One of the newest problem in the eld of planar graphs is to nd a connected subgraph of a plane graph such that all the faces of that plane graph are covered. The faces of a plane graph are the maximal regions of the plane that contain no point used in the embedding. A face is said to be covered or spanned if at least one of the vertices of that face boundary is visited. We denote this type of subgraph as a face spanning subgraph. The minimum face spanning subgraph is the face spanning subgraph with minimum cost. Cost can be measured by number vertices or total weight of the edges. These kind...