Minimum Face-Spanning Subgraphs of Plane Graphs

A plane graph is a planar graph with a fixed embedding. Let G = (V,E) be an edge weighted connected plane graph, where V and E are the set of vertices and edges, respectively. Let F be the set of faces of G. For each edge e ∈ E, let w(e) ≥ 0 be the weight of the edge e of G. A face-spanning subgraph of G is a connected subgraph H induced by a set of edges S ⊆ E such that the vertex set of H contains at least one vertex from the boundary of each face f ∈ F of G. A minimum face-spanning subgraph H of G is a face-spanning subgraph of G, where P