The Realizable Extension Problem and the Weighted Graph $(K_{3,3},l)$

TL;DR

This paper investigates the realizable extension problem for weighted graphs, specifically analyzing the case of $(K_{3,3},l)$, and explores implications for the connectedness of their moduli spaces of planar realizations.

Contribution

It provides a detailed analysis of the realizable extension problem for $(K_{3,3},l)$ and relates this to the connectedness of its moduli space, highlighting limitations of extending cycle results.

Findings

Connectedness of the moduli space of $(K_{3,3},l)$ is characterized.

Realizability results for cycles do not generalize to larger graphs.

Examples demonstrate limitations in extending moduli space properties.

Abstract

This note outlines the realizable extension problem for weighted graphs and provides results of a detailed analysis of this problem for the weighted graph $(K_{3,3},l)$. This analysis is then utilized to provide a result relating to the connectedness of the moduli space of planar realizations of $(K_{3,3},l)$. 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.