  In this paper we further our understanding of the structure of class two
cubic graphs, or snarks, as they are commonly known. We do this by
investigating their 3-critical subgraphs, or as we will call them, minimal
conflicting subgraphs. We consider how the minimal conflicting subgraphs of a
snark relate to its possible minimal 4-edge-colourings. We fully characterise
the relationship between the resistance of a snark and the set of minimal
conflicting subgraphs. That is, we show that the resistance of a snark is equal
to the minimum number of edges which can be selected from the snark, such that
the selection contains at least one edge from each minimal conflicting
subgraph. We similarly characterise the relationship between what we call
\textit{the critical subgraph} of a snark and the set of minimal conflicting
subgraphs. The critical subgraph being the set of all edges which are
conflicting in some minimal colouring of the snark. Further to this, we define
groups, or \textit{clusters}, of minimal conflicting subgraphs. We then
highlight some interesting properties and problems relating to clusters of
minimal conflicting subgraphs.
