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22
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 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...
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 ..."
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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...
PARTITION REFINEMENT TECHNIQUES: AN INTERESTING ALGORITHMIC TOOL KIT
 INTERNATIONAL JOURNAL OF FOUNDATIONS OF COMPUTER SCIENCE
, 1999
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Collective tree spanners and routing in ATfree related graphs (Extended Abstract)
 IN GRAPHTHEORETIC CONCEPTS IN COMPUTER SCIENCE, LECTURE NOTES IN COMPUT. SCI. 3353
, 2004
"... In this paper we study collective additive tree spanners for families of graphs that either contain or are contained in ATfree graphs. We say that a graph G = (V, E) admits a system of µ collective additive tree rspanners if there is a system T (G) of at most µ spanning trees of G such that for any ..."
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Cited by 9 (8 self)
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In this paper we study collective additive tree spanners for families of graphs that either contain or are contained in ATfree graphs. We say that a graph G = (V, E) admits a system of µ collective additive tree rspanners if there is a system T (G) of at most µ spanning trees of G such that for any two vertices x, y of G a spanning tree T ∈ T (G) exists such that dT(x, y) ≤ dG(x, y) + r. Among other results, we show that ATfree graphs have a system of two collective additive tree 2spanners (whereas there are trapezoid graphs that do not admit any additive tree 2spanner). Furthermore, based on this collection, we derive a compact and efficient routing scheme. Also, any DSPgraph (there exists a dominating shortest path) admits an additive tree 4spanner, a system of two collective additive tree 3spanners and a system of five collective additive tree 2spanners.
On the Power of BFS to Determine a Graph's Diameter
 Networks
, 2003
"... this paper, we show that, in some cases, the full power of LBFS is not required and that other variations of Breadth First Search (BFS) suffice. The restricted graph classes that are amenable to this approach all have a small constant upper bound on the maximumsized cycle that may appear as an indu ..."
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Cited by 9 (0 self)
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this paper, we show that, in some cases, the full power of LBFS is not required and that other variations of Breadth First Search (BFS) suffice. The restricted graph classes that are amenable to this approach all have a small constant upper bound on the maximumsized cycle that may appear as an induced subgraph. We show that, on graphs that have no induced cycle of size greater than k, BFS finds an estimate of the diameter that is no worse than diam(G) # #k/2#. 2003 Wiley Periodicals, Inc
Good Maps are Straight
 Proc. 4th Int. Conf. on Intelligent Systems for Molecular Biology
, 1996
"... This paper proposes a simplified approach to the assembly of large physical genome maps. The approach focuses on two key problems: (i) the integration of diverse forms of data from numerous sources, and (ii) the detection and removal of errors and anomalies in the data. The approach simplifies map a ..."
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Cited by 5 (1 self)
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This paper proposes a simplified approach to the assembly of large physical genome maps. The approach focuses on two key problems: (i) the integration of diverse forms of data from numerous sources, and (ii) the detection and removal of errors and anomalies in the data. The approach simplifies map assembly by dividing it into three phasesoverlap, linkage and ordering. In the first phase, all forms of overlap data are integrated into a simple abstract structure, called clusters, where each cluster is a set of mutuallyoverlapping DNA segments. This phase filters out many questionable overlaps in the mapping data. In the second phase, clusters are linked together into a weighted intersection graph. False links between widely separated regions of the genome show up as crooked, branching structures in the graph. Removing these false links produces graphs that are straight, reflecting the linear structure of chromosomes. From these straight graphs, the third phase constructs a physical m...
Asteroidal Triples of Moplexes
 Discrete Applied Mathematics
, 2000
"... An asteroidal triple is an independent set of vertices such that each pair is joined by a path that avoids the neighborhood of the third, and a moplex is an extension to an arbitrary graph of a simplicial vertex in a triangulated graph. The main result of this paper is that the investigation of the ..."
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Cited by 4 (4 self)
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An asteroidal triple is an independent set of vertices such that each pair is joined by a path that avoids the neighborhood of the third, and a moplex is an extension to an arbitrary graph of a simplicial vertex in a triangulated graph. The main result of this paper is that the investigation of the set of moplexes of a graph is sufficient to conclude as to its having an asteroidal triple. Specifically, we show that a graph has an asteroidal triple of vertices if and only if it has an asteroidal triple of moplexes. We also examine the behavior of an asteroidal triple of moplexes in the course of a minimal triangulation process, and give some related properties.
Linear Time Algorithms for Hamiltonian Problems on (claw,net)Free Graphs
 SIAM J. COMPUTING
, 2000
"... We prove that clawfree graphs, containing an induced dominating path, have a Hamiltonian path, and that 2connected clawfree graphs, containing an induced doubly dominating cycle or a pair of vertices such that there exist two internally disjoint induced dominating paths connecting them, have a Ha ..."
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Cited by 4 (2 self)
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We prove that clawfree graphs, containing an induced dominating path, have a Hamiltonian path, and that 2connected clawfree graphs, containing an induced doubly dominating cycle or a pair of vertices such that there exist two internally disjoint induced dominating paths connecting them, have a Hamiltonian cycle. As a consequence, we obtain linear time algorithms for both problems if the input is restricted to (claw,net)free graphs. These graphs enjoy those interesting structural properties.
On Linear and Circular Structure of (claw, net)Free Graphs
, 2003
"... We prove that every (claw, net)free graph contains an induced doubly dominating cycle or a dominating pair. Moreover, using LexBFS we present alS[SE timealen##ES which, for a given (claw, net)free graph, finds either a dominating pair or an induceddoubl dominatingcycln We show aln how one can uses ..."
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Cited by 4 (3 self)
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We prove that every (claw, net)free graph contains an induced doubly dominating cycle or a dominating pair. Moreover, using LexBFS we present alS[SE timealen##ES which, for a given (claw, net)free graph, finds either a dominating pair or an induceddoubl dominatingcycln We show aln how one can usestructural properties of (claw, net)free graphs tosolI efficiently the domination, independent domination, and independent set problems on these graphs.