Before revision:  This leads to a deterministic hierarchical construction method that yields an efficient topological way to control both the number of connected pieces (T) and the total degrees of freedom (DoF). We then turn to a statistical approach to the question by studying the rigidity and connectivity of kirigami with random cuts , and find that both the T and DoF can be exquisitely controlled by the density of cuts (links) in the neighborhood of percolation transitions in the connectivity and rigidity.
After revision:  This leads to a deterministic hierarchical construction method that yields an efficient topological way to control both the number of connected pieces (T) and the total degrees of freedom . A statistical approach to the question by studying the rigidity and connectivity of kirigami with random cuts , and find that both the T and DoF can be exquisitely controlled by the density of cuts (links) in the neighborhood of percolation transitions in the connectivity and rigidity.