. 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 triple-free (AT-free, for short) if it contains no asteroidal triples. The motivation for this investigation was provided, in part, by the fact that the asteroidal triple-free 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 AT-free graphs. Specifically, we show that every connected AT-free 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 AT-free graphs in terms of dominating pairs and minimal triangulations. Subsequently, we state and prove a decomposition theorem for AT-free graphs. An assortment of other properties of AT-free graphs is also p...

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 triple-free (AT-free, for short) if it contains no asteroidal triple. The motivation for this work is provided, in part, by the fact that AT-free 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 AT-free 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 AT-free graphs. The resulting simple algorithm, based on the well-known Lexicographic Breadth-First Search, can be implemented to run in time linear in the size of the input, whereas the best algorithm previousl...

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 AT-free if it does not contain an asteroidal triple. We present a simple linear-time algorithm to compute a dominating path in a connected AT-free graph. Keywords. asteroidal triple-free 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 triple-free graphs (AT-free graphs, for short). An independent triple fx; y; zg is called an asteroidal triple if between any p...

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 triple-free (AT-free, for short) if it contains no asteroidal triple. The motivation for this work is provided, in part, by the fact that AT-free 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 AT-free 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 AT-free 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...

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.

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 NP-complete [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].

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.

Let D = (V (D), A(D)) be a digraph. The competition graph of D, is the graph with vertex set V (D) and edge set {uv ∈ � � V (D)

We introduce and study the donation center location problem, which has an additional application in network testing and may also be of independent interest as a general graphtheoretic problem. Given a set of agents and a set of centers, where agents have preferences over centers and centers have capacities, the goal is to open a subset of centers and to assign a maximum-sized subset of agents to their most-preferred open centers, while respecting the capacity constraints. We prove that in general, the problem is hard to approximate within n 1/2−ɛ for any ɛ> 0. In view of this, we investigate two special cases. In one, every agent has a bounded number of centers on her preference list, and in the other, all preferences are induced by a line-metric. We present constant-factor approximation algorithms for the former and exact polynomial-time algorithms for the latter. Of particular interest among our techniques are an analysis of the greedy algorithm for a variant of the maximum coverage problem called frugal coverage, the use of maximum matching subroutine with subsequent modification, analyzed using a counting argument, and a reduction to the independent set problem on terminal intersection graphs, which we show to be a subclass of trapezoid graphs. 1

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 simple-triangle graphs – also known as PI graphs (for Point-Interval) – 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 simple-triangle recognition problems (namely, the problems of deciding whether a given graph is a triangle or a simple-triangle graph, respectively) have been the most fundamental open problems on these classes of graphs since their introduction two decades ago. Moreover, since triangle and simple-triangle 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 simple-triangle 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 NP-complete, even in the case where the input graph is known to be a trapezoid graph.