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1-Skeleton

The 1-skeleton is the p=1 case of the p-skeleton of a simplicial complex or CW-complex. For a simplicial complex K, it is the simplicial subcomplex K^((1)) consisting of its vertices and edges. For a CW-complex X, it is the union X^1 of the zero- and one-dimensional cells (Hatcher 2002).

For a map represented by a cellular embedding of a graph in a surface, the 1-skeleton is the embedded graph formed by the vertices and edges. Its underlying graph is obtained by forgetting the interiors of the faces and the supporting surface (Gross and Tucker 1987).

The 1-skeleton does not in general determine a simplicial complex, since a finite clique in its graph need not span a simplex. In a flag complex, every finite clique in the 1-skeleton spans a simplex. Consequently, a flag complex is the clique complex of its 1-skeleton and is determined by that graph (Jonsson 2008, Kahle 2009).

See also

Cellular Embedding, Clique Complex, CW-Complex, Flag Complex, Graph Embedding, p-Skeleton, Simplicial Complex, Skeleton

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References

Gross, J. L. and Tucker, T. W. Topological Graph Theory. New York: Wiley, 1987.Hatcher, A. Algebraic Topology. Cambridge, England: Cambridge University Press, 2002.Jonsson, J. Simplicial Complexes of Graphs. Berlin, Germany: Springer-Verlag, 2008. https://doi.org/10.1007/978-3-540-75859-4.Kahle, M. "Topology of Random Clique Complexes." Disc. Math. 309, 1658-1671, 2009. https://doi.org/10.1016/j.disc.2008.02.037.

Cite this as:

Weisstein, Eric W. "1-Skeleton." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/1-Skeleton.html

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