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Fixed Parameter Algorithms for Dominating Set and Related Problems on Planar Graphs
, 2002
"... We present an algorithm that constructively produces a solution to the kdominating set problem for planar graphs in time O(c . To obtain this result, we show that the treewidth of a planar graph with domination number (G) is O( (G)), and that such a tree decomposition can be found in O( (G)n) time. ..."
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Cited by 105 (23 self)
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We present an algorithm that constructively produces a solution to the kdominating set problem for planar graphs in time O(c . To obtain this result, we show that the treewidth of a planar graph with domination number (G) is O( (G)), and that such a tree decomposition can be found in O( (G)n) time. The same technique can be used to show that the kface cover problem ( find a size k set of faces that cover all vertices of a given plane graph) can be solved in O(c n) time, where c 1 = 3 and k is the size of the face cover set. Similar results can be obtained in the planar case for some variants of kdominating set, e.g., kindependent dominating set and kweighted dominating set.
Parameterized Complexity: Exponential SpeedUp for Planar Graph Problems
 in Electronic Colloquium on Computational Complexity (ECCC
, 2001
"... A parameterized problem is xed parameter tractable if it admits a solving algorithm whose running time on input instance (I; k) is f(k) jIj , where f is an arbitrary function depending only on k. Typically, f is some exponential function, e.g., f(k) = c k for constant c. We describe general techniqu ..."
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Cited by 61 (21 self)
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A parameterized problem is xed parameter tractable if it admits a solving algorithm whose running time on input instance (I; k) is f(k) jIj , where f is an arbitrary function depending only on k. Typically, f is some exponential function, e.g., f(k) = c k for constant c. We describe general techniques to obtain growth of the form f(k) = c p k for a large variety of planar graph problems. The key to this type of algorithm is what we call the "Layerwise Separation Property" of a planar graph problem. Problems having this property include planar vertex cover, planar independent set, and planar dominating set.
Fixed parameter algorithms for planar dominating set and related problems
, 2000
"... We present an algorithm that constructively produces a solution to the kdominating set problem for planar graphs in time O(c √ kn), where c = 36√34. To obtain this result, we show that the treewidth of a planar graph with domination number γ(G) is O ( � γ(G)), and that such a tree decomposition ca ..."
Abstract

Cited by 35 (10 self)
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We present an algorithm that constructively produces a solution to the kdominating set problem for planar graphs in time O(c √ kn), where c = 36√34. To obtain this result, we show that the treewidth of a planar graph with domination number γ(G) is O ( � γ(G)), and that such a tree decomposition can be found in O ( � γ(G)n) time. The same technique can be used to show that the kface cover problem (find a size k set of faces that cover all vertices of a given plane graph) can be solved √ k in O(c1 n + n2) time, where c1 = 236√34 and k is the size of the face cover set. Similar results can be obtained in the planar case for some variants of kdominating set, e.g., kindependent dominating set and kweighted dominating set. Keywords. NPcomplete problems, fixed parameter tractability, planar graphs, planar dominating set, face cover, outerplanarity, treewidth.
Undirected VertexConnectivity Structure and Smallest FourVertexConnectivity Augmentation
 Proc. 6th ISAAC
, 1995
"... In this paper, we study properties for the structure of an undirected graph that is not 4vertexconnected. We also study the evolution of this structure when an edge is added to optimally increase the vertexconnectivity of the underlying graph. Several properties reported here can be extended t ..."
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Cited by 4 (0 self)
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In this paper, we study properties for the structure of an undirected graph that is not 4vertexconnected. We also study the evolution of this structure when an edge is added to optimally increase the vertexconnectivity of the underlying graph. Several properties reported here can be extended to the case of a graph that is not kvertex connected, for an arbitrary k. Using properties obtained here, we solve the problem of finding a smallest set of edges whose addition 4vertexconnects an undirected graph. This is a fundamental problem in graph theory and has applications in network reliability and in statistical data security. We give an O(n \Delta log n + m)time algorithm for finding a set of edges with the smallest cardinality whose addition 4vertexconnects an undirected graph, where n and m are the number of vertices and edges in the input graph, respectively. This is the first polynomial time algorithm for this problem when the input graph is not 3vertexconnecte...