On the Path Separability of Planar Graphs

It is known that every weighted planar graph with n vertices contains three shortest paths whose removal halves the graph into connected components of at most n/2 vertices. Whether this property remains true with the use of two shortest paths only is an open problem.We show that two shortest paths are enough for a large family of planar graphs, called face-separable, composed of graphs for which every induced subgraph can be halved by removing the border of a face in some suitable embedding of the subgraph. We also show that this non-trivial family of graphs includes unbounded treewidth graphs.