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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 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...
On the Interplay between Interval Dimension and Dimension
, 1991
"... This paper investigates a transformation P ! Q between partial orders P; Q that transforms the interval dimension of P to the dimension of Q, i.e., idim(P ) = dim(Q). Such a construction has been shown before in the context of Ferrer's dimension by Cogis [2]. Our construction can be shown to be equ ..."
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Cited by 3 (2 self)
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This paper investigates a transformation P ! Q between partial orders P; Q that transforms the interval dimension of P to the dimension of Q, i.e., idim(P ) = dim(Q). Such a construction has been shown before in the context of Ferrer's dimension by Cogis [2]. Our construction can be shown to be equivalent to his, but it has the advantage of (1) being purely ordertheoretic, (2) providing a geometric interpretation of interval dimension similar to that of Ore [15] for dimension, and (3) revealing several somewhat surprising connections to other ordertheoretic results. For instance, the transformation P ! Q can be seen as almost an inverse of the wellknown split operation, it provides a theoretical background for the influence of edge subdivision on dimension (e.g., the results of Spinrad [17]) and interval dimension, and it turns out to be invariant with respect to changes of P that do not alter its comparability graph, thus providing also a simple new proof for the comparability in...
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].
Vertex Splitting and the Recognition of Trapezoid Graphs
"... 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 co ..."
Abstract
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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].