WADS 1995: Algorithms and Data Structures pp 358-368

# Computing a dominating pair in an asteroidal triple-free graph in linear time

• Derek G. Corneil
• Stephan Olariu
• Lorna Stewart
Invited Presentation
Part of the Lecture Notes in Computer Science book series (LNCS, volume 955)

## Abstract

An independent set of three of vertices is called an asteroidal triple if between each pair in the triple there exists a path that avoids the neighborhood of the third. A graph is asteroidal triple-free (AT-free, for short) if it contains no asteroidal triple. The motivation for this work is provided, in part, by the fact that AT-free graphs offer a common generalization of interval, permutation, trapezoid, and cocomparability graphs. Previously, the authors have given an existential proof of the fact that every connected AT-free graph contains a dominating pair, that is, a pair of vertices such that every path joining them is a dominating set in the graph. The main contribution of this paper is a constructive proof of the existence of dominating pairs in connected AT-free graphs. The resulting simple algorithm can be implemented to run in time linear in the size of the input, whereas the best algorithm previously known for this problem has complexity OV¦3) for input graph G=(V, E).

## References

1. 1.
A. V. Aho, J. E. Hopcroft and J. D. Ullman, Data Structures and Algorithms, Addison-Wesley, Reading, Massachusetts, 1983.Google Scholar
2. 2.
K. A. Baker, P. C. Fishburn and F. S. Roberts, Partial orders of dimension two, Networks, 2, (1971), 11–28.Google Scholar
3. 3.
H. Balakrishnan, A. Rajaraman and C. Pandu Rangan, Connected domination and Steiner set on asteroidal triple-free graphs, Proc. Workshop on Algorithms and Data Structures, WADS'93, Montreal, Canada, August 1993, LNCS, Vol. 709, F. Dehne, J.-R. Sack, N. Santoro, S. Whitesides (Eds.), Springer-Verlag, Heidelberg, Berlin, 1993, 131–141.Google Scholar
4. 4.
J. A. Bondy and U. S. R. Murty, Graph Theory with Applications, North-Holland, Amsterdam, 1976.Google Scholar
5. 5.
K. S. Booth and G. S. Lueker, Testing for the consecutive ones property, interval graphs and graph planarity using PQ-tree algorithms. Journal of Comput. Syst. Sci., 13 (1976), 335–379.Google Scholar
6. 6.
K. S. Booth and G. S. Lueker, A linear time algorithm for deciding interval graph isomorphism, Journal of the ACM, 26 (1979), 183–195.
7. 7.
F. Cheah, A recognition algorithm for II-graphs, Doctoral thesis, Department of Computer Science, University of Toronto, (available as TR 246/90), 1990.Google Scholar
8. 8.
D.G. Corneil and P.A. Kamula Extensions of permutation and interval graphs, Proceedings 18th Southeastern Conference on Combinatorics, Graph Theory and Computing (1987), 267–276.Google Scholar
9. 9.
D.G. Corneil, S. Olariu and L. Stewart, Asteroidal triple-free graphs, Proc. 19th International Workshop on Graph Theoretic Concepts in Computer Science, WG'93, Utrecht, The Netherlands, June 1993, LNCS, Vol. 790, J. van Leeuwen (Ed.), Springer-Verlag, Berlin, 1994, 211–224.Google Scholar
10. 10.
D.G. Corneil, S. Olariu and L. Stewart, A linear time algorithm to compute a dominating path in an AT-free graph, Information Processing Letters, to appear.Google Scholar
11. 11.
D.G. Corneil, S. Olariu and L. Stewart, Asteroidal triple-free graphs, Technical Report TR-94-31, Department of Computer Science, Old Dominion University, November, 1994.Google Scholar
12. 12.
D.G. Corneil, S. Olariu and L. Stewart, Linear time algorithms for dominating pairs in asteroidal triple-free graphs, submitted for publication, (available as TR 294/95, Department of Computer Science, University of Toronto), extended abstract to appear in Proceedings of ICALP Conference, July 1995.Google Scholar
13. 13.
I. Dagan, M.C. Golumbic and R.Y. Pinter, Trapezoid graphs and their coloring, Discrete Applied Mathematics 21 (1988), 35–46.
14. 14.
S. Even, A. Pnueli and A. Lempel, Permutation graphs and transitive graphs, Journal of the ACM 19 (1972), 400–410.
15. 15.
M.C. Golumbic. Algorithmic Graph Theory and Perfect Graphs. Academic Press, New York, 1980.Google Scholar
16. 16.
M.C. Golumbic, C.L. Monma and W.T. Trotter Jr., Tolerance graphs, Discrete Applied Mathematics 9 (1984), 157–170.
17. 17.
D. Kratsch and L. Stewart, Domination on cocomparability graphs, SIAM Journal on Discrete Mathematics, 6 (1993) 400–417.
18. 18.
C.G. Lekkerkerker and J.C. Boland, Representation of a finite graph by a set of intervals on the real line, Fundamenta Mathematicae 51 (1962), 45–64.Google Scholar

## Authors and Affiliations

• Derek G. Corneil
• 1
• Stephan Olariu
• 2
• Lorna Stewart
• 3
1. 1.Department of Computer ScienceUniversity of TorontoTorontoCanada
2. 2.Department of Computer ScienceOld Dominion UniversityNorfolkUSA
3. 3.Department of Computing ScienceUniversity of AlbertaEdmontonCanada