A characterization of infinite planar primitive graphs

It is shown that the automorphism group of an infinite, locally finite, planar graph acts primitively on its vertex set if and only if the graph has connectivity 1 and, for some integer m⩾2, every vertex is incident with exactly m lobes, all of which are finite. Specifically, either all of the lobes are isomorphic to K4 or all are circuits of length p for some odd prime p.