The realizable extension problem and the weighted graph (K 3,3, l)

Abstract

This note outlines the realizable extension problem for weighted graphs and provides a detailed analysis of this problem for the weighted graph (K3,3, l). The main result of this analysis is that the moduli space of planar realizations of (K3,3, l) can have one, two, four, six or eight connected components and explicit examples of each case are provided. 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.Mathematics Subject Classification. 55R80, 51-XX.