The Game of Cops and Robbers on Graphs

The Game of Cops and Robbers on Graphs

Author: Anthony Bonato

Publisher: American Mathematical Soc.

Published: 2011-08-16

Total Pages: 298

ISBN-13: 0821853473

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This book is the first and only one of its kind on the topic of Cops and Robbers games, and more generally, on the field of vertex pursuit games on graphs. The book is written in a lively and highly readable fashion, which should appeal to both senior undergraduates and experts in the field (and everyone in between). One of the main goals of the book is to bring together the key results in the field; as such, it presents structural, probabilistic, and algorithmic results on Cops and Robbers games. Several recent and new results are discussed, along with a comprehensive set of references. The book is suitable for self-study or as a textbook, owing in part to the over 200 exercises. The reader will gain insight into all the main directions of research in the field and will be exposed to a number of open problems.


Book Synopsis The Game of Cops and Robbers on Graphs by : Anthony Bonato

Download or read book The Game of Cops and Robbers on Graphs written by Anthony Bonato and published by American Mathematical Soc.. This book was released on 2011-08-16 with total page 298 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is the first and only one of its kind on the topic of Cops and Robbers games, and more generally, on the field of vertex pursuit games on graphs. The book is written in a lively and highly readable fashion, which should appeal to both senior undergraduates and experts in the field (and everyone in between). One of the main goals of the book is to bring together the key results in the field; as such, it presents structural, probabilistic, and algorithmic results on Cops and Robbers games. Several recent and new results are discussed, along with a comprehensive set of references. The book is suitable for self-study or as a textbook, owing in part to the over 200 exercises. The reader will gain insight into all the main directions of research in the field and will be exposed to a number of open problems.

The Game of Cops and Robbers on Graphs

The Game of Cops and Robbers on Graphs

Author: Anthony Bonato

Publisher: American Mathematical Soc.

Published:

Total Pages: 298

ISBN-13: 0821884778

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Book Synopsis The Game of Cops and Robbers on Graphs by : Anthony Bonato

Download or read book The Game of Cops and Robbers on Graphs written by Anthony Bonato and published by American Mathematical Soc.. This book was released on with total page 298 pages. Available in PDF, EPUB and Kindle. Book excerpt:

The Game of Cops and Robbers on Graphs

The Game of Cops and Robbers on Graphs

Author:

Publisher:

Published: 2011

Total Pages: 276

ISBN-13: 9781470416560

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Book Synopsis The Game of Cops and Robbers on Graphs by :

Download or read book The Game of Cops and Robbers on Graphs written by and published by . This book was released on 2011 with total page 276 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Graph Searching Games and Probabilistic Methods

Graph Searching Games and Probabilistic Methods

Author: Anthony Bonato

Publisher: CRC Press

Published: 2017-11-28

Total Pages: 346

ISBN-13: 135181477X

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Graph Searching Games and Probabilistic Methods is the first book that focuses on the intersection of graph searching games and probabilistic methods. The book explores various applications of these powerful mathematical tools to games and processes such as Cops and Robbers, Zombie and Survivors, and Firefighting. Written in an engaging style, the book is accessible to a wide audience including mathematicians and computer scientists. Readers will find that the book provides state-of-the-art results, techniques, and directions in graph searching games, especially from the point of view of probabilistic methods. The authors describe three directions while providing numerous examples, which include: • Playing a deterministic game on a random board. • Players making random moves. • Probabilistic methods used to analyze a deterministic game.


Book Synopsis Graph Searching Games and Probabilistic Methods by : Anthony Bonato

Download or read book Graph Searching Games and Probabilistic Methods written by Anthony Bonato and published by CRC Press. This book was released on 2017-11-28 with total page 346 pages. Available in PDF, EPUB and Kindle. Book excerpt: Graph Searching Games and Probabilistic Methods is the first book that focuses on the intersection of graph searching games and probabilistic methods. The book explores various applications of these powerful mathematical tools to games and processes such as Cops and Robbers, Zombie and Survivors, and Firefighting. Written in an engaging style, the book is accessible to a wide audience including mathematicians and computer scientists. Readers will find that the book provides state-of-the-art results, techniques, and directions in graph searching games, especially from the point of view of probabilistic methods. The authors describe three directions while providing numerous examples, which include: • Playing a deterministic game on a random board. • Players making random moves. • Probabilistic methods used to analyze a deterministic game.

Cops and Robber Game with a Fast Robber

Cops and Robber Game with a Fast Robber

Author: Abbas Mehrabian

Publisher:

Published: 2011

Total Pages: 57

ISBN-13:

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Graph searching problems are described as games played on graphs, between a set of searchers and a fugitive. Variants of the game restrict the abilities of the searchers and the fugitive and the corresponding search number (the least number of searchers that have a winning strategy) is related to several well-known parameters in graph theory. One popular variant is called the Cops and Robber game, where the searchers (cops) and the fugitive (robber) move in rounds, and in each round they move to an adjacent vertex. This game, defined in late 1970's, has been studied intensively. The most famous open problem is Meyniel's conjecture, which states that the cop number (the minimum number of cops that can always capture the robber) of a connected graph on n vertices is O(sqrt n). We consider a version of the Cops and Robber game, where the robber is faster than the cops, but is not allowed to jump over the cops. This version was first studied in 2008. We show that when the robber has speed s, the cop number of a connected n-vertex graph can be as large as Omega(n^(s/s+1)). This improves the Omega(n^(s-3/s-2)) lower bound of Frieze, Krivelevich, and Loh (Variations on Cops and Robbers, J. Graph Theory, to appear). We also conjecture a general upper bound O(n^(s/s+1)) for the cop number, generalizing Meyniel's conjecture. Then we focus on the version where the robber is infinitely fast, but is again not allowed to jump over the cops. We give a mathematical characterization for graphs with cop number one. For a graph with treewidth tw and maximum degree Delta, we prove the cop number is between (tw+1)/(Delta+1) and tw+1. Using this we show that the cop number of the m-dimensional hypercube is between c1 n / m sqrt(m) and c2 n / m for some constants c1 and c2. If G is a connected interval graph on n vertices, then we give a polynomial time 3-approximation algorithm for finding the cop number of G, and prove that the cop number is O(sqrt(n)). We prove that given n, there exists a connected chordal graph on n vertices with cop number Omega(n/log n). We show a lower bound for the cop numbers of expander graphs, and use this to prove that the random G(n, p) that is not very sparse, asymptotically almost surely has cop number between d1 / p and d2 log (np) / p for suitable constants d1 and d2. Moreover, we prove that a fixed-degree regular random graph with n vertices asymptotically almost surely has cop number Theta(n).


Book Synopsis Cops and Robber Game with a Fast Robber by : Abbas Mehrabian

Download or read book Cops and Robber Game with a Fast Robber written by Abbas Mehrabian and published by . This book was released on 2011 with total page 57 pages. Available in PDF, EPUB and Kindle. Book excerpt: Graph searching problems are described as games played on graphs, between a set of searchers and a fugitive. Variants of the game restrict the abilities of the searchers and the fugitive and the corresponding search number (the least number of searchers that have a winning strategy) is related to several well-known parameters in graph theory. One popular variant is called the Cops and Robber game, where the searchers (cops) and the fugitive (robber) move in rounds, and in each round they move to an adjacent vertex. This game, defined in late 1970's, has been studied intensively. The most famous open problem is Meyniel's conjecture, which states that the cop number (the minimum number of cops that can always capture the robber) of a connected graph on n vertices is O(sqrt n). We consider a version of the Cops and Robber game, where the robber is faster than the cops, but is not allowed to jump over the cops. This version was first studied in 2008. We show that when the robber has speed s, the cop number of a connected n-vertex graph can be as large as Omega(n^(s/s+1)). This improves the Omega(n^(s-3/s-2)) lower bound of Frieze, Krivelevich, and Loh (Variations on Cops and Robbers, J. Graph Theory, to appear). We also conjecture a general upper bound O(n^(s/s+1)) for the cop number, generalizing Meyniel's conjecture. Then we focus on the version where the robber is infinitely fast, but is again not allowed to jump over the cops. We give a mathematical characterization for graphs with cop number one. For a graph with treewidth tw and maximum degree Delta, we prove the cop number is between (tw+1)/(Delta+1) and tw+1. Using this we show that the cop number of the m-dimensional hypercube is between c1 n / m sqrt(m) and c2 n / m for some constants c1 and c2. If G is a connected interval graph on n vertices, then we give a polynomial time 3-approximation algorithm for finding the cop number of G, and prove that the cop number is O(sqrt(n)). We prove that given n, there exists a connected chordal graph on n vertices with cop number Omega(n/log n). We show a lower bound for the cop numbers of expander graphs, and use this to prove that the random G(n, p) that is not very sparse, asymptotically almost surely has cop number between d1 / p and d2 log (np) / p for suitable constants d1 and d2. Moreover, we prove that a fixed-degree regular random graph with n vertices asymptotically almost surely has cop number Theta(n).

Games on Graphs

Games on Graphs

Author: Shannon Dillman

Publisher:

Published: 2019

Total Pages: 58

ISBN-13:

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This project investigates various games on graphs. One game, cops and robbers, is played on a graph where one player plays as cops and the other plays as a robber. The goal of the cops is to capture the robber, and conversely, the goal of the robber is to evade the cops. In general, this problem is computationally hard to solve, and further, there is a large amount of theoretical interest on this topic. Another game is zero forcing. Loosely speaking, zero forcing is a propagation process on a graph where vertices become \colored" under specic rules. The goal is to color the entire graph using the minimal number of initially colored vertices. Zero forcing has applications to quantum computing and sensor allocation among other areas; however, this study will focus on the graph-theoretical aspects of the problem.


Book Synopsis Games on Graphs by : Shannon Dillman

Download or read book Games on Graphs written by Shannon Dillman and published by . This book was released on 2019 with total page 58 pages. Available in PDF, EPUB and Kindle. Book excerpt: This project investigates various games on graphs. One game, cops and robbers, is played on a graph where one player plays as cops and the other plays as a robber. The goal of the cops is to capture the robber, and conversely, the goal of the robber is to evade the cops. In general, this problem is computationally hard to solve, and further, there is a large amount of theoretical interest on this topic. Another game is zero forcing. Loosely speaking, zero forcing is a propagation process on a graph where vertices become \colored" under specic rules. The goal is to color the entire graph using the minimal number of initially colored vertices. Zero forcing has applications to quantum computing and sensor allocation among other areas; however, this study will focus on the graph-theoretical aspects of the problem.

Game Theory

Game Theory

Author: Steve Tadelis

Publisher: Princeton University Press

Published: 2013-01-06

Total Pages: 416

ISBN-13: 0691129088

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The definitive introduction to game theory This comprehensive textbook introduces readers to the principal ideas and applications of game theory, in a style that combines rigor with accessibility. Steven Tadelis begins with a concise description of rational decision making, and goes on to discuss strategic and extensive form games with complete information, Bayesian games, and extensive form games with imperfect information. He covers a host of topics, including multistage and repeated games, bargaining theory, auctions, rent-seeking games, mechanism design, signaling games, reputation building, and information transmission games. Unlike other books on game theory, this one begins with the idea of rationality and explores its implications for multiperson decision problems through concepts like dominated strategies and rationalizability. Only then does it present the subject of Nash equilibrium and its derivatives. Game Theory is the ideal textbook for advanced undergraduate and beginning graduate students. Throughout, concepts and methods are explained using real-world examples backed by precise analytic material. The book features many important applications to economics and political science, as well as numerous exercises that focus on how to formalize informal situations and then analyze them. Introduces the core ideas and applications of game theory Covers static and dynamic games, with complete and incomplete information Features a variety of examples, applications, and exercises Topics include repeated games, bargaining, auctions, signaling, reputation, and information transmission Ideal for advanced undergraduate and beginning graduate students Complete solutions available to teachers and selected solutions available to students


Book Synopsis Game Theory by : Steve Tadelis

Download or read book Game Theory written by Steve Tadelis and published by Princeton University Press. This book was released on 2013-01-06 with total page 416 pages. Available in PDF, EPUB and Kindle. Book excerpt: The definitive introduction to game theory This comprehensive textbook introduces readers to the principal ideas and applications of game theory, in a style that combines rigor with accessibility. Steven Tadelis begins with a concise description of rational decision making, and goes on to discuss strategic and extensive form games with complete information, Bayesian games, and extensive form games with imperfect information. He covers a host of topics, including multistage and repeated games, bargaining theory, auctions, rent-seeking games, mechanism design, signaling games, reputation building, and information transmission games. Unlike other books on game theory, this one begins with the idea of rationality and explores its implications for multiperson decision problems through concepts like dominated strategies and rationalizability. Only then does it present the subject of Nash equilibrium and its derivatives. Game Theory is the ideal textbook for advanced undergraduate and beginning graduate students. Throughout, concepts and methods are explained using real-world examples backed by precise analytic material. The book features many important applications to economics and political science, as well as numerous exercises that focus on how to formalize informal situations and then analyze them. Introduces the core ideas and applications of game theory Covers static and dynamic games, with complete and incomplete information Features a variety of examples, applications, and exercises Topics include repeated games, bargaining, auctions, signaling, reputation, and information transmission Ideal for advanced undergraduate and beginning graduate students Complete solutions available to teachers and selected solutions available to students

Cops and Robbers on Cayley Graphs and Embedded Graphs

Cops and Robbers on Cayley Graphs and Embedded Graphs

Author: Peter Bradshaw

Publisher:

Published: 2020

Total Pages: 67

ISBN-13:

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We consider the game of cops and robbers, which is a game played on a finite graph G by two players, Alice and Bob. Alice controls a team of cops, and Bob controls a robber, both of which occupy vertices of G. On Alice's turn, she may move each cop to an adjacent vertex or leave it at its current position. Similarly, on Bob's turn, he may move the robber to an adjacent vertex or leave it at its current position. Traditionally, Alice wins the game when a cop occupies the same vertex as the robber--that is, when a cop captures the robber. Conversely, Bob wins the game by letting the robber avoid capture forever. In a variation of the game, Alice wins the game when each neighbor of the robber's vertex is occupied by a cop--that is, when cops surround the robber. We will consider both of these winning conditions. The most fundamental graph invariant with regard to the game of cops and robbers is the cop number of a graph G, which denotes the minimum number of cops that Alice needs in order to have a winning strategy on G. We will introduce new techniques that may be used to calculate lower and upper bounds for the cop numbers of certain Cayley graphs. In particular, we will show that the well-known Meyniel's conjecture holds for both undirected and directed abelian Cayley graphs. We will also introduce new techniques for establishing upper bounds on the cop numbers of surface-embedded graphs bounded by the genus of the surface in the surrounding win condition.


Book Synopsis Cops and Robbers on Cayley Graphs and Embedded Graphs by : Peter Bradshaw

Download or read book Cops and Robbers on Cayley Graphs and Embedded Graphs written by Peter Bradshaw and published by . This book was released on 2020 with total page 67 pages. Available in PDF, EPUB and Kindle. Book excerpt: We consider the game of cops and robbers, which is a game played on a finite graph G by two players, Alice and Bob. Alice controls a team of cops, and Bob controls a robber, both of which occupy vertices of G. On Alice's turn, she may move each cop to an adjacent vertex or leave it at its current position. Similarly, on Bob's turn, he may move the robber to an adjacent vertex or leave it at its current position. Traditionally, Alice wins the game when a cop occupies the same vertex as the robber--that is, when a cop captures the robber. Conversely, Bob wins the game by letting the robber avoid capture forever. In a variation of the game, Alice wins the game when each neighbor of the robber's vertex is occupied by a cop--that is, when cops surround the robber. We will consider both of these winning conditions. The most fundamental graph invariant with regard to the game of cops and robbers is the cop number of a graph G, which denotes the minimum number of cops that Alice needs in order to have a winning strategy on G. We will introduce new techniques that may be used to calculate lower and upper bounds for the cop numbers of certain Cayley graphs. In particular, we will show that the well-known Meyniel's conjecture holds for both undirected and directed abelian Cayley graphs. We will also introduce new techniques for establishing upper bounds on the cop numbers of surface-embedded graphs bounded by the genus of the surface in the surrounding win condition.

A Course on the Web Graph

A Course on the Web Graph

Author: Anthony Bonato

Publisher: American Mathematical Soc.

Published: 2008

Total Pages: 200

ISBN-13: 0821844679

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"A Course on the Web Graph provides a comprehensive introduction to state-of-the-art research on the applications of graph theory to real-world networks such as the web graph. It is the first mathematically rigorous textbook discussing both models of the web graph and algorithms for searching the web. After introducing key tools required for the study of web graph mathematics, an overview is given of the most widely studied models for the web graph. A discussion of popular web search algorithms, e.g. PageRank, is followed by additional topics, such as applications of infinite graph theory to the web graph, spectral properties of power law graphs, domination in the web graph, and the spread of viruses in networks. The book is based on a graduate course taught at the AARMS 2006 Summer School at Dalhousie University. As such it is self-contained and includes over 100 exercises. The reader of the book will gain a working knowledge of current research in graph theory and its modern applications. In addition, the reader will learn first-hand about models of the web, and the mathematics underlying modern search engines."--Publisher's description.


Book Synopsis A Course on the Web Graph by : Anthony Bonato

Download or read book A Course on the Web Graph written by Anthony Bonato and published by American Mathematical Soc.. This book was released on 2008 with total page 200 pages. Available in PDF, EPUB and Kindle. Book excerpt: "A Course on the Web Graph provides a comprehensive introduction to state-of-the-art research on the applications of graph theory to real-world networks such as the web graph. It is the first mathematically rigorous textbook discussing both models of the web graph and algorithms for searching the web. After introducing key tools required for the study of web graph mathematics, an overview is given of the most widely studied models for the web graph. A discussion of popular web search algorithms, e.g. PageRank, is followed by additional topics, such as applications of infinite graph theory to the web graph, spectral properties of power law graphs, domination in the web graph, and the spread of viruses in networks. The book is based on a graduate course taught at the AARMS 2006 Summer School at Dalhousie University. As such it is self-contained and includes over 100 exercises. The reader of the book will gain a working knowledge of current research in graph theory and its modern applications. In addition, the reader will learn first-hand about models of the web, and the mathematics underlying modern search engines."--Publisher's description.

Divisors and Sandpiles: An Introduction to Chip-Firing

Divisors and Sandpiles: An Introduction to Chip-Firing

Author: Scott Corry

Publisher: American Mathematical Soc.

Published: 2018-07-23

Total Pages: 325

ISBN-13: 1470442183

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Divisors and Sandpiles provides an introduction to the combinatorial theory of chip-firing on finite graphs. Part 1 motivates the study of the discrete Laplacian by introducing the dollar game. The resulting theory of divisors on graphs runs in close parallel to the geometric theory of divisors on Riemann surfaces, and Part 1 culminates in a full exposition of the graph-theoretic Riemann-Roch theorem due to M. Baker and S. Norine. The text leverages the reader's understanding of the discrete story to provide a brief overview of the classical theory of Riemann surfaces. Part 2 focuses on sandpiles, which are toy models of physical systems with dynamics controlled by the discrete Laplacian of the underlying graph. The text provides a careful introduction to the sandpile group and the abelian sandpile model, leading ultimately to L. Levine's threshold density theorem for the fixed-energy sandpile Markov chain. In a precise sense, the theory of sandpiles is dual to the theory of divisors, and there are many beautiful connections between the first two parts of the book. Part 3 addresses various topics connecting the theory of chip-firing to other areas of mathematics, including the matrix-tree theorem, harmonic morphisms, parking functions, M-matrices, matroids, the Tutte polynomial, and simplicial homology. The text is suitable for advanced undergraduates and beginning graduate students.


Book Synopsis Divisors and Sandpiles: An Introduction to Chip-Firing by : Scott Corry

Download or read book Divisors and Sandpiles: An Introduction to Chip-Firing written by Scott Corry and published by American Mathematical Soc.. This book was released on 2018-07-23 with total page 325 pages. Available in PDF, EPUB and Kindle. Book excerpt: Divisors and Sandpiles provides an introduction to the combinatorial theory of chip-firing on finite graphs. Part 1 motivates the study of the discrete Laplacian by introducing the dollar game. The resulting theory of divisors on graphs runs in close parallel to the geometric theory of divisors on Riemann surfaces, and Part 1 culminates in a full exposition of the graph-theoretic Riemann-Roch theorem due to M. Baker and S. Norine. The text leverages the reader's understanding of the discrete story to provide a brief overview of the classical theory of Riemann surfaces. Part 2 focuses on sandpiles, which are toy models of physical systems with dynamics controlled by the discrete Laplacian of the underlying graph. The text provides a careful introduction to the sandpile group and the abelian sandpile model, leading ultimately to L. Levine's threshold density theorem for the fixed-energy sandpile Markov chain. In a precise sense, the theory of sandpiles is dual to the theory of divisors, and there are many beautiful connections between the first two parts of the book. Part 3 addresses various topics connecting the theory of chip-firing to other areas of mathematics, including the matrix-tree theorem, harmonic morphisms, parking functions, M-matrices, matroids, the Tutte polynomial, and simplicial homology. The text is suitable for advanced undergraduates and beginning graduate students.