Topological directions in Cops and Robbers

Bonato, Anthony; Mohar, Bojan

doi:10.4310/joc.2020.v11.n1.a3

Cited by 14 publications

(15 citation statements)

References 30 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We end this section with a result that does not directly apply to throttling, but is nonetheless an interesting fact about the game of Cops and Robbers on chordal graphs. This result resolves an open problem from [4]. Theorem 3.9.…”

Section: Cop Throttling For Chordal Graphssupporting

confidence: 65%

See 1 more Smart Citation

Optimizing the trade-off between number of cops and capture time in Cops and Robbers

Bonato¹,

Breen²,

Brimkov³

et al. 2019

Preprint

Self Cite

View full text Add to dashboard Cite

The cop throttling number th c (G) of a graph G for the game of Cops and Robbers is 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 in which both players play optimally. In this paper, we construct a family of graphs having th c (G) = Ω n 2/3 , 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. 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 G, th × c (G) = 1 + rad(G).

show abstract

Section: Cop Throttling For Chordal Graphssupporting

confidence: 65%

“…In Section 3, we prove that for any chordal graph of order n the k-capture time is equal to the k-radius and the cop throttling number is O( √ n). We also answer an open problem from [4] about classifying cop-win outerplanar graphs.…”

Section: Introductionmentioning

confidence: 98%

Optimizing the trade-off between number of cops and capture time in Cops and Robbers

Bonato¹,

Breen²,

Brimkov³

et al. 2019

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…In the game of Cops and Robbers, the notion of robber territory is a useful tool, especially when studying the cop number of planar graphs and graphs on surfaces; see [7]. Roughly put, the robber territory is an induced subgraph where the robber is safe from capture.…”

Section: Localization Number Of Bibdsmentioning

confidence: 99%

“…Two nonisomorphic BIBD (7,21,9,3,3) are given in Figure 1; see [15]. The incidence graph of the first has f G ( ) = 3 while the second has f G ( ) = 1.…”

mentioning

confidence: 99%

The localization number of designs

Bonato

Huggan

Marbach

2020

J of Combinatorial Designs

Self Cite

View full text Add to dashboard Cite

We study the localization number of incidence graphs of designs. In the localization game played on a graph, the cops attempt to determine the location of an invisible robber via distance probes. The localization number of a graph , written , is the minimum number of cops needed to ensure the robber's capture. We present bounds on the localization number of incidence graphs of balanced incomplete block designs. Exact values of the localization number are given for the incidence graphs of projective and affine planes. Bounds are given for Steiner systems and for transversal designs.

show abstract

“…The cops win if one can eventually occupy the same node as the robber and the robber wins if they can evade capture indefinitely. A graph G is cop-win if a single cop can always guarantee a win playing on G. Cops and Robber has been studied extensively since its first introduction; see [5] for an overview and [3,4,13,15,21] for some recent work on the game.…”

mentioning

confidence: 99%

The node cop‐win reliability of unicyclic and bicyclic graphs

Ahmed

Cameron

2021

Networks

View full text Add to dashboard Cite

Various models to quantify the reliability of a network have been studied where certain components of the graph may fail at random and the probability that the remaining graph is connected is the proxy for reliability. We introduce a strengthening of one of these models by considering the probability that the remaining graph is cop-win. A graph is cop-win if one cop can win the game Cops and Robber. More precisely, for a graph G with nodes that are operational independently with probability p, the node cop-win reliability of G, denoted NCRel(G, p), is the probability that the operational nodes induce a cop-win subgraph of G. It is then of interest to find graphs G with n nodes and m edges such that NCRel(G, p) ≥ NCRel(H, p) for all p ∈ [0, 1] and all graphs H with n nodes and m edges. We show that such graphs exist among unicyclic and bicyclic graphs, respectively.

show abstract

Topological directions in Cops and Robbers

Cited by 14 publications

References 30 publications

Optimizing the trade-off between number of cops and capture time in Cops and Robbers

Optimizing the trade-off between number of cops and capture time in Cops and Robbers

The localization number of designs

The node cop‐win reliability of unicyclic and bicyclic graphs

Contact Info

Product

Resources

About