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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. ..."
Abstract

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.
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 ..."
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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.
A Qualitative Formalization of Built Environments
 In DEXA2000 Conference Proceedings, Lecture Notes in Computer Science
, 2000
"... . In this paper I argue that a qualitative formalization of built environments needs to take into account: (1) the ontological distinction between bonafide and fiat boundaries and objects, (2) the different character of constraints on relations involving these different kinds of boundaries and ob ..."
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Cited by 3 (1 self)
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. In this paper I argue that a qualitative formalization of built environments needs to take into account: (1) the ontological distinction between bonafide and fiat boundaries and objects, (2) the different character of constraints on relations involving these different kinds of boundaries and objects, (3) the distinction between partition forming and nonpartition forming objects, and (4) the fundamental organizational structure of regional partitions. I discuss the notion of boundary sensitive rough location and show that a formalization based on this notion takes all these points into account. 1 Introduction Imagine a computer program (I) that generates configurations of lines in the Euclidean plane that look like plans of built environments [Lyn60]. Examples of built environments are shopping malls, airports, or parking lots (See, for example, the left part of Fig. 1.). Program (I) generates configurations of lines of different style and width. Suppose our task is to desig...
Towards a Model Theory of Figure Ground Locations
 Proc. 6th Symp. on Mathematics and AI, Fort
"... In this paper a model theory for figure ground locations is proposed. Figure ground locations are ntuples of predicates about relations between regional individuals and a set of regional individuals forming a regional partition of the plane. This language is based on the RCCtheory which is extende ..."
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Cited by 2 (0 self)
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In this paper a model theory for figure ground locations is proposed. Figure ground locations are ntuples of predicates about relations between regional individuals and a set of regional individuals forming a regional partition of the plane. This language is based on the RCCtheory which is extended by a number of definitions and axioms constraining the models to the set of regular closed sets in a 2dimensional T4 topological space. In this paper we give two models of figure ground locations: One model in point set topology referring to equivalence classes of regular closed sets. One graph theoretical model which is a set of directed and edgevalued versions of the nondirected nonlabeled dual graph of the planar graph representing the regional partition. Both models are shown to be isormorphic with respect to a set of union and intersection operations. 1. INTRODUCTION Much effort has been expended on the problem of constructing formal theories for qualitative spatial reasoning (QS...