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.
CITATION STYLE
Corneil, D. G., Olariu, S., & Stewart, L. (1995). Linear time algorithms for dominating pairs in asteroidal triple-free graphs. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 944, pp. 292–302). Springer Verlag. https://doi.org/10.1007/3-540-60084-1_82
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