Art gallery theorems and algorithms
Art gallery theorems and algorithms
Visibility problems for polyhedral terrains
Journal of Symbolic Computation
Coverage problems and visibility regions on topographic surfaces
Annals of Operations Research
Computational geometry in C
Computational Geometry: Theory and Applications
Edge guarding polyhedral terrains
Computational Geometry: Theory and Applications
Journal of Combinatorial Theory Series B - Special issue: dedicated to Professor W. T. Tutte on the occasion of his eightieth birthday
Efficient algorithms for Petersen's matching theorem
Journal of Algorithms
Graph Theory With Applications
Graph Theory With Applications
Polychromatic colorings of plane graphs
Proceedings of the twenty-fourth annual symposium on Computational geometry
Polychromatic Colorings of n-Dimensional Guillotine-Partitions
COCOON '08 Proceedings of the 14th annual international conference on Computing and Combinatorics
A note on the lower bound of edge guards of polyhedral terrains
International Journal of Computer Mathematics
Polychromatic 4-coloring of guillotine subdivisions
Information Processing Letters
Experimental study on approximation algorithms for guarding sets of line segments
ISVC'10 Proceedings of the 6th international conference on Advances in visual computing - Volume Part I
Guarding a set of line segments in the plane
Theoretical Computer Science
Complexity and approximability issues in combinatorial image analysis
IWCIA'11 Proceedings of the 14th international conference on Combinatorial image analysis
Approximation algorithms for a geometric set cover problem
Discrete Applied Mathematics
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We present an optimal Θ(n)-time algorithm for the selection of a subset of the vertices of an n-vertex plane graph G so that each of the faces of G is covered by (i.e., incident with) one or more of the selected vertices. At most ⌊n/2⌋ vertices are selected, matching the worst-case requirement. Analogous results for edge-covers are developed for two different notions of "coverage". In particular, our linear-time algorithm selects at most n - 2 edges to strongly cover G, at most ⌊n/3⌋ diagonals to cover G, and in the case where G has no quadrilateral faces, at most ⌊n/3⌋ edges to cover G. All these bounds are optimal in the worst-case. Most of our results flow from the study of a relaxation of the familiar notion of a 2-coloring of a plane graph which we call a face-respecting 2-coloring that permits monochromatic edges as long as there are no monochromatic faces. Our algorithms apply directly to the location of guards, utilities or illumination sources on the vertices or edges of polyhedral terrains, polyhedral surfaces, or planar subdivisions.