3 research outputs found
Optimizing the trade-off between number of cops and capture time in Cops and Robbers
The cop throttling number of a graph for the game of Cops and
Robbers is the minimum of , where is the number of cops and
is the minimum number of rounds needed for cops to capture the
robber on over all possible games in which both players play optimally. In
this paper, we construct a family of graphs having ,
establish a sublinear upper bound on the cop throttling number, and show that
the cop throttling number of chordal graphs is . We also introduce
the product cop throttling number as a parameter that
minimizes the person-hours used by the cops. This parameter extends the notion
of speed-up that has been studied in the context of parallel processing and
network decontamination. We establish bounds on the product cop throttling
number in terms of the cop throttling number, characterize graphs with low
product cop throttling number, and show that for a chordal graph ,
.Comment: 19 pages, 3 figure
Cop throttling number: Bounds, values, and variants
The cop throttling number thc(G) of a graph G for the game of Cops and Robbers is the minimum of k+captk(G), where k is the number of cops and captk(G) is the minimum number of rounds needed for k cops to capture the robber on G over all possible games in which both players play optimally. In this paper, we answer in the negative a question from [Breen et al., Throttling for the game of Cops and Robbers on graphs, {\em Discrete Math.}, 341 (2018) 2418--2430.] about whether the cop throttling number of any graph is O(n−−√) by constructing a family of graphs having thc(G)=Ω(n2/3). We establish a sublinear upper bound on the cop throttling number and show that the cop throttling number of chordal graphs is O(n−−√). We also introduce the product cop throttling number th×c(G) as a parameter that minimizes the person-hours used by the cops. We establish bounds on the product cop throttling number in terms of the cop throttling number, characterize graphs with low product cop throttling number, and show that for a chordal graph G, th×c(G)=1+rad(G)