What is an algoithm for graphing concentric perimeters of an arbitrary shape given a starting point?

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In an infinte and uniform graph with four connecting edges to each vertex (a spreadsheet where each cell represents a vertex and each edge an edge), each vertex is either include or excluded in a subsection of the graph.

Given some such arbitrary continuous subsection and any starting point in that subsection, what algorithm would form a set of new graphs whose nodes hug the perimeter of the original subsection with no overlap or gaps? and would that algorithm be a function from a graph to a set of graphs (regardless of the starting point choosen)?

EG: enter image description here