Showing papers on "Pathwidth published in 1992"

Graphs: Theory and Algorithms

Abstract: Basic Concepts. Trees, Cutsets, and Circuits. Eulerian and Hamiltonian Graphs. Graphs and Vector Spaces. Directed Graphs. Matrices of a Graph. Planarity and Duality. Connectivity and Matching. Covering and Coloring. Matroids. Graph Algorithms. Flows in Networks. Indexes.

Laying out graphs using queues

Abstract: The problem of laying out the edges of a graph using queues is studied. In a k-queue layout, vertices of the graph are placed in some linear order and each edge is assigned to exactly one of the k queues so that the edges assigned to each queue obey a first-in/first-out discipline. This layout problem abstracts a design problem of fault-tolerant processor arrays, a problem of sorting with parallel queues, and a problem of scheduling parallel processors. A number of basic results about queue layouts of graphs are established, and these results are contrasted with their analogues for stack layouts of graphs (the book-embedding problem). The 1-queue graphs (they are almost leveled-planar graphs) are characterized. It is proved that the problem of recognizing 1-queue graphs is NP-complete. Queue layouts for some specific classes of graphs are given. Relationships between the queuenumber of a graph and its bandwidth and separator size are presented. An apparent tradeoff between the queuewidth and the number of...

Supereulerian graphs: a survey

Abstract: A graph is supereulerian if it has a spanning eulerian subgraph. There is a rduction method to determine whether a graph is supereulerian, and it can also be applied to study other concepts, e.g., hamiltonian line graphs, a certain type of double cycle cover, and the total interval number of a graph. We outline the research on supereulerian graphs, the reduction method, and its applications.

Rotator graphs: an efficient topology for point-to-point multiprocessor networks

Abstract: Rotator graphs, a set of directed permutation graphs, are proposed as an alternative to star and pancake graphs. Rotator graphs are defined in a way similar to the recently proposed Faber-Moore graphs. They have smaller diameter, n-1 in a graph with n factorial vertices, than either the star or pancake graphs or the k-ary n-cubes. A simple optimal routing algorithm is presented for rotator graphs. The n-rotator graphs are defined as a subset of all rotator graphs. The distribution of distances of vertices in the n-rotator graphs is presented, and the average distance between vertices is found. The n-rotator graphs are shown to be optimally fault tolerant and maximally one-step fault diagnosable. The n-rotator graphs are shown to be Hamiltonian, and an algorithm for finding a Hamiltonian circuit in the graphs is given. >

Complexity results for well‐covered graphs

Abstract: 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.

A tourist guide through Treewidth

Abstract: A short overview is given of many recent results in algorithmic graph theory that deal with the notions treewidth, and pathwidth. We discuss algorithms that find tree-decompositions, algorithms that use tree-decompositions to solve hard problems efficiently, graph minor theory, and some applications. The paper contains an extensive bibliography.

Recognizing P 4 -sparse graphs in linear time

Abstract: A graph G is $P_4 $-sparse if no set of five vertices in G induces more than one chordless path of length three $P_4 $-sparse graphs generalize both the class of cographs and the class of $P_4 $-reducible graphs One remarkable feature of $P_4 $-sparse graphs is that they admit a tree representation unique up to isomorphism It has been shown that this tree representation can be obtained in polynomial time This paper gives a linear time algorithm to recognize $P_4 $-sparse graphs and shows how the data structures returned by the recognition algorithm can be used to construct the corresponding tree representation in linear time

Points, spheres, and separators: a unified geometric approach to graph partitioning

Abstract: Geometry is full of visual imagination and concrete intuition. Such imagination and intuition is of great value not only for the research worker, but also for anyone who wishes to study and appreciate the results in geometry. In this thesis, a unified geometric approach is presented for graph partitioning--a fundamental problem in computer science that has important applications in numerical analysis, VLSI design, computational geometry, complexity theory, and other fields. The main ingredient in obtaining this unified approach is a novel geometrical characterization of graphs that have small separators, where a separator of a graph is a relatively small subset of vertices whose removal divides the rest of the graph into two disconnected pieces of approximately equal size. The characterization is based on elementary geometric concepts such as points, balls, cubes, and spheres. More specifically, a new class of geometric graphs, overlap graphs, is proposed. This class has the following properties: (1) In two dimensions, planar graphs are special cases of overlap graphs. (2) In d dimensions (d $\ge$ 2), any finite subgraph of the infinite d-dimensional grid is an overlap graph. (3) Every overlap graph of n vertices in d dimensions has an $O(n\sp{(d-1)/d})$ separator. At the time of this writing, this is the first time that a class of graphs has been proposed with these three natural properties. The proof that planar graphs are special cases of overlap graphs relies on recent deep theorems by Andreev and Thurston characterizing all planar graphs in a geometric fashion. A consequence is a new geometric proof of a classical theorem of Lipton and Tarjan that every planar graph has an $O(\sqrt{n})$-separator. This is another beautiful illustration of the use of geometry in understanding combinatorial concepts.

A framework for dynamic graph drawing

Abstract: In this paper we give a model for dynamic graph algorithms, based on performing queries and updates on an implicit representation of the drawing. We present dynamic algorithms for drawing planar graphs that use a variety of drawing standards (such as polyline, straight-line, orthogonal, grid, upward, and visibility drawings), and address aesthetic criteria that are important for readability, such as the display of planarity, symmetry, and reachability. Also, we provide techniques that are especially tailored for important subclasses of planar graphs such as trees and series-parallel digraphs. Our dynamic drawing algorithms have the important property of performing “smooth updates” of the drawing. Of special geometric interest is the possibility of performing point-location and window queries on the implicit representation of the drawing.

Geometric symmetry in graphs

Abstract: This thesis investigates the general problem of constructing meaningful drawings of abstract graphs, in particular the application of axial and rotational symmetry, collectively known as geometric symmetry, to achieving this goal The problem of algorithmically constructing these visually-informative drawings presents two distinct challenges: firstly, a set of explicit and objective drawing criteria must be identified to direct their construction, and secondly, efficient computational techniques must then be developed to actually implement these criteria Accordingly, a comprehensive list of criteria is introduced, and conflicts between various criteria are revealed An extensive survey of existing work is then presented, making use of this list as a means of categorizing and comparing these known results The construction of symmetric drawings is identified as one of the foremost criteria, since such drawings enable an understanding of the entire graph to be built up from that of a smaller subgraph, replicated a number of times Formal definitions of drawings and geometric symmetry of graphs are presented, and some of their more important features are discussed The fundamental problems of determining if a graph has any geometric symmetry, along with several variations, are all shown to be NP-complete in the case of general graphs Consequently, attention focuses on symmetry in planar graphs In the case of trees, outerplanar graphs, and embedded graphs, algorithms are successfully developed for detecting geometric symmetry and constructing related drawings, exhibiting the maximum number of simultaneously-displayable symmetries All these algorithms run in time which is linear in the size of the graph, and hence are optimal The thesis concludes by examining several open problems and potential directions for future research in graph drawing In particular, the problem of drawing graphs which have no geometric symmetry is discussed, and modifications to the algorithms developed here are put forward to handle such cases

Finding Hamiltonian paths in cocomparability graphs using the bump number algorithm

Abstract: Hamiltonian Path/Cycle are well known NP-complete problems on general graphs, but their complexity status for permutation graphs has been an open question in algorithmic graph theory for many years. In this paper, we prove that theHamiltonian Path problem is solvable in polynomial time even for the larger class of cocomparability graphs. Our result is based on a nice relationship between Hamiltonian paths and the bump number of partial orders. As another consequence we get a new interpretation of the bump number in terms of path partitions, leading to polynomial time solutions of theHamiltonian Path/Cycle Completion problems in cocomparability graphs.

Conceptual graph matching: a flexible algorithm and experiments

Abstract: Graph matching is recognized as a central problem across a variety of application areas, and application-specific matchers have been developed with different simplifying assumptions to reduce the computational complexity. Graph matching is viewed as a form of plausible reasoning when conceptual information contained in graphs are considered, and thus requires an underlying algorithm flexible and general enough to accommodate application-specific matching heuristics and schemes that determine the degree of plausibility. This paper presents such an algorithm, based on the notion of association graphs, developed for matching Sowa's conceptual graphs (CG). While the general subgraph isomorphism problem is known to be NP-complete, matching graphs containing conceptual information appears to be computationally tractable. Following the detailed description of the algorithm, some experimental results are shown to discuss the time complexity of the algorithm and its practicality.

A Simple Test for Interval Graphs

Abstract: Interval graphs are a special class of chordal graphs. Several linear time algorithms have been designed to recognize interval graphs. All of these algorithms rely on the following fact: a graph is an interval graph iff there exists a linear order of its maximal cliques such that for each vertex v, all maximal cliques containing v are consecutive. We give a much simpler recogntion algorithm in this paper which directly place the intervals without even using maximal cliques. The key is to find a good ordering of intervals to be placed. An on-line version of the algorithm that takes O(mα(n)) time will be discussed in a separate paper.

An optimal algorithm to solve the all‐pair shortest path problem on interval graphs

Abstract: We present an O(n2) time-optimal algorithm for solving the unweighted all-pair shortest path problem on interval graphs, an important subclass of perfect graphs. An interesting structure called the neighborhood tree is studied and used in the algorithm. This tree is formed by identifying the successive neighborhoods of the vertex labeled last in the graph according to the IG-ordering.

Faster Algorithms for Finding Small Edge Cuts in Planar Graphs (Extended Abstract)

Abstract: In this paper, we consider partitioning a planar graph by removing either nodes or edges. In particular, we consider a cut to be either a set of nodes or edges whose removal divides the graph into two pieces. We define the balance of a cut as the ratio of the weight of the smaller side of the cut to the total weight in the graph. Thus, the best possible balance is 1/2. We define the quotient cost of a cut as the ratio of the cost of the cut to the size of the smaller side of the cut. Thus, a costly cut with a high balance may have lower quotient cost than a less costly cut with a lower balance. Finally, we define the flux of a graph as the minimum quotient cost of any cut in the graph. We are interested in finding small cuts with high balance; either finding the cut that minimizes the quotient cost of the cut, or finding the smallest cut of some specified balance. (For balance 1/3, this problem has been called the minimum separator problem.) We present a very simple algorithm for finding 1.5 times optimal quotient node or edge cuts in planar graphs in time O(n2 min (W, C)), where W is the total weight of the graph, and C is the total cost of the graph. In fact, these cuts are only nonoptimal when they have very good balance. Thus, the algorithms either find an optimal quotient cut or a nearly optimal quotient cut with very high balance. We can greatly improve the running time if we settle for finding 3.5 times optimal quotient node and edge cuts. We can accomplish this in O(n2(log W + log C)) time. And, if we are willing to settle for O(k) times optimal solutions, we can make the running time quite fast; O(n 1+1/k log n(log W + log C)log C). If k=log m, and W and C are polynomial in n, this becomes O(n log3n). Finally, we give approximation algorithms for finding nearly optimal balanced node or edge separators in planar graphs, based on the quotient cut algorithms. These algorithms improve upon previous algorithms by being much faster, much much simpler, and by applying to node separation as well as edge separation.

Tournament-like oriented graphs

Abstract: A local tournament is an oriented graph in which the inset as well as the outset of each vertex induces a tournament. Local tournaments possess many properties of tournaments and have interesting structure. In 1982, Skrien proved (in different terminology), using a deep structural characterization of proper circular arc graphs by Tucker, that a connected graph is local-tournament-orientable if and only if it is a proper circular arc graph. In Chapter 2, we shall give a simple O($m\Delta$) algorithm to decide if a graph can be oriented as a local tournament, and hence whether or not it is a proper circular arc graph. We analyze relationships among local tournaments, local transitive tournaments, and proper circular arc graphs. We obtain theorems to describe all possible local-tournament orientations of a proper circular arc graph. In Chapter 3, we shall present an O($m\Delta$) algorithm to recognize comparability graphs and to calculate transitive orientations. Our method can be applied to recognize proper circular arc graphs and to find local-transitive-tournament orientations, and can also be applied to recognize proper interval graphs and to find acyclic local-tournament orientations. We shall give a simple proof of Skrien's theorem, which does not depend on Tucker's result. In Chapter 4, we shall present two O(m + n) time algorithms. One is for recognizing proper interval graphs and for finding an associated interval family. The other is for recognizing proper circular arc graphs and for finding an associated circular arc family. In Chapter 5, we shall obtain two additional O(m + n) time algorithms for proper circular arc graphs by using the auxiliary local-tournament orientations. One is for finding maximum cliques, and the other is for determining c-colourability. In Chapter 6, we shall introduce a new class of oriented graphs, namely, in-tournaments, which contains the class of local tournaments. We shall show that some of the basic and very nice properties of tournaments extend not only to local tournaments, but also to this more general class of digraphs. Our results imply a polynomial time algorithm for finding hamiltonian paths and cycles in the class of in-tournaments. We shall also investigate the class of graphs which are orientable as in-tournaments. Finally, in Chapter 7, we shall introduce another class of oriented graphs, i.e., those of Moon type. We shall find a close relationship between the class of oriented graphs of Moon type and the class of local tournaments. In fact, oriented graphs of Moon type can be characterized in terms of local transitive tournaments.

The pagenumber of genus g graphs is O(g)

Abstract: In 1979, Bernhart and Kainen conjectured that graphs of fixed genus g ≥ 1 have unbounded pagenumber. In this paper, it is proven that genus g graphs can be embedded in O(g) pages, thus disproving the conjecture. An O(g1/2) lower bound is also derived. The first algorithm in the literature for embedding an arbitrary graph in a book with a non-trivial upper bound on the number of pages is presented. First, the algorithm computes the genus g of a graph using the algorithm of Filotti, Miller, Reif (1979), which is polynomial-time for fixed genus. Second, it applies an optimal-time algorithm for obtaining an O(g)-page book embedding. Separate book embedding algorithms are given for the cases of graphs embedded in orientable and nonorientable surfaces. An important aspect of the construction is a new decomposition theorem, of independent interest, for a graph embedded on a surface. Book embedding has application in several areas, two of which are directly related to the results obtained: fault-tolerant VLSI and complexity theory.

The Power and the Limitations of Local Computations on Graphs

Abstract: This paper is a contribution to understanding the power and the limitations of asynchronous local computations on graphs and networks. We use local computations to define a notion of graph recognition, in particular our model enables a simulation of finite automata on words and on trees. We introduce the notion of k-covering to examine limitations of such systems. For example we prove that we cannot recognize the families of series parallel graphs and planar graphs by means of local computations.

A combinatorial approach to complexity

Abstract: We present a problem of construction of certain intersection graphs. If these graphs were explicitly constructed, we would have an explicit construction of Boolean functions of large complexity.