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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...
Complexity classification of some edge modification problems
, 2001
"... In an edge modification problem one has to change the edge set of a given graph as little as possible so as to satisfy a certain property. We prove the NPhardness of a variety of edge modification problems with respect to some wellstudied classes of graphs. These include perfect, chordal, chain, c ..."
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Cited by 41 (2 self)
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In an edge modification problem one has to change the edge set of a given graph as little as possible so as to satisfy a certain property. We prove the NPhardness of a variety of edge modification problems with respect to some wellstudied classes of graphs. These include perfect, chordal, chain, comparability, split and asteroidal triple free. We show that some of these problems become polynomial when the input graph has bounded degree. We also give a general constant factor approximation algorithm for deletion and editing problems on bounded degree graphs with respect to properties that can be characterized by a finite set of forbidden induced subgraphs.
The Ultimate Interval Graph Recognition Algorithm? (Extended Abstract)
 Proceedings of the Ninth Annual ACMSIAM Symposium on Discrete Algorithms
, 1998
"... ) Derek G. Corneil Stephan Olariu y Lorna Stewart z Summary of Results An independent set of three vertices is called an asteroidal triple if between every two vertices in the triple there exists a path avoiding the neighbourhood of the third. A graph is asteroidal triplefree (ATfree, for ..."
Abstract

Cited by 36 (0 self)
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) Derek G. Corneil Stephan Olariu y Lorna Stewart z Summary of Results An independent set of three vertices is called an asteroidal triple if between every two vertices in the triple there exists a path avoiding the neighbourhood of the third. A graph is asteroidal triplefree (ATfree, for short) if it contains no asteroidal triple. A classic result states that a graph is an interval graph if and only if it is chordal and ATfree. Our main contribution is to exhibit a very simple, lineartime, recognition algorithm for interval graphs involving four sweeps of the wellknown Lexicographic Breadth First Search. Unlike the vast majority of existing algorithms, we do not use maximal cliques in our algorithm  we rely, instead, on a less wellknown characterization by a linear order of the vertices. 1 Introduction Interval graphs arise naturally in the process of modeling reallife situations, especially those involving time dependencies or other restrictions that are linear...