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NP-Completeness and APX-Completeness of Restrained Domination in Graphs

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NP-Completeness and APX-Completeness of Restrained Domination in Graphs View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Theoretical Computer Science 448 (2012) 1–8 Contents lists available at SciVerse ScienceDirect Theoretical Computer Science journal homepage: www.elsevier.com/locate/tcs NP-completeness and APX-completeness of restrained domination in graphs Lei Chen a, Weiming Zeng a, Changhong Lu b,∗ a College of Information Engineering, Shanghai Maritime University, Shanghai 201306, China b Department of Mathematics, East China Normal University, Shanghai 200241, China article info a b s t r a c t Article history: Let G D .V ; E/ be a simple graph. A vertex set S ⊆ V is a restrained dominating set Received 21 October 2011 if every vertex not in S is adjacent to a vertex in S and to a vertex in V − S. In this Received in revised form 18 February 2012 paper, we investigate the NP-completeness of the restrained domination problem in planar Accepted 3 May 2012 graphs and split graphs. Meanwhile, it is proved that the restrained domination problem is Communicated by D.-Z. Du APX-complete for bounded-degree graphs. Crown Copyright ' 2012 Published by Elsevier B.V. All rights reserved. Keywords: NP-complete APX-complete Approximation ratio Domination problems Restrained domination 1. Introduction Let G D .V ; E/ be a simple graph. For a vertex v 2 V , the neighborhood of v in G is defined as NG.v/ D fu 2 V j uv 2 Eg. The degree of v in G, denoted by dG.v/, is defined as jNG.v/j. We use N.v/ for NG.v/ and d.v/ for dG.v/ if there is no ambiguity. All graphs in this paper are connected. Some other notations and terminology not introduced here can be found in [28]. Domination and its variations in graphs have been extensively studied [3,17,18]. A vertex set S ⊆ V is a restrained dominating set if every vertex not in S is adjacent to a vertex in S and to a vertex in V − S. The restrained domination number of G, denoted by γr .G/, is the minimum cardinality of a restrained dominating set. The restrained domination problem is to determine the restrained domination number of any graph. The concept of restrained domination was introduced by Telle and Proskurowski [27] in 1997, albeit indirectly, as a vertex partitioning problem. The restrained domination has been widely studied in the past ten years, please see [5,6,12–16,19,29]. It has been proved that the restrained domination problem is NP-complete even for bipartite graphs and chordal graphs [5]. In terms of the complexity of the restrained domination problem in graphs, a linear-time algorithm has been proposed for the restrained domination problem in trees [5]. In this paper, we continue studying the complexity of the restrained domination problem in some special graphs and extend these studies by investigating the approximation hardness of the restrained domination problem in graphs. An approximation algorithm is a polynomial time algorithm that outputs a solution whose cost can be compared to the optimal one. The ratio (more precisely, the worst-case ratio over all input instances) between these two costs is called the approximation ratio. General references on approximation algorithms can be found in [2,21]. The paper is organized as follows. In Section2, we prove that the restrained domination problem is NP-complete for planar graphs and split graphs. In Section3, a lower bound and an upper bound on the approximation ratio for the restrained ∗ Corresponding author. Tel.: +86 21 54863276. E-mail address: [email protected] (C. Lu). 0304-3975/$ – see front matter Crown Copyright ' 2012 Published by Elsevier B.V. All rights reserved. doi:10.1016/j.tcs.2012.05.005 2 L. Chen et al. / Theoretical Computer Science 448 (2012) 1–8 a ✟a H 1 ✟ H 2 se e H✟ s ✟H e H ✟ b s b✟HH✟ s 1 ✟ H 2 sf f H✟ s ✟H f H) H ✟ c s c ✟HH✟ s 1 ✟ H 2 sg g H✟ s ✟H g H ✟ d s dH✟ s s s Fig. 1. A reduction from the vertex cover problem to the restrained domination problem in planar graph. domination problem in general graphs are provided. In Section4, it is proved that the restrained domination problem is APX-complete for bounded-degree graphs. 2. NP-completeness of restrained domination problem A graph is called a planar graph if it can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their vertices. A graph is chordal if every cycle of length greater than three has a chord, i.e. an edge jointing two nonconsecutive vertices in the cycle. A graph is split if its vertex set can be partitioned into a stable set and a clique. It is easy to know that split graphs is a subclass of chordal graphs. It has been proved that the restrained domination problem is NP-complete even for bipartite graphs and chordal graphs [5]. We continue investigating its NP-completeness in other special graphs. In detail, we prove that it is also NP-complete for planar graphs and split graphs, which in part extend the results in [5]. We need to employ a well-known NP-completeness result, called the vertex cover problem, which is defined as follows. The vertex cover problem is for a given nontrivial graph and a positive integer k to answer if there is a vertex set of size at most k such that each edge of the graph has at least one end vertex in this set. Theorem 1 ([8,23]). The vertex cover problem is NP-complete even for planar graphs. Theorem 2. The restrained domination problem is NP-complete for planar graphs. Proof. For a planar graph G D .V ; E/, a positive integer k, and an arbitrary subset S ⊆ V with jSj ≤ k, it is easy to verify in polynomial time whether S is a restrained dominating set of G. Hence, the restrained domination problem in planar graphs is in NP. We construct a reduction from the vertex cover problem to the restrained domination problem. Given a nontrivial planar D D f g D f g D f i i i g D graph G .V ; E/, where V v1; v2; : : : ; vn and E e1; e2;:::; em , let Ei e1; e2;:::; em .i 1; 2/. Construct a 0 D 0 0 0 D [ [ 0 D [ f i j D graph G .V ; E / with vertex set V V E1 E2, and edge set E E vjek i 1; 2 and vj is incident to ek in g [ f 1 2 j D g 0 G ej ej j 1; 2;:::; m . Since G is a planar graph, it is easy to verify that G is also a planar graph. Fig.1 shows an example of the reduction. Next, we will show that G has a vertex cover of size at most k if and only if G0 has a restrained dominating set of size at most k. Let VC D fvi ; vi ; : : : ; vi g be a vertex cover of G. Then it is obvious that VC is a restrained dominating set of 0 1 2 k 0 G and its size is k. For the converse, let RD be a restrained dominating set of G with jRDj ≤ k. Suppose RD1 D RD \ V and E∗ D fe 2 E j e1 2 RD or e2 2 RDg. Take one end vertex of each edge in E∗ to compose a vertex set V ∗. Obviously, ∗ ∗ ∗ jV j ≤ jE j ≤ jRD − RD1j. Let VC D RD1 [ V , then it is easy to verify that VC is a vertex cover of G. Furthermore, ∗ jVCj ≤ jRD1j C jV j ≤ jRD1j C jRD − RD1j ≤ jRDj ≤ k. Finally, one can construct G0 from G in polynomial time. By Theorem1, we conclude that the restrained domination problem is NP-complete for planar graphs. Theorem 3. The restrained domination problem is NP-complete for split graphs. Proof. We still construct a reduction from the vertex cover problem to the restrained domination problem. Given a D D f g D f g D f i i i g D nontrivial graph G .V ; E/, where V v1; v2; : : : ; vn and E e1; e2;:::; em , let Ei e1; e2;:::; em .i 1; 2/. 0 D 0 0 0 D [ [ 0 D f 1j Construct a graph G .V ; E / with vertex set V V E1 E2, and edge set E vjek vj is incident to ek in g [ f 1 2 j D g [ f j 2 [ g 0 [ G ej ej j 1; 2;:::; m xy x; y V E2 . Then G is a split graph, where the vertices in V E2 form a clique and vertices in E1 form an independent set. Fig.2 shows an example of the reduction. With the similar argument of Theorem2, G has a vertex cover of size at most k if and only if G0 has a restrained dominating set of size at most k. On the other hand, one can construct G0 from G in polynomial time. Therefore, we conclude that the restrained domination problem is NP-complete for split graphs by Theorem1. Remark. A particular case of the restrained domination, say total restrained domination, was proposed in 2005 by Ma and Chen [24]. A set S ⊆ V is a total restrained dominating set if every vertex is adjacent to a vertex in S and every vertex L.
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