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Asteroidal TripleFree Graphs
, 1997
"... . An independent set of three vertices such that each pair is joined by a path that avoids the neighborhood of the third is called an asteroidal triple. A graph is asteroidal triplefree (ATfree, for short) if it contains no asteroidal triples. The motivation for this investigation was provided, in ..."
Abstract

Cited by 55 (10 self)
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. An independent set of three vertices such that each pair is joined by a path that avoids the neighborhood of the third is called an asteroidal triple. A graph is asteroidal triplefree (ATfree, for short) if it contains no asteroidal triples. The motivation for this investigation was provided, in part, by the fact that the asteroidal triplefree graphs provide a common generalization of interval, permutation, trapezoid, and cocomparability graphs. The main contribution of this work is to investigate and reveal fundamental structural properties of ATfree graphs. Specifically, we show that every connected ATfree graph contains a dominating pair, that is, a pair of vertices such that every path joining them is a dominating set in the graph. We then provide characterizations of ATfree graphs in terms of dominating pairs and minimal triangulations. Subsequently, we state and prove a decomposition theorem for ATfree graphs. An assortment of other properties of ATfree graphs is also p...
Independent Sets In Asteroidal TripleFree Graphs
, 1999
"... An asteroidal triple (AT) is a set of three vertices such that there is a path between any pair of them avoiding the closed neighborhood of the third. A graph is called ATfree if it does not have an AT. We show that there is an O(n 4 ) ..."
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Cited by 11 (2 self)
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An asteroidal triple (AT) is a set of three vertices such that there is a path between any pair of them avoiding the closed neighborhood of the third. A graph is called ATfree if it does not have an AT. We show that there is an<F3.502e+05><F3.817e+05><F3.502e+05> O(n<F2.756e+05> 4<F3.817e+05> ) time algorithm to compute the maximum weight of an independent set for ATfree graphs. Furthermore, we obtain<F3.502e+05><F3.817e+05><F3.502e+05> O(n<F2.756e+05> 4<F3.817e+05> ) time algorithms to solve the<F3.728e+05> independent dominating set<F3.817e+05> and the<F3.728e+05> independent perfect dominating set<F3.817e+05> problems on ATfree graphs. We also show how to adapt these algorithms such that they solve the corresponding problem for graphs with bounded asteroidal number in polynomial time. Finally, we observe that the problems clique and partition into cliques remain NPcomplete when restricted to ATfree graphs.