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 determined by inclusion, we call such linear extensions {\it
construction sequences} for the graph as each edge follows both of its
endpoints. The number of such sequences for paths, cycles, stars, double-stars,
and complete graphs is found. For paths, we agree with Stanley (the Tangent
numbers) and get formulas for the other classes. Structure and applications are
also studied.Comment: 18 page