Infinite paths in planar graphs II, structures and ladder nets

A graph is k-indivisible, where k is a positive integer, if the deletion of any finite set of vertices results in at most k − 1 infinite components. In 1971, Nash-Williams conjectured that a 4-connected infinite planar graph contains a spanning 2-way infinite path if, and only if, it is 3-indivisible. In this paper, we prove a structural result for 2-indivisible infinite planar graphs. This structural result is then used to prove Nash-Williams conjecture for all 4-connected 2-indivisible infinite planar graphs