Planar embeddings with infinite faces

We investigate vertex-transitive graphs that admit planar embeddings having infinite faces, i.e., faces whose boundary is a double ray. In the case of graphs with connectivity exactly 2, we present examples wherein no face is finite. In particular, the planar embeddings of the Cartesian product of the r-valent tree with K2 are comprehensively studied and enumerated, as are the automorphisms of the resulting maps, and it is shown for r A? 3 that no vertex-transitive group of graph automorphisms is extendable to a group of homeomorphisms of the plane. We present all known families of infinite, locally finite, vertex-transitive graphs of connectivity 3 and an infinite family of 4-connected graphs that admit planar embeddings wherein each vertex is incident with an infinite face.