# Linear time algorithms for dominating pairs in asteroidal triple-free graphs

## 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 neighbourhood of the third. A graph is asteroidal triple-free (AT-free, for short) if it contains no asteroidal triple. The motivation for this investigation 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, based on the well-known Lexicographic Breadth-First Search, 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 *O*(¦V¦^{3}) for input graph *G=(V, E)*. In addition, we indicate how our algorithm can be extended to find, in time linear in the size of the input, all dominating pairs in a connected AT-free graph with diameter greater than three. A remarkable feature of the extended algorithm is that, even though there may be *O*(¦*V*¦^{2}) dominating pairs, the algorithm can compute and represent them in linear time.

## Preview

Unable to display preview. Download preview PDF.

## References

- 1.K. A. Baker, P. C. Fishburn, and F. S. Roberts, Partial orders of dimension two,
*Networks*, 2 (1971), 11–28.Google Scholar - 2.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, Berlin, 1993, 131–141.Google Scholar - 3.J. A. Bondy, U. S. R. Murty,
*Graph Theory with Applications*, North-Holland, Amsterdam, 1976.Google Scholar - 4.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 - 5.K. S. Booth and G. S. Lueker, A linear time algorithm for deciding interval graph isomorphism,
*Journal of the ACM*, 26 (1979), 183–195.Google Scholar - 6.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 - 7.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 - 8.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 - 9.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, submitted for publication.Google Scholar
- 10.D.G. Corneil, S. Olariu and L. Stewart, Linear time algorithms for dominating pairs in asteroidal triple-free graphs, Technical Report 294/95, Department of Computer Science, University of Toronto, January 1995, submitted for publication.Google Scholar
- 11.I. Dagan, M.C. Golumbic and R.Y. Pinter, Trapezoid graphs and their coloring,
*Discrete Applied Mathematics*, 21 (1988), 35–46.Google Scholar - 12.S. Even, A. Pnueli and A. Lempel, Permutation graphs and transitive graphs,
*Journal of the ACM*, 19 (1972), 400–410.Google Scholar - 13.M.C. Golumbic.
*Algorithmic Graph Theory and Perfect Graphs*. Academic Press, New York, 1980.Google Scholar - 14.M.C. Golumbic, C.L. Monma and W.T. Trotter Jr., Tolerance graphs,
*Discrete Applied Mathematics*, 9 (1984), 157–170.Google Scholar - 15.D. Kratsch and L. Stewart, Domination on cocomparability graphs,
*SIAM Journal on Discrete Mathematics*, 6 (1993), 400–417.Google Scholar - 16.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 - 17.D.J. Rose, R.E. Tarjan, G.S. Lueker, Algorithmic aspects of vertex elimination on graphs,
*SIAM Journal on Computing*, 5 (1976), 266–283.Google Scholar - 18.R.E. Tarjan and M. Yannakakis, Simple linear-time algorithms to test chordality of graphs, test acyclicity of hypergraphs and selectively reduce acyclic hypergraphs,
*SIAM Journal on Computing*, 13 (1984), 566–579.Google Scholar