238 research outputs found

    How to Guard a Graph?

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    We initiate the study of the algorithmic foundations of games in which a set of cops has to guard a region in a graph (or digraph) against a robber. The robber and the cops are placed on vertices of the graph; they take turns in moving to adjacent vertices (or staying). The goal of the robber is to enter the guarded region at a vertex with no cop on it. The problem is to find the minimum number of cops needed to prevent the robber from entering the guarded region. The problem is highly non-trivial even if the robber's or the cops' regions are restricted to very simple graphs. The computational complexity of the problem depends heavily on the chosen restriction. In particular, if the robber's region is only a path, then the problem can be solved in polynomial time. When the robber moves in a tree (or even in a star), then the decision version of the problem is NP-complete. Furthermore, if the robber is moving in a directed acyclic graph, the problem becomes PSPACE-complet

    On the Cop Number of String Graphs

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    Cops and Robber is a well-studied two-player pursuit-evasion game played on a graph, where a group of cops tries to capture the robber. The cop number of a graph is the minimum number of cops required to capture the robber. We show that the cop number of a string graph is at most 13, improving upon a result of Gaven?iak et al. [Eur. J. of Comb. 72, 45-69 (2018)]. Using similar techniques, we also show that four cops have a winning strategy for a variant of Cops and Robber, named Fully Active Cops and Robber, on planar graphs, addressing an open question of Gromovikov et al. [Austr. J. Comb. 76(2), 248-265 (2020)]

    Dagstuhl Reports : Volume 1, Issue 2, February 2011

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    Online Privacy: Towards Informational Self-Determination on the Internet (Dagstuhl Perspectives Workshop 11061) : Simone Fischer-Hübner, Chris Hoofnagle, Kai Rannenberg, Michael Waidner, Ioannis Krontiris and Michael Marhöfer Self-Repairing Programs (Dagstuhl Seminar 11062) : Mauro Pezzé, Martin C. Rinard, Westley Weimer and Andreas Zeller Theory and Applications of Graph Searching Problems (Dagstuhl Seminar 11071) : Fedor V. Fomin, Pierre Fraigniaud, Stephan Kreutzer and Dimitrios M. Thilikos Combinatorial and Algorithmic Aspects of Sequence Processing (Dagstuhl Seminar 11081) : Maxime Crochemore, Lila Kari, Mehryar Mohri and Dirk Nowotka Packing and Scheduling Algorithms for Information and Communication Services (Dagstuhl Seminar 11091) Klaus Jansen, Claire Mathieu, Hadas Shachnai and Neal E. Youn

    Walks and games on graphs

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    Herrman, Rebekah Ph.D. The University of Memphis, May 2020. Walks and Games on Graphs. Major Professor: B\\u27ela Bollob\\u27as, Ph.D.Chapter 1 is joint work with Dr. Travis Humble and appears in the journal Physical Review A. In this work, we consider continuous-time quantum walks on dynamic graphs. Continuous-time quantum walks have been well studied on graphs that do not change as a function of time. We offer a mathematical formulation for how to express continuous-time quantum walks on graphs that can change in time, find a universal set of walks that can perform any operation, and use them to simulate basic quantum circuits. This work was supported in part by the Department of Energy Student Undergraduate Laboratory Internship and the National Science Foundation Mathematical Sciences Graduate Internship programs.The (t,r)(t,r) broadcast domination number of a graph GG, γt,r(G)\gamma_{t,r}(G), is a generalization of the domination number of a graph. In Chapter 2, we consider the (t,r)(t,r) broadcast domination number on graphs, specifically powers of cycles, powers of paths, and infinite grids. This work is joint with Peter van Hintum and has been submitted to the journal Discrete Applied Mathematics.Bridge-burning cops and robbers is a variant of the cops and robbers game on graphs in which the robber removes an edge from the graph once it is traversed. In Chapter 3, we study the maximum time it takes the cops to capture the robber in this variant. This is joint with Peter van Hintum and Dr. Stephen Smith.In Chapter 4, we study a variant of the chip-firing game called the \emph{diffusion game}. In the diffusion game, we begin with some integer labelling of the vertices of a graph, interpreted as a number of chips on each vertex, and for each subsequent step every vertex simultaneously fires a chip to each neighbour with fewer chips. In general, this could result in negative vertex labels. Long and Narayanan asked whether there exists an f(n)f(n) for each nn, such that whenever we have a graph on nn vertices and an initial allocation with at least f(n)f(n) chips on each vertex, then the number of chips on each vertex will remain non-negative. We answer their question in the affirmative, showing further that f(n)=n−2f(n)=n-2 is the best possible bound. We also consider the existence of a similar bound g(d)g(d) for each dd, where dd is the maximum degree of the graph. This work is joint with Andrew Carlotti and has been submitted to the journal Discrete Mathematics.In Chapter 5, we consider the eternal game chromatic number of random graphs. The eternal graph colouring problem, recently introduced by Klostermeyer and Mendoza \cite{klostermeyer}, is a version of the graph colouring game, where two players take turns properly colouring a graph. In this chapter, we show that with high probability χg∞(Gn,p)=(p2+o(1))n\chi_{g}^{\infty}(G_{n,p}) = (\frac{p}{2} + o(1))n for odd nn, and also for even nn when p=1kp=\frac{1}{k} for some k∈Nk \in \N. This work is joint with Vojt\u{e}ch Dvo\u{r}\\u27ak and Peter van Hintum, and has been submitted to the European Journal of Combinatorics

    Cops and Robber -- When Capturing is not Surrounding

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    We consider "surrounding" versions of the classic Cops and Robber game. The game is played on a connected graph in which two players, one controlling a number of cops and the other controlling a robber, take alternating turns. In a turn, each player may move each of their pieces: The robber always moves between adjacent vertices. Regarding the moves of the cops we distinguish four versions that differ in whether the cops are on the vertices or the edges of the graph and whether the robber may move on/through them. The goal of the cops is to surround the robber, i.e., occupying all neighbors (vertex version) or incident edges (edge version) of the robber's current vertex. In contrast, the robber tries to avoid being surrounded indefinitely. Given a graph, the so-called cop number denotes the minimum number of cops required to eventually surround the robber. We relate the different cop numbers of these versions and prove that none of them is bounded by a function of the classical cop number and the maximum degree of the graph, thereby refuting a conjecture by Crytser, Komarov and Mackey [Graphs and Combinatorics, 2020]
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