Let a graph
be defined on vertex set and edge set . Then a construction sequence (or c-sequence) for is a linear order on in which each edge appears after both of its endpoints.
The construction number of is then defined as the number of distinct construction sequences
for
(Kainen 2023).

For example, for the path graph with vertex set 1, 2, 3 and
edge set 12, 23, there are 16 possible construction sequences
(so ),
one of which is (1, 2, 3, 23, 13) (Kainen 2023).

The construction numbers for a number of parametrized graph are summarized in the table below (cf. Kainen 2023), where is a Bernoulli number.

## See also

Assembly Number
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## References

Kainen, P. C. "Construction Numbers: How to Build a Graph?" 25 Feb, 2023. https://arxiv.org/abs/2302.13186.Sloane,
N. J. A. Sequences A000142, A000182,
A024255, A055546,
A098694, and A210277
in "The On-Line Encyclopedia of Integer Sequences."
## Cite this as:

Weisstein, Eric W. "Construction Number."
From *MathWorld*--A Wolfram Web Resource. https://mathworld.wolfram.com/ConstructionNumber.html