Planarizing Gadgets for (k, l)-tight Graphs Do Not Exist

TL;DR

This paper proves that planarizing gadgets cannot be used to recognize (k, l)-tight graphs, highlighting fundamental limitations in reducing the problem to planar cases.

Contribution

It establishes the non-existence of planarizing gadgets for recognizing (k, l)-tight graphs, a key step in graph rigidity and planarity algorithms.

Findings

Planarizing gadgets for recognizing (k, l)-tight graphs do not exist.

The result applies unconditionally, without additional assumptions.

Implications for algorithm design in graph rigidity and planarity recognition.

Abstract

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.