Lower bound for realization increase under planar 1-extensions

Establish a universal lower bound greater than a constant for the ratio of complex realization counts under every minimally rigid planar graph 1-extension.

Background

The paper analyzes how Henneberg construction steps change the number of complex realizations of minimally rigid planar graphs. Although some subclasses have known lower bounds, the general behavior of 1-extensions remains poorly understood.

The authors explicitly identify as open the problem of proving a general lower bound that grows beyond a constant factor.