xk } and Y 1 = { y 1 , y 2 , . . . , yk }. • Add all possible the edges xi y j so that G [ X 1 ∪ Y 1 ] becomes a complete bipartite graph.
Similar to the proof of Theorem 5.2, we can choose the positive integer l such that 2k < ε2 , where ε > 0 and can l prove easily that Min k-Tuple Total Dom Set for bipartite graphs cannot be approximated within ratio (1 − ε ) ln | V | for any ε > 0, unless NP ⊆ DTIME(| V | O (log log | V |) ). 2 Note that if G is a chordal graph, then the constructed graph G as in Theorem 5.2 is also a chordal graph. So we immediately have the following corollary. Corollary 5.4. Min k-Tuple Total Dom Set for chordal graphs cannot be approximated within (1 − ε ) ln | V | for any ε > 0, unless NP ⊆ DTIME(| V | O (log log | V |) ). Next we show that Min k-Tuple Total Dom Set problem is in APX. For this, we need to define some terminologies which are introduced in [16]. Let X be any nonempty set and F be a family of subsets of X . Let k > 0 be an integer.
• An element x ∈ X is k-covered in a set C ⊆ F of subsets of X if x is in at least k sets of C . • A k-cover of ( X , F ) is a subset C of F such that x is k-covered in C for each x ∈ X . • A k-cover of ( X , F ) of minimum cardinality is called a minimum k-cover. Algorithm 2: GEN-SET-COVER
• For each x ∈ X , add the edges xy j , 1 j k − 1. • For each y ∈ Y , add the edges yxi , 1 i k − 1. Notice that G = ( X , Y , E ) is a bipartite graph with X = X ∪ X 1 and Y = Y ∪ Y 1 . Clearly, it can be seen that G can be constructed in polynomial time as | V | = n + 2k. Next, we show that γ×k,t (G ) = γt (G ) + 2k. Let D be a minimum total dominating set of G. Then D k = D ∪ {xi , y i | 1 i k} is a k-tuple total dominating set of G . So γ×k,t (G ) | D k | = | D | + 2k = γt (G ) + 2k. On the other hand, suppose that D k∗ is a k-tuple total dominating set of G . Then it is clear that xi , y i ∈ D k∗ for 1 i k
as d G (xk ) = d G ( yk ) = k. We prove that D ∗ = D k∗ \ {xi , y i | 1 i k} is a total dominating set of G. Let x ∈ X . Since | N G (x) ∩ D k∗ | k and | N G (x) ∩ { y 1 , y 2 , . . . , yk }| = k − 1, there exists a vertex y ∈ Y such that y ∈ D ∗ . This implies that | N G (x) ∩ D ∗ | 1 for each x ∈ X . Similar arguments can be made for each y ∈ Y . Hence D ∗ is a total dominating set of G. Therefore, γt (G ) γ×k,t (G ) − 2k implying the equality. Now assume that Min k-Tuple Total Dom Set can be approximated within ratio αk 1 by using an algorithm A k . Let l > 0 be an integer. Modifying the algorithm A k,l by making the set S = {xi , y i | 1 i k}. Let D k∗ be an optimal k-tuple total dominating set in
G and D ∗ be an optimal total dominating set in G. It is clear that | D ∗ | l. Let D be the total dominating set computed by algorithm A k,l . Then | D | = | D k | − 2k | D k | αk | D k∗ | = αk (1 + | D2k∗ | )| D ∗ | αk (1 + 2kl )| D ∗ |. Hence, algorithm A k,l approximates Min Total Dom Set within the ra). tio αk (1 + 2k l
1 2 3 4 5 6 7 8 9 10
Input: An integer k > 0, a set X and a family F of subsets of X such that F is a k-cover of ( X , F ); Output: A minimum k-cover C of ( X , F ); C = ∅; i = 0; while ( X \ ( S 1 ∪ S 2 ∪ · · · ∪ S i ) = ∅) do i + +; Choose S ∈ F \ C such that | S \ ( S 1 ∪ S 2 ∪ · · · ∪ S i −1 )| is maximized; S i = S; S i = set of elements of X , k-covered in C ∪ { S i } but not in C ; C = C ∪ { S i }; end Output (C );
Theorem 5.5. (See [16].) Let X be a set, F be a family of subsets of X and k > 0 be an integer. Let S M be one set of F with maximum cardinality. Then (i) Algorithm GEN-SET-COVER outputs a k-cover C of ( X , F ) in polynomial time. (ii) The cardinality of the output of Algorithm GEN-SET-COVER with input ( X , F ) is at most ln(| S M |) + 1 times the cardinality of an optimal k-cover set of ( X , F ). Theorem 5.6. Min k-Tuple Total Dom Set problem in any graph G = ( V , E ) with maximum degree (G ) can be approximated with an approximation ratio of ln((G )) + 1. Proof. Clearly, G contains a k-tuple total dominating set if and only if δ(G ) k. If this is the case, then let X = V and F = { N G ( v ) | v ∈ V }. Suppose the algorithm GEN-SETCOVER on ( X , F ) outputs C . Finally, D = { v ∈ V | N G ( v ) ∈ C } is a k-tuple total dominating set of G. In this case, | S M | = (G ). So by Theorem 5.5, the theorem follows. 2
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5.1. k-Tuple total domination in bounded degree graphs By Theorem 5.6, it is clear that if the degree of the graph is bounded by a constant, the approximation ratio is constant. Hence Min k-Tuple Total Dom Set for bounded degree graphs is in APX. Next we prove that Min k-Tuple Total Dom Set is APX-complete even if the degree of the graph is bounded by k + 2. Hereafter, we denote Min kTuple Total Dom Set-(B) as the Min k-Tuple Total Dom Set restricted to graphs (bipartite graphs) of degree at most . Similarly, we denote Min Total Dom Set-(B) as the Min Total Dom Set restricted to graphs (bipartite graphs) of degree at most . To show that Min k-Tuple Total Dom Set-k + 2 is APXcomplete for k 2, we require the following theorem that we are going to prove. Theorem 5.7. Min Total Dom Set-3 is APX-complete. Proof. It is well known that the Min Total Dom Set-3 is in APX. Since Min Vertex Cover-3 is APX-complete [1,19], it is enough to establish an L-reduction [19] from Min Vertex Cover-3. Given a graph G = ( V , E ) of degree at most 3, an instance of Min Vertex Cover-3, we construct a graph G = ( V , E ) as an instance of Min Total Dom Set-3 in polynomial time as follows. For each edge e = uv, we subdivide the edge uv by introducing a new vertex e such that ue , ve ∈ E . Then we introduce a pendant vertex e such that e e ∈ E . Note that G is also a bipartite graph. Next, we prove that γt (G ) = β(G ) + m, where β(G ) is the cardinality of a minimum vertex cover of G. If S is a vertex cover of G, then S ∪ {e | e ∈ E (G )} is a total dominating set of G . So we have γt (G ) β(G ) + m. On the other hand, let D be a minimum total dominating set of G . Suppose that e = uv ∈ E (G ). To dominate the vertex e , either e ∈ D or e ∈ D. If e ∈ D, then e ∈ D. If u ∈ D or v ∈ D, then D \ {e } is a smaller total dominating set of G . This is a contradiction. So assume that u , v ∈ / D. Then ( D \ {e }) ∪ {e } is also a minimum total dominating set of G . So without loss of generality, we assume that / D and e ∈ D for each e ∈ E (G ). Again we have seen e ∈ that for each edge uv ∈ E (G ), either u or v is contained in D. So it is clear that D \ {e | e ∈ E (G )} is a vertex cover of G and hence β(G ) γt (G ) − m. This implies the equality. Since (G ) = 3, we have m 3n and β(G ) n4 . Now 2
γt (G ) β(G ) + m β(G ) + 3n β(G ) + 6β(G ) = 7β(G ). 2 Since e belongs to any total dominating set D of G for each e ∈ E (G ), D \ {e | e ∈ E (G )} is always a vertex cover of G. So |C | − |C ∗ | | D | − m − (| D ∗ | − m) = | D | − | D ∗ |. From these two inequalities, it is clear that the above reduction is an L-reduction with a = 7 and b = 1. Therefore, Min Total Dom Set-3 is APX-complete. 2
Due to the reduction used in Theorem 5.7, it can be seen easily that G is also a bipartite graph and hence the following corollary directly follows from Theorem 5.7. Corollary 5.8. Min Total Dom Set-B3 is APX-complete.
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In the following theorem, we show that Min k-Tuple Total Dom Set is APX-complete for graphs of degree at most k + 2 for any k 2. Theorem 5.9. Min k-Tuple Total Dom Set-(k + 2) is APXcomplete for any k 2. Proof. Clearly, by Theorem 5.6, Min k-Tuple Total Dom Set-(k + 2) is in APX. Since by Theorem 5.7, Min Total Dom Set-3 is APX-complete, it is enough to establish an L-reduction from Min Total Dom Set-3 to Min k-Tuple Total Dom Set-(k + 2). Given a graph G = ( V , E ), an instance of Min Total Dom Set-3, we construct a graph G = ( V , E ) of degree at most k + 2 as follows. For each vertex v ∈ V , we take a complete graph G ( v ) on k + 1 vertices, say v 1 , v 2 , . . . , v k+1 and connect v to v i , 1 i k − 1. Note that (G ) = k + 2. Next, we prove that γ×k,t (G ) = γt (G ) + (k + 1)n. If D is a total dominating set of G, then D ∪ { v 1 , v 2 , . . . , v k+1 | v ∈ V } is a k-tuple total dominating set of G . So γ×k,t (G ) γt (G ) + (k + 1)n. On the other hand, suppose that D k is a k-tuple total dominating set of G . Then it is clear that { v 1 , v 2 , . . . , v k+1 | v ∈ V } ⊆ D k . Since v is adjacent to v i , 1 i k − 1, we have | N G ( v )∩ D k | 1. Let D = D k \ { v 1 , v 2 , . . . , v k+1 | v ∈ V }. It is clear that D is a total dominating set of G and hence γt (G ) γ×k,t (G )−(k + 1)n. This implies the equality and the claim follows. Since (G ) = 3 and γt (G ) γ (G ) n4 , we have γ×k,t (G ) γt (G ) + (k + 1)n γt (G ) + 4(k + 1)γt (G ) = (4k + 5)γt (G ). Again γt (G ) − | D | = γ×k,t (G ) − (k + 1)n − | D k | + (k + 1)n γ×k,t (G ) − | D k |. From these two inequalities, it is clear that the above reduction is an L-reduction with a = 4k + 5 and b = 1. Therefore, Min k-Tuple Total Dom Set-(k + 2) is APX-complete. 2 Theorem 5.9 can be improvised by showing that Min k-Tuple Total Dom Set-B(k + 2) is APX-complete. We prove this in the following theorem. Theorem 5.10. Min k-Tuple Total Dom Set-B(k + 2) is APXcomplete for any k 2. Proof. By Corollary 5.8, Min Total Dom Set-B3 is APXcomplete. So it is enough to establish an L-reduction from Min Total Dom Set-B3 to Min k-Tuple Total Dom SetB(k + 2). Given a bipartite graph G = ( X , Y , E ), an instance of Min Total Dom Set-B3, we construct a bipartite graph G = ( X , Y , E ) of degree at most k + 2, an instance of Min kTuple Total Dom Set-B(k + 2) as follows. For each vertex v ∈ X ∪ Y , we take a complete bipartite graph G ( v ) = K k,k on 2k vertices with bipartite sets X v and Y v , where X v = {x1v , x2v , . . . , xkv } and Y v = { y 1v , y 2v , . . . , ykv }. Connect v to xiv , 1 i k − 1 if v ∈ Y ; otherwise connect v to y iv , 1 i k − 1. Note that (G ) = k + 2. We can proved similarly as proved in Theorem 5.9 that γ×k,t (G ) = γt (G ) + 2kn. Again we can easily proved that the above reduction is an L-reduction with a = 8k + 1 and b = 1. Therefore, Min k-Tuple Total Dom Set-B(k + 2) is APX-complete. 2
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6. Conclusion In this paper, we have shown that Decide Min k-Tuple Total Dom Set problem is NP-complete for split graphs, doubly chordal graphs and bipartite graphs. We have then proved that Min k-Tuple Total Dom Set problem can be solved in polynomial time for chordal bipartite graphs. Apart from these, we have presented inapproximability results of Min k-Tuple Total Dom Set problem for general graphs and bipartite graphs. We have also shown that for k 2, Min k-Tuple Total Dom Set problem is APXcomplete for graphs (bipartite graphs) with maximum degree k + 2. Acknowledgements The author would like to thank the anonymous referees for their helpful comments leading to improvements in the presentation of the paper. References [1] P. Alimonti, V. Kann, Hardness of approximating problems on cubic graphs, in: Proc. of 3rd Italian Conference on Algorithms and Complexity, Rome, in: Lecture Notes in Comput. Sci., vol. 1203, 1997, pp. 288–298. [2] G. Ausiello, P. Crescenzi, G. Gambosi, V. Kann, A. MarchettiSpaccamela, M. Protasi, Complexity and Approximation: Combinatorial Optimization Problems and Their Approximability Properties, Springer-Verlag, Berlin, 1999. [3] A. Brandstädt, Classes of bipartite graphs related to chordal graphs, Discrete Appl. Math. 32 (1) (1991) 51–60. [4] A. Brandstädt, F.F. Dragan, V. Chepoi, V. Voloshin, Dually chordal graphs, SIAM J. Discrete Math. 11 (3) (1998) 437–455.
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