Lorna Stewart

A LINEAR RECOGNITION ALGORITHM FOR COGRAPHS

Cographs are the graphs formed from a single vertex under the closure of the operations of union and complement. Another characterization of cographs is that they are the undirected graphs with no induced paths on four vertices. Cographs arise naturally in such application areas as examination scheduling and automatic clustering of index terms. Furthermore, it is known that cographs have a unique tree representation called a cotree. Using the cotree it is possible to design very fast polynomial time algorithms for problems which are intractable for graphs in general. Such problems include chromatic number, clique determination, clustering, minimum weight domination, isomorphism, minimum fill-in and Hamiltonicity. In this paper we present a linear time algorithm for recognizing cographs and constructing their cotree representation.

Bipartite permutation graphs

This paper examines the class of bipartite permutation graphs. Two characterizations of graphs in this class are presented. These characterizations lead to a linear time recognition algorithm, and to polynomial time algorithms for a number of NP-complete problems when restricted to graphs in this class.

Asteroidal Triple-Free Graphs

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) if it contains no asteroidal triples. The motivation for this investigation was provided, in part, by the fact that the AT-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 provided. These properties generalize known structural properties of interval, permutation, trapezoid, and cocomparability graphs.

Domination on cocomparability graphs

The authors determine the algorithmic complexity of domination and variants on cocomparability graphs, a class of perfect graphs containing both the interval and the permutation graphs. Minimum dominating, total dominating, connected dominating, and independent dominating sets can be constructed in polynomial time. On the other hand, DOMINATING CLIQUE and MINIMUM DOMINATING CLIQUE remain NP-complete on cocomparability graphs.

The LBFS Structure and Recognition of Interval Graphs

A graph is an interval graph if it is the intersection graph of intervals on a line. Interval graphs are known to be the intersection of chordal graphs and asteroidal triple-free graphs, two families where the well-known lexicographic breadth first search (LBFS) plays an important algorithmic and structural role. In this paper we show that interval graphs have a very rich LBFS structure and that by exploiting this structure one can design a linear time, easily implementable, interval graph recognition algorithm.

Complexity results for well-covered graphs

A graph with n vertices is well covered if every maximal independent set is a maximum independent set and very well covered if every maximal independent set has size n/2. In this work, we study these graphs from an algorithmic complexity point of view. We show that well-covered graph recognition is co-NP-complete and that several other problems are NP-complete for well-covered graphs. A number of these problems remain NP-complete on very well covered graphs, while some admit polynomial time solutions for the smaller class. For both families, the isomorphism problem is as hard as general graph isomorphism.

Permutation graphs: connected domination and Steiner trees

Efficient algorithms are developed for finding a minimum cardinality connected dominating set and a minimum cardinality Steiner tree in permutation graphs. This contrasts with the known NP-completeness of both problems on comparability graphs in general.

Dominating sets in perfect graphs

In this paper, we review the complexity of the minimum cardinality dominating set problem and some of its variations on several families of perfect graphs. We describe the techniques which are used to attain these complexity results, with emphasis on the dynamic programming approach to the design of algorithms.

Total domination and transformation

Abstract Using a linear time many-one reduction from the problem total dominating set to the problem dominating set we show how to obtain efficient algorithms to compute a minimum cardinality total dominating set on a variety of graph classes, among them permutation graphs, dually chordal graphs and k -polygon graphs.

A linear time algorithm to compute a dominating path in an AT-free graph

Abstract An independent set { x , y , z } 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.