  Counting the number of linear extensions of a partial order was considered by
Stanley about 50 years ago. For the partial order on the vertices and edges of
a graph given by inclusion, we call a linear extension a {\it construction
sequence} for the graph as each edge follows the vertices where it is attached.
The number of these c-sequences is counted for various graph families. We also
consider the set of all length-$n$ c-sequences produced by the graphs with $n$
elements, simplified to their structural skeleton: vertex or edge, and further
allow the generating graph to be structurally constrained. Efficiency is
analyzed.
