Throttling for the game of Cops and Robbers on graphs

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2018-09-01
Authors
Breen, Jane
Brimkov, Boris
Carlson, Joshua
Perry, K. E.
Reinhart, Carolyn
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Hogben, Leslie
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Electrical and Computer Engineering

The Department of Electrical and Computer Engineering (ECpE) contains two focuses. The focus on Electrical Engineering teaches students in the fields of control systems, electromagnetics and non-destructive evaluation, microelectronics, electric power & energy systems, and the like. The Computer Engineering focus teaches in the fields of software systems, embedded systems, networking, information security, computer architecture, etc.

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The Department of Electrical Engineering was formed in 1909 from the division of the Department of Physics and Electrical Engineering. In 1985 its name changed to Department of Electrical Engineering and Computer Engineering. In 1995 it became the Department of Electrical and Computer Engineering.

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1909-present

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  • Department of Electrical Engineering (1909-1985)
  • Department of Electrical Engineering and Computer Engineering (1985-1995)

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Mathematics
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Abstract

We consider the cop-throttling number of a graph G for the game of Cops and Robbers, which is defined to be the minimum of (k + capt(k)(G)), where k is the number of cops and capt(k)(G) is the minimum number of rounds needed for k cops to capture the robber on G over all possible games. We provide some tools for bounding the cop-throttling number, including showing that the positive semidefinite (PSD) throttling number, a variant of zero forcing throttling, is an upper bound for the cop-throttling number. We also characterize graphs having low cop-throttling number and investigate how large the cop-throttling number can be for a given graph. We consider trees, unicyclic graphs, incidence graphs of finite projective planes (a Meyniel extrema] family of graphs), a family of cop-win graphs with maximum capture time, grids, and hypercubes. All the upper bounds on the cop throttling number we obtain for families of graphs are O(root n).

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This is a manuscript of an article published as Breen, Jane, Boris Brimkov, Joshua Carlson, Leslie Hogben, K. E. Perry, and Carolyn Reinhart. "Throttling for the game of Cops and Robbers on graphs." Discrete Mathematics 341, no. 9 (2018): 2418-2430. DOI: 10.1016/j.disc.2018.05.017. Posted with permission.

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Mon Jan 01 00:00:00 UTC 2018
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