This note outlines the realizable extension problem for weighted graphs and provides results of a detailed analysis of this problem for the weighted graph (K3,3,l). This analysis is then utilized to provide a result relating to the connectedness of the moduli space of planar realizations of (K3,3,l). The note culminates with two examples which show that in general, realizability and connectedness results relating to the moduli spaces of weighted cycles which are contained in a larger weighted graph cannot be extended to similar results regarding the moduli space of the larger weighted graph.
@article{arxiv.1009.5626,
title = {The Realizable Extension Problem and the Weighted Graph $(K_{3,3},l)$},
author = {Jonathan McLaughlin},
journal= {arXiv preprint arXiv:1009.5626},
year = {2010}
}