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 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 arborescence fixed-charge flow problems.