# 선분 그래프의 정점 연결성에 대한 완전 동적 알고리즘

Kim, Jae-hoon
김재훈

• Accepted : 2015.11.30
• Published : 2016.02.29
• 17 15

#### Abstract

A graph G=(V,E) is called an interval graph with a set V of vertices representing intervals on a line such that there is an edge $(i,j){\in}E$ if and only if intervals i and j intersect. In this paper, we are concerned in the vertex connectivity, one of various characteristics of the graph. Specifically, the vertex connectivity of an interval graph is represented by the overlapping of intervals. Also we propose an efficient algorithm to compute the vertex connectivity on the fully dynamic environment in which the vertices or the edges are inserted or deleted. Using a special kind of interval tree, we show how to compute the vertex connectivity and to maintain the tree in O(logn) time when a new interval is added or an existing interval is deleted.

#### Keywords

interval graph;vertex connectivity;fully dynamic;interval tree;algorithm;interval

#### References

1. M. C. Golumbic, Algorithmic Graph Theory and Perfect Graphs, New York, NY: Academic Press, 1980.
2. J. M. Keil, "Finding hamiltonian circuits in interval graphs," Information Processing Letters, vol. 20, pp. 201-206, May 1985. https://doi.org/10.1016/0020-0190(85)90050-X
3. G. Ramalingam and C. Pandu Rangan, "A unified approach to domination problems on interval graphs," Information Processing Letters, vol. 27, pp. 271-274, April 1988. https://doi.org/10.1016/0020-0190(88)90091-9
4. A. Srinivasa Rao and C. Pandu Rangan, "Linear algorithm for domatic number problem on interval graphs," Information Processing Letters, vol. 33, pp. 29-33, Oct. 1989. https://doi.org/10.1016/0020-0190(89)90184-1
5. H. Broersma, Jiri Fiala, P. A. Golovach, T. Kaiser, D. Paulusma, and A. Proskurowski, "Linear-time algorithms for scattering number and hamilton-connectivity of interval graphs," Journal of Graph Theory, vol. 79, pp. 282-299, Aug. 2015. https://doi.org/10.1002/jgt.21832
6. S. Even and R. E. Tarjan, "Network flow and testing graph connectivity," SIAM Journal on Computing, vol. 4, pp. 507-518, Oct. 1975. https://doi.org/10.1137/0204043
7. P. K. Ghosh and M. Pal, "An Efficient algorithm to solve connectivity problem on trapezoid graphs," Journal of Applied Mathematics and Computing, vol. 24, pp. 141-154, May 2007. https://doi.org/10.1007/BF02832306
8. A. Ilic, "Efficient algorithm for the vertex connectivity of trapezoid graph," Information Processing Letters, vol. 113, pp. 398-404, May 2013. https://doi.org/10.1016/j.ipl.2013.02.012
9. T. W. Kao and S. J. Horng, "Computing k-vertex connectivity on an interval graph," in Proceeding of the 13th International Conference on Parallel Processing, pp. 218-221, 1994.
10. D. Eppstein, Z. Galil, G, and F. Italiano, Agorithms and Theoretical Computing Handbook, CRC Press, 1999.
11. J. Holm, K. de Lichtenberg, and M. Thorup, "Polylogarithmic deterministic fully dynamic algorithms for connectivity, minimum spanning tree, 2-edge, and biconnectivity," Journal of ACM, vol. 48, pp. 723-760, July 2001. https://doi.org/10.1145/502090.502095
12. C. Crespelle, "Fully dynamic representations of interval graphs," in Proceeding of the 35th International Workshop on Graph-Theoretic Concepts in Computer Science, pp. 77-87, 2009.

#### Acknowledgement

Supported by : Busan University of Foreign Studies