Douglas B. West

Bio: Douglas B. West is an academic researcher from University of Illinois at Urbana–Champaign. The author has contributed to research in topics: Vertex (geometry) & Degree (graph theory). The author has an hindex of 34, co-authored 292 publications receiving 11587 citations. Previous affiliations of Douglas B. West include Urbana University & Princeton University.

Introduction to Graph Theory

Abstract: 1. Fundamental Concepts. Definitions and examples. Paths and proofs. Vertex degrees and counting. Degrees and algorithmic proof. 2. Trees and Distance. Basic properties. Spanning trees and enumeration. Optimization and trees. Eulerian graphs and digraphs. 3. Matchings and Factors. Matchings in bipartite graphs. Applications and algorithms. Matchings in general graphs. 4. Connectivity and Paths. Cuts and connectivity. k-connected graphs. Network flow problems. 5. Graph Coloring. Vertex colorings and upper bounds. Structure of k-chromatic graphs. Enumerative aspects. 6. Edges and Cycles. Line graphs and edge-coloring. Hamiltonian cycles. Complexity. 7. Planar Graphs. Embeddings and Eulers formula. Characterization of planar graphs. Parameters of planarity. 8. Additional Topics. Perfect graphs. Matroids. Ramsey theory. More extremal problems. Random graphs. Eigenvalues of graphs. Glossary of Terms. Glossary of Notation. References. Author Index. Subject Index.

A Graph-Theoretic Game and its Application to the $k$-Server Problem

Abstract: This paper investigates a zero-sum game played on a weighted connected graph $G$ between two players, the tree player and the edge player. At each play, the tree player chooses a spanning tree $T$ and the edge player chooses an edge $e$. The payoff to the edge player is $cost(T,e)$, defined as follows: If $e$ lies in the tree $T$ then $cost(T,e)=0$; if $e$ does not lie in the tree then $cost(T,e) = cycle(T,e)/w(e)$, where $w(e)$ is the weight of edge $e$ and $cycle(T,e)$ is the weight of the unique cycle formed when edge $e$ is added to the tree $T$. The main result is that the value of the game on any $n$-vertex graph is bounded above by $\exp(O(\sqrt{\log n \log\log n}))$. It is conjectured that the value of the game is $O(\log n)$. The game arises in connection with the $k$-server problem on a road network; i.e., a metric space that can be represented as a multigraph $G$ in which each edge $e$ represents a road of length $w(e)$. It is shown that, if the value of the game on $G$ is $Val(G,w)$, then there is a randomized strategy that achieves a competitive ratio of $k(1 + Val(G,w))$ against any oblivious adversary. Thus, on any $n$-vertex road network, there is a randomized algorithm for the $k$-server problem that is $k\cdot\exp(O(\sqrt{\log n \log\log n}))$ competitive against oblivious adversaries. At the heart of the analysis of the game is an algorithm that provides an approximate solution for the simple network design problem. Specifically, for any $n$-vertex weighted, connected multigraph, the algorithm constructs a spanning tree $T$ such that the average, over all edges $e$, of $cost(T,e)$ is less than or equal to $\exp(O(\sqrt{\log n \log\log n}))$. This result has potential application to the design of communication networks. It also improves substantially known estimates concerning the existence of a sparse basis for the cycle space of a graph.

Spanning trees with many leaves

Abstract: A connected graph having large minimum vertex degree must have a spanning tree with many leaves. In particular, let $l( n,k )$ be the maximum integer m such that every connected n-vertex graph with minimum degree at least k has a spanning tree with at least m leaves. Then $l( n,3 )\geqq n/4 + 2, l( n,4 )\geqq ( 2n + 8 )/5,$ and $l( n,k )\leqq n - 3\lfloor n/( k + 1 ) \rfloor + 2$ for all k. The lower bounds are proved by an algorithm that constructs a spanning tree with at least the desired number of leaves. Finally, $l( n,k )\geqq ( 1 - b \ln k/k )n$ for large k, again proved algorithmically, where b is any constant exceeding 2.5.

Extremal Problems for Roman Domination

Abstract: A Roman dominating function of a graph $G$ is a labeling $f\colon\,V(G)\to\{0,1,2\}$ such that every vertex with label 0 has a neighbor with label 2. The Roman domination number $\gamma_R(G)$ of $G$ is the minimum of $\sum_{v\in V(G)}f(v)$ over such functions. Let $G$ be a connected $n$-vertex graph. We prove that $\gamma_R(G)\leq4n/5$, and we characterize the graphs achieving equality. We obtain sharp upper and lower bounds for $\gamma_R(G)+\gamma_R(\overline{G})$ and $\gamma_R(G)\gamma_R(\overline{G})$, improving known results for domination number. We prove that $\gamma_R(G)\leq8n/11$ when $\delta(G)\geq2$ and $n\geq9$, and this is sharp.

Extremal Problems for Game Domination Number

Abstract: In the domination game on a graph $G$, two players called Dominator and Staller alternately select vertices of $G$. Each vertex chosen must strictly increase the number of vertices dominated; the game ends when the chosen set becomes a dominating set of $G$. Dominator aims to minimize the size of the resulting dominating set, while Staller aims to maximize it. When both players play optimally, the size of the dominating set produced is the game domination number of $G$, denoted by $\gamma_g(G)$ when Dominator plays first and by $\gamma_g^\prime(G)$ when Staller plays first. We prove that $\gamma_g(G) \le 7n/11$ when $G$ is an isolate-free $n$-vertex forest and that $\gamma_g(G) \le \left\lceil7n/10\right\rceil$ for any isolate-free $n$-vertex graph. In both cases we conjecture that $\gamma_g(G) \le 3n/5$ and prove it when $G$ is a forest of nontrivial caterpillars. We also resolve conjectures of Bresar, Klavžar, and Rall by showing that always $\gamma_g^\prime(G)\le\gamma_g(G)+1$, that for $k\ge2$ there a...

I and i

Abstract: There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality. Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

Random graphs

Abstract: We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Introduction to Econophysics: Correlations and Complexity in Finance

Abstract: This book concerns the use of concepts from statistical physics in the description of financial systems. The authors illustrate the scaling concepts used in probability theory, critical phenomena, and fully developed turbulent fluids. These concepts are then applied to financial time series. The authors also present a stochastic model that displays several of the statistical properties observed in empirical data. Statistical physics concepts such as stochastic dynamics, short- and long-range correlations, self-similarity and scaling permit an understanding of the global behaviour of economic systems without first having to work out a detailed microscopic description of the system. Physicists will find the application of statistical physics concepts to economic systems interesting. Economists and workers in the financial world will find useful the presentation of empirical analysis methods and well-formulated theoretical tools that might help describe systems composed of a huge number of interacting subsystems.

Analysis of weighted networks.

Abstract: The connections in many networks are not merely binary entities, either present or not, but have associated weights that record their strengths relative to one another. Recent studies of networks have, by and large, steered clear of such weighted networks, which are often perceived as being harder to analyze than their unweighted counterparts. Here we point out that weighted networks can in many cases be analyzed using a simple mapping from a weighted network to an unweighted multigraph, allowing us to apply standard techniques for unweighted graphs to weighted ones as well. We give a number of examples of the method, including an algorithm for detecting community structure in weighted networks and a simple proof of the maximum-flow--minimum-cut theorem.