  A cut of a graph can be represented in many different ways. Here we propose
to represent a cut through a ``relation tree'', which is a spanning tree with
signed edges. We show that this picture helps to classify the main greedy
heuristics for the maximum cut problem, in analogy with the minimum spanning
tree problem. Namely, all versions of the Sahni-Gonzalez~(SG) algorithms could
be classified as the Prim class, while various Edge-Contraction~(EC) algorithms
are of the Kruskal class. We further elucidate the relation of this framework
to the stabilizer formalism in quantum computing, and point out that the
recently proposed \textit{ADAPT-Clifford} algorithm is a reformulation of a
refined version of the SG algorithm, SG3. Numerical performance of the typical
algorithms from the two classes are studied with various kinds of graphs. It
turns out that, the Prim-class algorithms perform better for general dense
graphs, and the Kruskal-class algorithms performs better when the graphs are
sparse enough.
