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

Extended abstract
  • Derek G. Corneil
  • Stephan Olariu
  • Lorna Stewart
Algorithms III
Part of the Lecture Notes in Computer Science book series (LNCS, volume 944)


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 OV¦2) dominating pairs, the algorithm can compute and represent them in linear time.


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Copyright information

© Springer-Verlag Berlin Heidelberg 1995

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

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