Algorithmica

, Volume 62, Issue 3–4, pp 754–766

# An O(n+m) Certifying Triconnnectivity Algorithm for Hamiltonian Graphs

• Amr Elmasry
• Kurt Mehlhorn
• Jens M. Schmidt
Article

## Abstract

A graph is triconnected if it is connected, has at least 4 vertices and the removal of any two vertices does not disconnect the graph. We give a certifying algorithm deciding triconnectivity of Hamiltonian graphs with linear running time (this assumes that the cycle is given as part of the input). If the input graph is triconnected, the algorithm constructs an easily checkable proof for this fact. If the input graph is not triconnected, the algorithm returns a separation pair.

## Keywords

Graph algorithms Certifying algorithms Triconnectivity Hamiltonian graph

## References

1. 1.
Alstrup, S., Brodal, G., Rauhe, T.: New data structures for orthogonal range searching. In: FOCS, pp. 198–207 (2000) Google Scholar
2. 2.
Ando, K., Enomoto, H., Saito, A.: Contractible edges in 3-connected graphs. J. Comb. Theory, Ser. B 42, 87–93 (1987)
3. 3.
Brandes, U.: Eager st-ordering. In: ESA, pp. 247–256 (2002) Google Scholar
4. 4.
de Berg, M., van Kreveld, M., Overmars, M., Schwarzkopf, O.: Computational Geometry: Algorithms and Applications. Springer, Berlin (1997)
5. 5.
Elmasry, A., Mehlhorn, K., Schmidt, J.M.: Every DFS-tree of a 3-connected graph contains a contractible edge. Available at the second author’s web-page, February 2010 Google Scholar
6. 6.
Even, S.: An algorithm for determining whether the connectivity of a graph is at least k. SIAM J. Comput. 4(3), 393–396 (1975)
7. 7.
Galil, Z.: Finding the vertex connectivity of graphs. SIAM J. Comput. 9(1), 197–199 (1980)
8. 8.
Gabow, H., Bentley, J., Tarjan, R.: Scaling and related techniques for geometry problems. In: STOC, pp. 135–143 (1984) Google Scholar
9. 9.
Gutwenger, C., Mutzel, P.: A linear time implementation of SPQR-trees. In: Graph Drawing. LNCS, vol. 1984, pp. 77–90 (2000)
10. 10.
Gabow, H.N., Tarjan, R.E.: A linear-time algorithm for a special case of disjoint set union. J. Comput. Syst. Sci. 30(2), 209–221 (1985)
11. 11.
Hopcroft, J., Tarjan, R.E.: Efficient planarity testing. J. ACM 21(4), 549–568 (1974)
12. 12.
Imai, T., Asano, T.: Dynamic segment intersection with applications. J. ACM 8, 1–18 (1987)
13. 13.
Kriesell, M.: A survey on contractible edges in graphs of a prescribed vertex connectivity. Graphs Comb. 18(1), 1–30 (2002)
14. 14.
Linial, N., Lovász, L., Wigderson, A.: Rubber bands, convex embeddings, and graph connectivity. Combinatorica 8(1), 91–102 (1988)
15. 15.
Mehlhorn, K.: Data Structures and Efficient Algorithms, vol. 3: Multi-dimensional Searching and Computational Geometry. Springer, Berlin (1984) Google Scholar
16. 16.
McConnell, R., Mehlhorn, K., Näher, S., Schweitzer, P.: Certifying algorithms. Comput. Sci. Rev. (2010). doi: Google Scholar
17. 17.
Miller, G.L., Ramachandran, V.: A new graph triconnectivity algorithm and its parallelization. Combinatorica 12(1), 53–76 (1992)
18. 18.
Nagamochi, H., Ibaraki, T.: A linear-time algorithm for finding a sparse k-connected spanning subgraph of a k-connected graph. Algorithmica 7, 583–596 (1992)
19. 19.
Schmidt, J.M.: Construction sequences and certifying 3-connectedness. In: 27th International Symposium on Theoretical Aspects of Computer Science (STACS’10), Nancy, France (2010). http://drops.dagstuhl.de/portals/extern/index.php?conf=STACS10 Google Scholar
20. 20.
Thomassen, C.: Nonseparating cycles in k-connected graphs. J. Graph. Theory 5, 351–354 (1981)
21. 21.
Tutte, W.: A theory of 3-connected graphs. Indag. Math. 23, 441–455 (1961)