# The Common Structure of the Curves Having a Same Gauss Word

• Bruno Courcelle
Chapter
Part of the Emergence, Complexity and Computation book series (ECC, volume 12)

## Abstract

Gauss words are finite sequences of letters associated with self-intersecting closed curves in the plane. (These curves have no “triple” self-intersection). These sequences encode the order of intersections on the curves. We characterize, up to homeomorphism, all curves having a given Gauss word. We extend this characterization to the n-tuples of closed curves having a given n-tuple of words, that we call a Gauss multiword. These words encode the self-intersections of the curves and their pairwise intersections. Our characterization uses decompositions of strongly connected graphs in 3-edge-connected components and algebraic terms formalizing these decompositions.

## Keywords

Directed Graph Regular Graph Atomic Decomposition Plane Embedding Biconnected Component
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. 1.
2. 2.
Bollobas, B.: Modern graph theory. Springer (2001)Google Scholar
3. 3.
Chaves, N., Weber, C.: Plombages de rubans et problème des mots de Gauss. Exp. Math. 12, 53–77 (1994)
4. 4.
Courcelle, B.: The monadic second-order logic of graphs XII: Planar graphs and planar maps. Theoret. Comput. Sci. 237, 1–32 (2000)
5. 5.
Courcelle, B.: The atomic decomposition of strongly connected graphs, Research report (June 2013), http://hal.archives-ouvertes.fr/hal-00875661
6. 6.
Courcelle, B., Dussaux, V.: Map genus, forbidden maps and monadic second-order logic. The Electronic Journal of Combinatorics 9(1), 40 (2002), http://www.combinatorics.org/Volume_9/Abstracts/v9i1r40.html
7. 7.
Diestel, R.: Graph theory, 4th edn. Springer (2010), http://diestel-graph-theory.com
8. 8.
de Fraysseix, H., Ossona de Mendez, P.: On a characterization of Gauss codes. Discrete Comput. Geom. 22, 287–295 (1999)
9. 9.
Galil, Z., Italiano, G.: Maintaining the 3-edge-connected components of a graph on-line. SIAM J. Comput. 22, 11–28 (1993)
10. 10.
Godsil, C., Royle, G.: Algebraic graph theory. Springer (2001)Google Scholar
11. 11.
Lins, S., Richter, B., Shank, H.: The Gauss code problem off the plane. Aequationes Mathematicae 33, 81–95 (1987)
12. 12.
Lovasz, L., Marx, M.: A forbidden substructure characterization of Gauss codes. Bulletin of the AMS 82, 121–122 (1976)
13. 13.
Mohar, B.: A linear time algorithm for embedding graphs in an arbitrary surface. SIAM J. Discrete Math. 12, 6–26 (1999)
14. 14.
Mohar, B., Thomassen, C.: Graphs on surfaces. The Johns Hopkins University Press (2001)Google Scholar
15. 15.
Naor, D., Gusfield, D., Martel, C.: A fast algorithm for optimally increasing the edge connectivity. SIAM J. Comput. 26, 1139–1165 (1997)
16. 16.
Rosenstiehl, P.: Solution algébrique du problème Gauss sur la permutation des points d’intersection d’une ou plusieurs courbes fermées du plan. C. R. Acad. Sc. Paris, Sér. A 283, 551–553 (1976)
17. 17.
Tsin, Y.H.: A simple 3-edge-connected component algorithm. Theory Comput. Syst. 40, 125–142 (2007)