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34 Int'l Conf. Foundations of Computer Science | FCS'12 | The Parameterized Complexity of Perfect Code in Graphs without Small Cycles Yong Zhang Department of Computer Science, Kutztown University of Pennsylvania, Kutztown, PA 19530 Abstract— We study the parameterized complexity of k- PERFECT CODE in graphs without small cycles. We show that k-PERFECT CODE is W[1]-hard in bipartite graphs and thus in graphs with girth 4. On the other hand, we show that k-PERFECT CODE admits a k 2 + k kernel in graphs with girth ≥ 5. Keywords: parameterized complexity, perfect code 1. Introduction Parameterized complexity is a powerful framework that deals with hard computational problems. A parameterized problem is a set of instances of the form (x,k), where x is the input instance and k is a nonnegative integer called the parameter. A parameterized problem is said to be fixed parameter tractable if there is an algorithm that solves the problem in time f(k)|x| O(1) , where f is a computable function solely dependent on k, and |x| is the size of the input instance. The kernelization of a parameterized problem is a reduction to a problem kernel, that is, to apply a polynomial-time algorithm to transform any input instance (x,k) to an equivalent reduced instance (x ′ ,k ′ ) with k ′ ≤ k and |x ′ | ≤ g(k) for some function g solely dependent on k . A parameterized problem is fixed parameter tractable if and only if it is kernelizable. On the other hand, many fixed parameter intractable problems can be classified in a hierarchy of complexity classes W[1] ⊆ W[2] ... ⊆ W[t]. For example, k-INDEPENDENT SET and k-CLIQUE are known to be W[1]-complete and k-DOMINATING SET is known to be W[2]-complete. We refer the readers to [1], [2] for more details. Let G = (V,E) be an undirected graph. For a vertex v ∈ V , let N(v) and N[v] be the open neighborhood and closed neighborhood of v, respectively. A perfect code in G is a subset of vertices D ⊆ V such that for every vertex v ∈ V , there is exactly one vertex in N[v]∩D. Definition 1.1: Given an input graph G and a positive integer k, the k-PERFECT CODE problem is to determine whether G has a perfect code of size at most k. In the literatures k-PERFECT CODE is also known as EF- FICIENT DOMINATING SET, PERFECT DOMINATING SET, and INDEPENDENT PERFECT DOMINATING SET. It is a well-known NP-complete problem. Its computational complexity in various classes of graphs has been extensively studied. See Lu and Tang [3] for an overview. In terms of parameterized complexity, k-PERFECT CODE is known to be W[1]-complete [4], [5] in general graphs. Guo and Niedermeier [6] showed that k-PERFECT CODE is fixed parameter tractable in planar graphs by giving a 84k kernel. Dawar and Kreutzer [7] showed that it is fixed parameter tractable in effectively nowhere-dense classes of graphs. The girth of a graph is the length of the shortest cycle contained in the graph. In this paper we study the parameterized complexity of k-PERFECT CODE in graphs with certain girths, i.e., graphs without small cycles. The parameterized complexity of several related problems, including k-DOMINATING SET and k-INDEPENDENT SET [8], and k- CONNECTED DOMINATING SET [9] in graphs without small cycles has been studied. In this paper we show that k- PERFECT CODE is W[1]-hard in bipartite graphs, and thus in triangle-free graphs or graphs with girth 4. Then we show that k-PERFECT CODE admits a k 2 +k kernel in graphs with girth ≥ 5 and is therefore fixed parameter tractable. 2. Main results 2.1 Bipartite Graphs To show the W[1]-hardness of k-PERFECT CODE in bipartite graphs, we give a reduction from the problem k- MULTICOLORED CLIQUE: Given a graph G = (V,E) and a vertex-coloring κ : V → {1,2,...,k}, decide whether G has a clique of k vertices containing exactly one vertex of each color. k-MULTICOLORED CLIQUE is shown to be W[1]-complete by Fellows et al. [10]. Theorem 2.1: k-PERFECT CODE is W[1]-complete in bipartite graphs. Proof: Let (G = (V,E),κ) be an input instance of k-MULTICOLORED CLIQUE. For each color i, 1 ≤ i ≤ k, let Vi be the set of vertices in G with color i. Let ni be the number of vertices in Vi. Without loss of generality, we assume that ni > 1 for all i. We fixed an ordering of the vertices in each Vi. To simplify the presentation, we abuse notations here: for two vertices u,v ∈ Vi, u > v means u is in front of v with respect to the fixed ordering. Without loss of generality, we also assume that no edge in G connects two vertices of the same color. For any two colors i and j, 1 ≤ i < j ≤ k, let Eij be the set of edges in G that connect
Int'l Conf. Foundations of Computer Science | FCS'12 | 35 vertices in Vi and Vj. Let mij be the number of edges in Eij. We construct a graph G ′ = (V ′ ,E ′ ). The vertex set V ′ is a union of the following sets of vertices: S1 = {a[i,v],b[i,v] | 1 ≤ i ≤ k,v ∈ Vi} ∪{x[i] | 1 ≤ i ≤ k} S2 = {c[i,j,e],d[i,j,e] | 1 ≤ i < j ≤ k,e ∈ Eij} ∪{y[i,j] | 1 ≤ i < j ≤ k} S3 = {f[i,j,v] | 1 ≤ i < j ≤ k,v ∈ Vi} S4 = {g[i,j,v] | 1 ≤ i < j ≤ k,v ∈ Vj} The edge set E ′ is a union of the following set of edges: E1 = {(a[i,v1],b[i,v2]) | 1 ≤ i ≤ k,v1,v2 ∈ Vi and v1 �= v2} E2 = {(x[i],b[i,v]) | 1 ≤ i ≤ k,v ∈ Vi} E3 = {(c[i,j,e1],d[i,j,e2]) | 1 ≤ i < j ≤ k, e1,e2 ∈ Eij and e1 �= e2} E4 = {(y[i,j],c[i,j,e]) | 1 ≤ i < j ≤ k,e ∈ Eij} E5 = {(b[i,v1],f[i,j,v2]) | 1 ≤ i < j ≤ k, v1,v2 ∈ Vi and v1 ≥ v2} E5 = {(b[j,v1],g[i,j,v2]) | 1 ≤ i < j ≤ k, v1,v2 ∈ Vj and v1 ≥ v2} E6 = {(c[i,j,e],f[i,j,v]) | 1 ≤ i < j ≤ k, e = (v1,v2) ∈ Eij,v,v1 ∈ Vi and v1 < v} E7 = {(c[i,j,e],g[i,j,v]) | 1 ≤ i < j ≤ k, e = (v1,v2) ∈ Eij,v,v2 ∈ Vj and v2 < v} Informally speaking, for each Vi, 1 ≤ i ≤ k, we construct a vertex selection gadget that contains 2ni+1 vertices. For each v ∈ Vi, there are two verticesa[i,v] and b[i,v]. For two vertices a[i,v1] and b[i,v2], v1,v2 ∈ Vi, they are adjacent if and only if v1 �= v2. There is a dummy vertex x[i] which is adjacent to all vertices {b[i,v] | v ∈ Vi} and none of the vertices {a[i,v] | v ∈ Vi}. Then, for each edge set Eij, 1 ≤ i < j ≤ k, we construct an edge selection gadget that contains 2mij + 1. For each edge e ∈ Eij, there are two vertices c[i,j,e] and d[i,j,e]. There is also a dummy vertex y[i,j]. They are connected in a similar fashion as in the vertex selection gadget. Finally, for each pair of colors i and j with 1 ≤ i < j ≤ k, we also construct a validation gadget that contains ni +nj vertices, namely {f[i,j,v] | v ∈ Vi} and{g[i,j,v] | v ∈ Vj}. The vertices in the validation gadget are not adjacent to each other, instead they are adjacent to vertices in the vertex selection gadgets for Vi and Vj, and the edge selection gadget for Eij. For a vertex v1 ∈ Vi, the corresponding vertex b[i,v1] is adjacent to f[i,j,v2] for all v2 ∈ Vi such that v1 ≥ v2 with respect to the fixed vertex ordering of Vi. Similarly, for a vertex v1 ∈ Vj, the corresponding vertex b[j,v1] is adjacent to g[i,j,v2] for all v2 ∈ Vj such that v1 ≥ v2 with respect to the fixed vertex ordering ofVj. On the other hand, for an edgee = (v1,v2) ∈ Eij with v1 ∈ Vi and v2 ∈ Vj, the corresponding vertex c[i,j,e] in the edge selection gadget is adjacent to f[i,j,v] for all v ∈ Vj such that v > v1, and to g[i,j,v] for all v ∈ Vj such that v > v2. See Figure 1 for an illustration of the construction. Clearly G ′ is a bipartite graph. a[i,v1] a[i,v2] a[i,v3] x[i] a[j,w1] a[j,w2] x[j] b[i,v1] b[i,v3] b[j,w1] b[j,w2] f[i,j,v1] f[i,j,v3] g[i,j,w1] g[i,j,w2] c[i,j,e1] c[i,j,e4] d[i,j,e1] d[i,j,e2] d[i,j,e3] d[i,j,e4] y[i,j] Fig. 1: A partial illustration of the construction of G ′ . Let Vi = {v1,v2,v3} and Vj = {w1,w2}. Let Eij = {e1,e2,e3,e4} where e1 = (v1,w1),e2 = (v1,w2),e3 = (v2,w1),e4 = (v3,w2). The figure shows how the vertex selection gadgets for Vi and Vj (left), the edge selection gadget Eij (right), and the validation gadgets (middle) are constructed. Lemma 2.2: G has a k-multicolored clique if and only if G ′ has a perfect code of size k ′ = 2k+2 � � k . 2 Proof: For the direct implication, suppose G = (V,E) has a k-multicolored clique K ⊆ V such that K = {vi | 1 ≤ i ≤ k,vi ∈ Vi}, then it is easy to verify that the following set D of vertices in V ′ is a perfect code of size k ′ for G ′ : D = {a[i,vi],b[i,vi] | vi ∈ K}∪{c[i,j,e],d[i,j,e] | 1 ≤ i < j ≤ k,e = (vi,vj) and vi,vj ∈ K}. For the reverse implication, suppose D is a perfect code of size k ′ for G ′ . First observe that the dummy vertex x[i] in the vertex selection gadget for Vi cannot be in D since otherwise vertices {a[i,v] | v ∈ Vi} cannot be dominated. To dominate x[i], D must contain exactly one vertex from the set {b[i,v] | v ∈ Vi}. Let b[i,vs] be such a vertex,b[i,vs] dominates all vertices {a[i,v] | v ∈ Vi} except a[i,vs], this implies that a[i,vs] must also be in D. By this argument, we see that D must contain exactly two vertices from each vertex selection gadget and each edge selection gadget. In another word, the following 2k +2 � � k 2 vertices must be in D: {a[i,vi],b[i,vi] | 1 ≤ i ≤ k,vi ∈ Vi}∪ {c[i,j,eij],d[i,j,eij] | 1 ≤ i < j ≤ k,eij ∈ Eij}.
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