Examining each connected pixel to select its neighbors to separate clusters using ArcGIS for Desktop? [duplicate]

Question

I am trying to develop a methodology for separating pixels from one another to determine whether they belong to a city cluster or not. Starting at the center pixel of the city I want to select all pixels that fall under a certain pixel neighborhood. I want to examine a few different ones 3x3, 4x4, etc.. The selection would expand outward, examining each connected pixel to select its neighbors, and so on, until there are no neighbors for any of the pixels. I have tried focal statistics in arcgis, but I cannot find out if there is a way to use the tool on just one pixel. The picture below shows the issue. I cannot tell if the pixels circled in black are actually connected to the center pixel. This is the output of focal statistics showing number of neighbors in 3x3 rectangular neighborhood. I would prefer a python solution if possible.