The theory of directed graphs has developed enormously over recent decades, yet this book (first published in 2000) remains the only book to cover more than a small fraction of the results. New research in the field has made a second edition a necessity. Substantially revised, reorganised and updated, the book now comprises eighteen chapters, carefully arranged in a straightforward and logical manner, with many new results and open problems. As well as covering the theoretical aspects of the subject, with detailed proofs of many important results, the authors present a number of algorithms, and whole chapters are devoted to topics such as branchings, feedback arc and vertex sets, connectivity augmentations, sparse subdigraphs with prescribed connectivity, and also packing, covering and decompositions of digraphs. Throughout the book, there is a strong focus on applications which include quantum mechanics, bioinformatics, embedded computing, and the travelling salesman problem. Detailed indices and topic-oriented chapters ease navigation, and more than 650 exercises, 170 figures and 150 open problems are included to help immerse the reader in all aspects of the subject. Digraphs is an essential, comprehensive reference for undergraduate and graduate students, and researchers in mathematics, operations research and computer science. It will also prove invaluable to specialists in related areas, such as meteorology, physics and computational biology.

Digraphs Theory Algorithms And Applications

Urban scholars have studied street networks in various ways, but there are data availability and consistency limitations to the current urban planning/street network analysis literature. To address these challenges, this article presents OSMnx, a new tool to make the collection of data and creation and analysis of street networks simple, consistent, automatable and sound from the perspectives of graph theory, transportation, and urban design. OSMnx contributes five significant capabilities for researchers and practitioners: first, the automated downloading of political boundaries and building footprints; second, the tailored and automated downloading and constructing of street network data from OpenStreetMap; third, the algorithmic correction of network topology; fourth, the ability to save street networks to disk as shapefiles, GraphML, or SVG files; and fifth, the ability to analyze street networks, including calculating routes, projecting and visualizing networks, and calculating metric and topological measures. These measures include those common in urban design and transportation studies, as well as advanced measures of the structure and topology of the network. Finally, this article presents a simple case study using OSMnx to construct and analyze street networks in Portland, Oregon.

/pdf/osmnx-new-methods-for-acquiring-constructing-analyzing-and-3956mgentt.pdf

OSMnx: New Methods for Acquiring, Constructing, Analyzing, and Visualizing Complex Street Networks

A set of vertices $S$ resolves a graph $G$ if every vertex is uniquely determined by its vector of distances to the vertices in $S$. The metric dimension of $G$ is the minimum cardinality of a resolving set of $G$. This paper studies the metric dimension of cartesian products $G\,\square\,H$. We prove that the metric dimension of $G\,\square\,G$ is tied in a strong sense to the minimum order of a so-called doubly resolving set in $G$. Using bounds on the order of doubly resolving sets, we establish bounds on $G\,\square\,H$ for many examples of $G$ and $H$. One of our main results is a family of graphs $G$ with bounded metric dimension for which the metric dimension of $G\,\square\,G$ is unbounded.

https://users.monash.edu.au/~davidwo/papers/MetricDimension-arXiv.pdf

On the Metric Dimension of Cartesian Products of Graphs

We study generators of metric spaces--sets of points with the property that every point of the space is uniquely determined by the distances from their elements. Such generators put a light on seemingly different kinds of problems in combinatorics that are not directly related to metric spaces. The two applications we present concern combinatorial search: problems on false coins known from the borderline of extremal combinatorics and information theory; and a problem known from combinatorial optimization--connected joins in graphs.We use results on the detection of false coins to approximate the metric dimension (minimum size of a generator for the metric space defined by the distances) of some particular graphs for which the problem was known and open. In the opposite direction, using metric generators, we show that the existence of connected joins in graphs can be solved in polynomial time, a problem asked in a survey paper of Frank. On the negative side we prove that the minimization of the number of components of a join is NP-hard.We further explore the metric dimension with some problems. The main problem we are led to is how to extend an isometry given on a metric generator of a metric space.

/pdf/on-metric-generators-of-graphs-1mpyv2xr2o.pdf

On Metric Generators of Graphs

: Significant progress has been made with solution of location problems and in preprocessing and decomposition for discrete optimization. There has also been research on the application of techniques from combinational optimization to nonlinear problems.

http://www.dtic.mil/dtic/tr/fulltext/u2/a273552.pdf

Discrete Applied Mathematics

Resolvability in graphs and the metric dimension of a graph

The strong metric dimension of graphs and digraphs

The average connectivity of a graph

The metric dimension of Cayley digraphs

I have lived in Michigan nearly all my life, having beenbrought up in Sault Ste. Marie, educated at Michigan StateUniversity, studying at the University of Michigan, and em-ployed by Western Michigan University. I've enjoyed graphtheory from the day I heard my eventual thesis advisor E. A.Nordhaus speak on it. I've had eleven doctoral students, themost recent being Ortrud Oellermann, and the two of us arenow very pleased to have Erdos number 1. I enjoy musicalsand many sports.The misfortune of birth overtook me on March 26, 1913. Myparents were both mathematicians, and I learned a greatdeal from them. I got my Ph.D. at the University of Budapestin 1934. 1 spent the years 1938-1948 in Manchester, En-gland, and I have been in the United States since 1954. 1travel constantly around the world: Hungary, Israel, theU.S.A., England, etc. My subjects are number theory, combi-natorics, set theory and probability.I was born in Vryheid, South Africa, and have been inter-ested in mathematics since childhood. After receiving mypredoctoral degrees from the University of Natal in Durban, Icame to the United States to study graph theory at WesternMichigan University, where I received my Ph .D. in 1986, andwhere I am now on the faculty. I enjoy playing the piano,violin and recorder. I speak German, Afrikaans and Zulu, inaddition to English.

Ortrud R. Oellermann

Papers

Resolvability in graphs and the metric dimension of a graph

The strong metric dimension of graphs and digraphs

The average connectivity of a graph

The metric dimension of Cayley digraphs

How to Define an Irregular Graph