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21
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...
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 26 (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 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.
Vertex Splitting and the Recognition of Trapezoid Graphs
, 2009
"... Trapezoid graphs are the intersection family of trapezoids where every trapezoid has a pair of opposite sides lying on two parallel lines. These graphs have received considerable attention and lie strictly between permutation graphs (where the trapezoids are lines) and cocomparability graphs (the c ..."
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Cited by 3 (2 self)
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Trapezoid graphs are the intersection family of trapezoids where every trapezoid has a pair of opposite sides lying on two parallel lines. These graphs have received considerable attention and lie strictly between permutation graphs (where the trapezoids are lines) and cocomparability graphs (the complement has a transitive orientation). The operation of “vertex splitting”, introduced in [3], first augments a given graph G and then transforms the augmented graph by replacing each of the original graph’s vertices by a pair of new vertices. This “splitted graph” is a permutation graph with special properties if and only if G is a trapezoid graph. Recently vertex splitting has been used to show that the recognition problems for both tolerance and bounded tolerance graphs is NPcomplete [11]. Unfortunately, the vertex splitting trapezoid graph recognition algorithm presented in [3] is not correct. In this paper, we present a new way of augmenting the given graph and using vertex splitting such that the resulting algorithm is simpler and faster than the one reported in [3].
Computing a Dominating Pair in an Asteroidal Triplefree Graph in Linear Time
 in Algorithms and Data Structures WADS '95, Lecture
, 1998
"... 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 neighborhood of the third. A graph is asteroidal triplefree (ATfree, for short) if it contains no asteroidal triple. The motivation for this work is prov ..."
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Cited by 3 (2 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 neighborhood 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 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(jV j 3 ) for input...
Triangle graphs and simple trapezoid graphs
 Journal of Information Science andEngineering
, 2002
"... In this paper, we present results on two subclasses of trapezoid graphs, including simple trapezoid graphs and triangle graphs (also known as PI graph in [3]). Simple trapezoid graphs and triangle graphs are proper subclasses of trapezoid graphs [3, 5]. Here we show that simple trapezoid graphs and ..."
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Cited by 3 (0 self)
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In this paper, we present results on two subclasses of trapezoid graphs, including simple trapezoid graphs and triangle graphs (also known as PI graph in [3]). Simple trapezoid graphs and triangle graphs are proper subclasses of trapezoid graphs [3, 5]. Here we show that simple trapezoid graphs and triangle graphs are also two distinct subclasses of trapezoid graphs.
The Recognition of Triangle Graphs
"... Trapezoid graphs are the intersection graphs of trapezoids, where every trapezoid has a pair of opposite sides lying on two parallel lines L1 and L2 of the plane. Strictly between permutation and trapezoid graphs lie the simpletriangle graphs – also known as PI graphs (for PointInterval) – where t ..."
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Cited by 2 (2 self)
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Trapezoid graphs are the intersection graphs of trapezoids, where every trapezoid has a pair of opposite sides lying on two parallel lines L1 and L2 of the plane. Strictly between permutation and trapezoid graphs lie the simpletriangle graphs – also known as PI graphs (for PointInterval) – where the objects are triangles with one point of the triangle on L1 and the other two points (i.e. interval) of the triangle on L2, and the triangle graphs – also known as PI ∗ graphs – where again the objects are triangles, but now there is no restriction on which line contains one point of the triangle and which line contains the other two. The complexity status of both triangle and simpletriangle recognition problems (namely, the problems of deciding whether a given graph is a triangle or a simpletriangle graph, respectively) have been the most fundamental open problems on these classes of graphs since their introduction two decades ago. Moreover, since triangle and simpletriangle graphs lie naturally between permutation and trapezoid graphs, and since they share a very similar structure with them, it was expected that the recognition of triangle and simpletriangle graphs is polynomial, as it is also the case for permutation and trapezoid graphs. In this article we surprisingly prove that the recognition of triangle graphs is NPcomplete, even in the case where the input graph is known to be a trapezoid graph.
The Recognition of SimpleTriangle Graphs and of LinearInterval Orders is Polynomial
"... Intersection graphs of geometric objects have been extensively studied, both due to their interesting structure and their numerous applications; prominent examples include interval graphs and permutation graphs. In this paper we study a natural graph class that generalizes both interval and permutat ..."
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Intersection graphs of geometric objects have been extensively studied, both due to their interesting structure and their numerous applications; prominent examples include interval graphs and permutation graphs. In this paper we study a natural graph class that generalizes both interval and permutation graphs, namely simpletriangle graphs. Simpletriangle graphs – also known as PI graphs (for PointInterval) – are the intersection graphs of triangles that are defined by a point on a line L1 and an interval on a parallel line L2. They lie naturally between permutation and trapezoid graphs, which are the intersection graphs of line segments between L1 and L2 and of trapezoids between L1 and L2, respectively. Although various efficient recognition algorithms for permutation and trapezoid graphs are well known to exist, the recognition of simpletriangle graphs has remained an open problem since their introduction by Corneil and Kamula three decades ago. In this paper we resolve this problem by proving that simpletriangle graphs can be recognized in polynomial time. As a consequence, our algorithm also solves a longstanding open problem in the area of partial orders, namely the recognition of linearinterval orders, i.e. of partial orders P = P1 ∩ P2, where P1 is a linear order and P2 is an interval order. This is one of the first results on recognizing partial orders P that are the intersection of orders from two different classes P1 and P2. In contrast, partial orders P which are the intersection of orders from the same class P have been extensively investigated, and in most cases the complexity status of these recognition problems has been established.
c ○ Uma Publicação da Sociedade Brasileira de Matemática Aplicada e Computacional. On the Representation of a PIGraph 1
"... Abstract. Consider two parallel lines (denoted r1 and r2). A graph is a PI graph (PointInterval graph) if it is an intersection graph of a family F of triangles between r1 and r2 such that each triangle has an interval with two endpoints on r1 and a vertex (a point) on r2. The family F is the PI re ..."
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Abstract. Consider two parallel lines (denoted r1 and r2). A graph is a PI graph (PointInterval graph) if it is an intersection graph of a family F of triangles between r1 and r2 such that each triangle has an interval with two endpoints on r1 and a vertex (a point) on r2. The family F is the PI representation of G. The PI graphs are an extension of interval and permutation graphs and they form a subclass of trapezoid graphs. In this paper, we characterize the PI graphs in terms of its trapezoid representation. Also we show how to construct a family of trapezoid graphs but not PI graphs from a trapezoid representation of a known graph in this class. 1.