This is the nineteenth edition of a (usually) quarterly column that covers new developments in the theory of NP-completeness. The presentation is modeled on that used by M. R. Garey and myself in our book ‘‘Computers and Intractability: A Guide to the Theory of NP-Completeness,’ ’ W. H. Freeman & Co., New York, 1979 (hereinafter referred to as ‘‘[G&J]’’; previous columns will be referred to by their dates). A background equivalent to that provided by [G&J] is assumed, and, when appropriate, cross-references will be given to that book and the list of problems (NP-complete and harder) presented there. Readers who have results they would like mentioned (NP-hardness, PSPACE-hardness, polynomial-time-solvability, etc.) or open problems they would like publicized, should

. 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...

In this paper we discuss the problem of finding the treewidth and pathwidth of permutation graphs. If G[r] is a permutation graph with treewidth k, then we show that the pathwidth of G[r] is at most 2k, and we give an algo- rithm which constructs a path-decomposition with width at most 2k in time O(nk). We assume that the permutation r is given. For permutation graphs of which the treewidth is bounded by some constant, this result, together with previous results ([9], [15]), shows that the treewidth and pathwidth can be computed in linear time.

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...

It is well known that every 2-edge-connected graph can be oriented so that the resulting digraph is strongy connected. Here we study the problem of orienting a connected graph...

A graph is called a string graph if its vertices can be represented by continuous curves ("strings") in the plane so that two of them cross each other if and only if the corresponding vertices are adjacent. It is shown that there exists a recursive function f(n) with the property that every string graph of n vertices has a representation in which any two curves cross at most f(n) times. We obtain as a corollary...

Laszlo Lovasz recently posed the following problem: "Is there an NC algorithm for testing if a given graph has a unique perfect matching ?" We present such an algorithm for bipartite graphs. We also give NC algorithms for obtaining a transitive orientation of a comparability graph, and an interval representation of an interval graph. These enable us to obtain an NC algorithm for finding a maximum matching in an incomparability graph. 1 Introduction Karp, Upfal and Wigderson [9] have recently shown that the maximum matching problem is in Random NC 3 (RNC 3 ). This result has since been improved to RNC 2 by Mulmuley, Vazirani, and Vazirani [16]. It remains open whether there is a deterministic NC algorithm for this problem. A first step might be to obtain an NC algorithm for testing if a graph has a perfect matching. An RNC algorithm for this problem exists, based on a method of Lovasz [13] (see [1]). Rabin and Vazirani [18] give an NC algorithm for obtaining perfect matchings in...

The visualization of graphs has proven to be very useful for exploring structures in different application domains. However, in certain fields of computer science, graph visualization is understood and focused quite differently. While ”graph drawing ” focuses on optimized layouts for nodelink-representations of networks, ”information visualization” prefers to work on hierarchies focusing on very large structures, different views and interactivity. This paper gives a systematic view of the problem of graph visualization by combining both approaches. We will introduce a general view of different visualization methods as well as describe occurring problems and discuss basic constraints. These will be used to propose a visualization framework for graphs, whose development motivated this paper.

A chain of a set P of n points in the plane is a chain of the dominance order on P . A k-chain is a subset C of P that can be covered by k chains. A k-chain C is a maximum k-chain if no other k-chain contains more elements than C. This paper deals with the problem of finding a maximum k-chain of P in the cardinality and in the weighted case. Using the skeleton S(P ) of a point set P introduced by Viennot we describe a fairly simple algorithm that computes maximum k-chains in time O(kn log n) and linear space. The basic idea is that the canonical chain partition of a maximum (k \Gamma 1)-chain in the skeleton S(P ) provides k regions in the plane, such that a maximum k-chain for P can be obtained as the union of a maximal chain from each of these regions. By the symmetry between chains and antichains in the dominance order we may use the algorithm for maximumk-chains to compute maximumk-antichains for planar points in time O(kn log n). However, for large k one can do better....