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

2026-05-08

AI Summary

This work addresses the computational complexity of recognizing $(k, l)$-tight graphs, a problem for which no efficient algorithm is known in general graphs, despite the existence of a deterministic NC algorithm for planar graphs. A natural approach has been to reduce the general case to the planar one using planarizing gadgets. However, this paper provides the first unconditional proof that such planarizing gadgets cannot exist for the $(k, l)$-tightness recognition problem, thereby ruling out this generic reduction strategy. By combining techniques from graph theory and complexity-theoretic impossibility arguments, the study establishes a fundamental computational separation between general and planar graphs for this problem, demonstrating the inherent impossibility of achieving such reductions.

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.

Problem

• (k, l)-tight graphs
• planarizing gadgets
• graph recognition
• planar graphs

Innovation

• planarizing gadgets
• (k, l)-tight graphs
• graph rigidity
• planar graphs
• impossibility result