  Which graphs, in the class of all graphs with given numbers n and m of edges
and vertices respectively, minimizes or maximizes the value of some graph
parameter? In this paper we develop a technique which provides answers for
several different parameters: the numbers of edges in the line graph, acyclic
orientations, cliques, and forests. (We minimize the first two and maximize the
third and fourth.)
  Our technique involves two moves on the class of graphs. A compression move
converts any graph to a form we call fully compressed: the fully compressed
graphs are split graphs in which the neighbourhoods of points in the
independent set are nested. A second consolidation move takes each fully
compressed graph to one particular graph which we call H(n,m). We show
monotonicity of the parameters listed for these moves in many cases, which
enables us to obtain our results fairly simply.
  The paper concludes with some open problems and future directions.
