The pagenumber of k-trees is O(k)

AbstractA k-tree is a graph defined inductively in the following way: the complete graph Kk is a k-tree, and if G is a k-tree, then the graph resulting from adding a new vertex adjacent to k vertices inducing a Kk in G is also a k-tree. This paper examines the book-embedding problem for k-trees. A book embedding of a graph maps the vertices onto a line along the spine of the book and assigns the edges to pages of the book such that no two edges on the same page cross. The pagenumber of a graph is the minimum number of pages in a valid book embedding. In this paper, it is proven that the pagenumber of a k-tree is at most k+1. Furthermore, it is shown that there exist k-trees that require k pages. The upper bound leads to bounds on the pagenumber of a variety of classes of graphs for which no bounds were previously known

Edge Coloring Planar Graphs with Two Outerplanar Subgraphs

The standard problem of edge coloring a graph with k colors is equivalent to partitioning the edge set of the graph into k matchings. Here edge coloring is generalized by replacing matchings with outerplanar graphs. We give a polynomial-time algorithm that edge colors any planar graph with two outerplanar subgraphs. Two is clearly minimal for the class of planar graphs

Graph Embeddings and Simplicial Maps

An undirected graph is viewed as a simplicial complex. The notion of a graph embedding of a guest graph in a host graph is generalized to the realm of simplicial maps. Dilation is redefined in this more general setting. Lower bounds on dilation for various guest and host graphs are considered. Of particular interest are graphs that have been proposed as communication networks for parallel architectures. Bhatt et al. provide a lower bound on dilation for embedding a planar guest graph in a butterfly host graph. Here, this lower bound is extended in two directions. First, a lower bound that applies to arbitrary guest graphs is derived, using tools from algebraic topology. Second, this lower bound is shown to apply to arbitrary host graphs through a new graph-theoretic measure, called bidecomposability. Bounds on the bidecomposability of the butterfly graph and of the k-dimensional torus are determined. As corollaries to the main lower bound theorem, lower bounds are derived for embedding arbitrary planar graphs, genus g graphs, and k-dimensional meshes in a butterfly host graph

Computing Genomic Signatures Using de Bruijn Chains

Genomic DNA sequences have both deterministic and random aspects and exhibit features at numerous scales, from codons to regions of conserved or divergent gene order. Genomic signatures work by capturing one or more such features efficiently into a compact mathematical structure. We examine the unique manner in which oligonucleotides constitute a genome, within a graph-theoretic setting. A de Bruijn chain (DBC) is a kind of de Bruijn graph that includes a finite Markov chain. By representing a DNA sequence as a walk over a DBC and retaining specific information at nodes and edges, we obtain the de Bruijn chain genomic signature θdbc, based on graph structure and the stationary distribution of the DBC. We demonstrate that the θdbc signature is information-rich, efficient, sufficiently representative of the sequence from which it is derived, and superior to existing genomic signatures such as the dinucleotide odds ratio and word frequency based signatures. We develop a mathematical framework to elucidate the power of the θdbc signature to distinguish between sequences hypothesized to be generated by DBCs of distinct parameters. We study the effect of order of the θdbc signature, genome size, and variation within a genome on accuracy. We illustrate its superior performance over existing genomic signatures in predicting the origin of short DNA sequences.</p

LEND and Faster Algorithms for Constructing Minimal Perfect Hash Functions

The Large External object-oriented Network Database (LEND) system has been developed to provide efficient access to large collections of primitive or multimedia objects, semantic networks, thesauri, hypertexts, and information retrieval collections. An overview of LEND is given, emphasizing aspects that yield efficient operation. In particular, a new algorithm is described for quickly finding minimal perfect hash functions whose specification space is very close to the theoretical lower bound, i.e., around 2 bits per key. The various stages of processing are detailed, along with analytical and empirical results, including timing for a set of over 3.8 million keys that was processed on a NeXTstation in about 6 hours

The Pagenumber of Genus g Graph is 0(g)

In 1979, Berhart and Kainen conjectured that graphs of fixed genus g greater than or equal to 1 have unbounded pagenumber. This proves that genus g graphs can be embedded in 0(g) pages, thus disproving the conjecture. An Omega(square root of g) 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 0(g)-page book embedding. We give separate book embedding algorithms for the cases of graphs embedded in orientable and nonorientable surfaces. An important aspect of the construction is a new decomposition algorithm, 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

Optimal and Random Partitions of Random Graphs

The behavior of random graphs with respect to graph partitioning is considered. Conditions are identified under which random graphs cannot be partitioned well, i.e., a random partition is likely to be almost as good as an optimal partition

Graph Layout Using Queues

We study the problem of laying out the edges of a graph using queues. 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 and a problem of sorting with parallel queues. We relate the queue layout problem to the corresponding stack layout problem using stacks (the book embedding problem) and immediately derive some asymptomic bounds for d-valent graph. We show that every 1-queue graph is a 2-stack graph and that every 1-stack graph is a 2-queue graph. We characterize the 1-queue graphs (they are almost leveled-planar graphs) and prove that the problem of recognizing 1-queue graphs is NP-complete. We give some queue layouts for specific classes of graphs. Relationships to cutwidth, bandwidth, and bifurcators are presented. We show a tradeoff between queuenumber and stacknumber for a fixed linear order of the vertices of G

Development of New Heuristics for the Euclidean Traveling SalesmanProblem

Many heuristics have been developed to approximate optimal tours for the Euclidean Traveling Salesman Problem (ETSP). While much progress has been made, there are few quick heuristics which consistently produce tours within 4 percent of the optimal solution. This project examines a few of the well known heuristics and introduces two improvements, Maxdiff and Checks. Most algorithms, during tour constrution, add a city to the subtour because the city best satisfies some criterion. Maxdiff, applied to an algorithm, ranks a city according to its effect (based on the algorithm's criterion) if it is not added to the subtour

New Results for the Minimum Weight Triangulation Problem

The current best polynomial time approximation algorithm produces a triangulation that can be O(log n) times the weight of the optimal triangulation. We propose an algorithm that triangulates a set P of n points in a plane in O(n3) time and that never does worse than the greedy triangulation. We investigate issues of local optimality pertaining to known triangulation algorithms and suggest an interesting new approach to studying triangulation algorithms. We restate the minimum weight triangulation problem as a graph problem and show the NP-hardness of a closely related graph problem. Finally, we show that the constrained problem of computing the minimum weight triangulation, given a set of points in a plane and enough edges to form a triangulation, is NP-hard. These results are an advance towards a proof that the minimum weight triangulation problem is NP-hard