Auto comment: topic has been updated by Sammmmmmm (previous revision, new revision, compare).

As far as i understand this graph will be always connected)

If you have consecutive columns filled only with zeros, you can compress them into one column (which doesn't change reachability). After that, group the zeros into maximal vertical segments. The graph constructed by adding edges between segments that are adjacent to each other has the same number of components as our original graph. We can construct it "brutally" with some sort of sweep-line, because it is planar and therefore sparse.