Minimal connected enclosures on an embedded planar graph

Minimal connected enclosures on an embedded planar graph

We study five problems of finding minimal enclosures comprised of elements of a connected, planar graph with a plane embedding. The first three problems consider the identification of a shortest enclosing walk, cycle or trail surrounding a polygonal, simply connected obstacle on the plane. We propos...

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Bibliographic Details
Published in:Discrete Applied Mathematics Vol. 91; no. 1; pp. 25 - 38
Main Authors: Alexopoulos, Christos, Provan, J.Scott, Ratliff, H.Donald, Stutzman, Bryan R.
Format: Journal Article
Language:English
Published: Lausanne Elsevier B.V 1999
Amsterdam Elsevier
New York, NY
Subjects:
Combinatorics
Combinatorics. Ordered structures
Exact sciences and technology
Graph theory
Mathematics
Sciences and techniques of general use
Geography
Optimal partition
Minimal graph
Steiner arborescence
Steiner tree
Planar graph
Connected graph
Embedding
Embedded planar graph
Graph theory
Distribution system
Logistics
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Summary:We study five problems of finding minimal enclosures comprised of elements of a connected, planar graph with a plane embedding. The first three problems consider the identification of a shortest enclosing walk, cycle or trail surrounding a polygonal, simply connected obstacle on the plane. We propose polynomial algorithms that improve on existing algorithms. The last two problems consider the formation of minimal zones (sets of adjacent regions such that any pair of points in a zone can be connected by a non-zero width curve that lies entirely in the zone). Specifically, we assume that the regions of the graph have nonnegative weights and seek the formation of minimum weight zones containing a set of points or a set of regions. We prove that the last two problems are NP-hard and transform them to Steiner arboresccnce/fixed-charge flow problems.
ISSN:0166-218X
1872-6771
DOI:10.1016/S0166-218X(98)00095-X