  We introduce the notion of a standard weighted graph and show that every
weighted graph has an essentially unique standard model. Moreover we classify
birational transformations between such models. Our central result shows that
these are composed of elementary transformations. The latter ones are defined
similarly to the well known elementary transformations of ruled surfaces.
  In a forthcoming paper, we apply these results in the geometric setup to
obtain standard equivariant completions of affine surfaces with an action of
certain algebraic groups. We show that these completions are unique up to
equivariant elementary transformations.
