Planarizing Gadgets for (k, l)-tight Graphs Do Not Exist
This resolves a fundamental question for researchers working on graph rigidity and recognition of (k, l)-tight graphs, showing that a standard reduction technique is impossible.
The paper proves that planarizing gadgets cannot exist for recognizing (k, l)-tight graphs, ruling out a common approach to reduce the problem to planar graphs.
The problem of recognizing (k, l)-tight graphs is a fundamental problem that has close connections to well studied problems like graph rigidity. The problem is better understood for planar graphs as compared to general graphs. For example, deterministic NC-algorithms for the problem are known for planar graphs, but no such algorithm is known for general graphs. A common approach to reduce a graph problem to the planar case is to use planarizing gadgets. Our main contribution is to show that, unconditionally, planarizing gadgets for the problem of recognizing (k, l)-tight graphs do not exist.