Near homeomorphisms and G-like continua

It is shown that for a given finite, connected, planar graph, G, containing a branch point, the set of surjective, continuous maps of G which cannot be extended to a near homeomorphism on any neighborhood of G is open and dense in the set of all such maps, with the C0-topology. Also, an example of a map which cannot be extended to a near homeomorphism for which the inverse limit can be embedded as an attractor in the plane is given.