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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 ..."
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Cited by 54 (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...
Linear Time Algorithms for Dominating Pairs in Asteroidal Triplefree Graphs
 SIAM J. Comput
, 1997
"... 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 triplefree (ATfree, for short) if it contains no asteroidal triple. The motivation for this work is pro ..."
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Cited by 24 (7 self)
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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 triplefree (ATfree, for short) if it contains no asteroidal triple. The motivation for this work is provided, in part, by the fact that ATfree 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 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. The main contribution of this paper is a constructive proof of the existence of dominating pairs in connected ATfree graphs. The resulting simple algorithm, based on the wellknown Lexicographic BreadthFirst Search, can be implemented to run in time linear in the size of the input, whereas the best algorithm previousl...
A Linear Time Algorithm to Compute a Dominating Path in an ATfree Graph
 Inform. Process. Lett
, 1998
"... An independent set fx; y; zg is called an asteroidal triple if between any pair in the triple there exists a path that avoids the neighborhood of the third. A graph is referred to as ATfree if it does not contain an asteroidal triple. We present a simple lineartime algorithm to compute a domina ..."
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Cited by 12 (3 self)
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An independent set fx; y; zg is called an asteroidal triple if between any pair in the triple there exists a path that avoids the neighborhood of the third. A graph is referred to as ATfree if it does not contain an asteroidal triple. We present a simple lineartime algorithm to compute a dominating path in a connected ATfree graph. Keywords. asteroidal triplefree graphs, domination, algorithms 1 Introduction A number of families of graphs including interval graphs [10], permutation graphs [6], trapezoid graphs [3, 5], and cocomparability graphs [8] feature a type of linear ordering of their vertex sets. It is precisely this linear ordering that is exploited in a search for efficient algorithms on these classes of graphs [2, 5, 7, 8, 9, 11, 12]. As it turns out, the classes mentioned above are all subfamilies of a class of graphs called the asteroidal triplefree graphs (ATfree graphs, for short). An independent triple fx; y; zg is called an asteroidal triple if between any p...
On the L(h, k)Labeling of CoComparability Graphs ⋆
"... Abstract. Given two non negative integers h and k, an L(h, k)labeling of a graph G = (V, E) is a map from V to a set of labels such that adjacent vertices receive labels at least h apart, while vertices at distance at most 2 receive labels at least k apart. The goal of the L(h, k)labeling problem ..."
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Abstract. Given two non negative integers h and k, an L(h, k)labeling of a graph G = (V, E) is a map from V to a set of labels such that adjacent vertices receive labels at least h apart, while vertices at distance at most 2 receive labels at least k apart. The goal of the L(h, k)labeling problem is to produce a legal labeling that minimizes the largest label used. Since the decision version of the L(h, k)labeling problem is NPcomplete, it is important to investigate classes of graphs for which the problem can be solved efficiently. Along this line of though, in this paper we deal with cocomparability graphs and two of its subclasses: interval graphs and unitinterval graphs. Specifically, we provide, in a constructive way, the first upper bounds on the L(h, k)number of cocomparability graphs and interval graphs. To the best of our knowledge, ours is the first reported result concerning the L(h, k)labeling of cocomparability graphs. In the special case where k = 1, our result improves on the best previouslyknown approximation ratio for interval graphs. Keywords: L(h, k)Labeling, cocomparability graphs, interval graphs, unitinterval graphs. 1