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## LAZY COPS AND ROBBERS PLAYED ON RANDOM GRAPHS AND GRAPHS ON SURFACES

### Citations

1034 | Random Graphs - Janson, Luczak, et al. - 2000 |

190 |
Introduction to Graph Theory, (2nd Edition
- West
- 2000
(Show Context)
Citation Context ...he Gilbert-Hutchinson-Tarjan separator theorem [15]. 1.1. Definitions and notation. We consider only finite, undirected graphs in this paper. For background on graph theory, the reader is directed to =-=[27]-=-. The game of Cops and Robbers was independently introduced in [20, 25] and the cop number was introduced in [1]. The game is played on a reflexive graph; that is, each vertex has at least one loop. M... |

121 | A game of cops and robbers
- Aigner, Fromme
- 1984
(Show Context)
Citation Context ...rected graphs in this paper. For background on graph theory, the reader is directed to [27]. The game of Cops and Robbers was independently introduced in [20, 25] and the cop number was introduced in =-=[1]-=-. The game is played on a reflexive graph; that is, each vertex has at least one loop. Multiple edges are allowed, but make no difference to the play of the game, so we always assume there is exactly ... |

92 |
Tarjan, A separation theorem for graphs of bounded genus
- Gilbert, Hutchinson, et al.
(Show Context)
Citation Context ...of p = p(n). We do this by examining typical expansion properties of such graphs. In Theorem 3.2, we provide an upper bound for graphs of genus g using the Gilbert-Hutchinson-Tarjan separator theorem =-=[15]-=-. 1.1. Definitions and notation. We consider only finite, undirected graphs in this paper. For background on graph theory, the reader is directed to [27]. The game of Cops and Robbers was independentl... |

65 |
Vertex-to-vertex pursuit in a graph, Discrete Mathematics 43
- Nowakowski, Winkler
- 1983
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Citation Context ...ns and notation. We consider only finite, undirected graphs in this paper. For background on graph theory, the reader is directed to [27]. The game of Cops and Robbers was independently introduced in =-=[20, 25]-=- and the cop number was introduced in [1]. The game is played on a reflexive graph; that is, each vertex has at least one loop. Multiple edges are allowed, but make no difference to the play of the ga... |

44 |
The Game of Cops and Robbers on Graphs
- Bonato, Nowakowski
- 2011
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Citation Context ... this parameter is the focus of Meyniel’s conjecture that the cop number of a connected n-vertex graph is O( √ n). For additional background on Cops and Robbers and Meyniel’s conjecture, see the book =-=[11]-=- and the surveys [3, 6, 7]. A number of variants of Cops and Robbers have been studied. For example, we may allow a cop to capture the robber from a distance k, where k is a non-negative integer [8, 9... |

21 |
Jeux et pointes fixes sur les graphes, Thèse de 3ème cycle
- Quilliot
- 1978
(Show Context)
Citation Context ...ns and notation. We consider only finite, undirected graphs in this paper. For background on graph theory, the reader is directed to [27]. The game of Cops and Robbers was independently introduced in =-=[20, 25]-=- and the cop number was introduced in [1]. The game is played on a reflexive graph; that is, each vertex has at least one loop. Multiple edges are allowed, but make no difference to the play of the ga... |

18 | Cops and robbers in a random graph
- Bollobás, Kun, et al.
(Show Context)
Citation Context ... n. A simple argument using dominating sets shows that Meyniel’s conjecture also holds a.a.s. if p tends to 1 as n goes to infinity (see [22] for this and stronger results). Bollobás, Kun and Leader =-=[5]-=- showed that if p(n) ≥ 2.1 log n/n, then a.a.s. 1 (pn)2 n1/2−9/(2 log log(pn)) ≤ c(G(n, p)) ≤ 160000√n log n . From these results, if np ≥ 2.1 log n and either np = no(1) or np = n1/2+o(1), then a.a.s... |

18 |
Pra lat, Chasing robbers on random graphs: zigzag theorem, preprint
- Luczak, P
(Show Context)
Citation Context ...g n . From these results, if np ≥ 2.1 log n and either np = no(1) or np = n1/2+o(1), then a.a.s. c(G(n, p)) = n1/2+o(1). Somewhat surprisingly, between these values it was shown bysLuczak and Pra lat =-=[19]-=- that the cop number has more complicated behaviour. It follows that a.a.s. logn c(G(n, nx−1)) is asymptotic to the function (denoted in blue) shown in Figure 1. The above results show that Meyniel’s ... |

18 |
When does a random graph have constant cop number
- lat
(Show Context)
Citation Context ...= n−o(1) we have that a.a.s. c(G(n, p)) = (1 + o(1)) log1/(1−p) n. A simple argument using dominating sets shows that Meyniel’s conjecture also holds a.a.s. if p tends to 1 as n goes to infinity (see =-=[22]-=- for this and stronger results). Bollobás, Kun and Leader [5] showed that if p(n) ≥ 2.1 log n/n, then a.a.s. 1 (pn)2 n1/2−9/(2 log log(pn)) ≤ c(G(n, p)) ≤ 160000√n log n . From these results, if np ≥... |

16 |
Meyniel’s conjecture holds for random graphs, Random Structures and Algorithms
- lat, Wormald
(Show Context)
Citation Context ...l’s conjecture holds a.a.s. for random graphs except perhaps when np = n1/(2k)+o(1) for some k ∈ N, or when np = no(1). Pra lat and Wormald showed recently that the conjecture holds a.a.s. in G(n, p) =-=[23]-=- as well as in random d-regular graphs [24]. In this paper, we investigate the lazy cop number of G(n, p). The main theorem of this section is the following. Theorem 2.1. Let 0 < α ≤ 1, let ε > 0, and... |

14 | Meyniel’s conjecture on the cop number: a survey, Submitted
- Baird, Bonato
(Show Context)
Citation Context ...e focus of Meyniel’s conjecture that the cop number of a connected n-vertex graph is O( √ n). For additional background on Cops and Robbers and Meyniel’s conjecture, see the book [11] and the surveys =-=[3, 6, 7]-=-. A number of variants of Cops and Robbers have been studied. For example, we may allow a cop to capture the robber from a distance k, where k is a non-negative integer [8, 9], play on edges [13], all... |

11 | Pursuit and evasion from a distance: algorithms and bounds
- Bonato, Chiniforooshan
- 2009
(Show Context)
Citation Context ... [11] and the surveys [3, 6, 7]. A number of variants of Cops and Robbers have been studied. For example, we may allow a cop to capture the robber from a distance k, where k is a non-negative integer =-=[8, 9]-=-, play on edges [13], allow one or both players to move with different speeds [2, 14] or to teleport, allow the robber to capture the cops [10], or make the robber invisible or drunk [17, 18]. See Cha... |

8 |
Meyniel’s conjecture holds for random d-regular graphs, manuscript
- lat, Wormald
(Show Context)
Citation Context ...hs except perhaps when np = n1/(2k)+o(1) for some k ∈ N, or when np = no(1). Pra lat and Wormald showed recently that the conjecture holds a.a.s. in G(n, p) [23] as well as in random d-regular graphs =-=[24]-=-. In this paper, we investigate the lazy cop number of G(n, p). The main theorem of this section is the following. Theorem 2.1. Let 0 < α ≤ 1, let ε > 0, and let d = d(n) = (n− 1)p = nα+o(1). (i) If α... |

7 | Catch me if you can: Cops and Robbers on graphs
- Bonato
- 2011
(Show Context)
Citation Context ...e focus of Meyniel’s conjecture that the cop number of a connected n-vertex graph is O( √ n). For additional background on Cops and Robbers and Meyniel’s conjecture, see the book [11] and the surveys =-=[3, 6, 7]-=-. A number of variants of Cops and Robbers have been studied. For example, we may allow a cop to capture the robber from a distance k, where k is a non-negative integer [8, 9], play on edges [13], all... |

7 |
Pra lat, Cops and Robbers from a distance, Theoretical Computer Science 411
- Bonato, Chiniforooshan, et al.
- 2010
(Show Context)
Citation Context ... [11] and the surveys [3, 6, 7]. A number of variants of Cops and Robbers have been studied. For example, we may allow a cop to capture the robber from a distance k, where k is a non-negative integer =-=[8, 9]-=-, play on edges [13], allow one or both players to move with different speeds [2, 14] or to teleport, allow the robber to capture the cops [10], or make the robber invisible or drunk [17, 18]. See Cha... |

6 | Network security in models of complex networks, submitted
- Bonato, PraÃlat, et al.
(Show Context)
Citation Context ...-lazy) cop number of G(n, p). Bonato, Wang, and Pra lat investigated such games in G(n, p) random graphs and in generalizations used to model complex networks with power-law degree distributions (see =-=[12]-=-). From their results it follows that if 2 log n/ √ n ≤ p < 1− ε for some ε > 0, then a.a.s. we have that c(G(n, p)) = Θ(log n/p), (1) so Meyniel’s conjecture holds a.a.s. for such p. In fact, for p =... |

5 | Chasing a fast robber on planar graphs and random graphs
- Alon, Mehrabian
- 2014
(Show Context)
Citation Context ...studied. For example, we may allow a cop to capture the robber from a distance k, where k is a non-negative integer [8, 9], play on edges [13], allow one or both players to move with different speeds =-=[2, 14]-=- or to teleport, allow the robber to capture the cops [10], or make the robber invisible or drunk [17, 18]. See Chapter 8 of [11] for a non-comprehensive survey of variants of Cops and Robbers. We are... |

5 | The robber strikes back
- Bonato, Finbow, et al.
- 2013
(Show Context)
Citation Context ...r from a distance k, where k is a non-negative integer [8, 9], play on edges [13], allow one or both players to move with different speeds [2, 14] or to teleport, allow the robber to capture the cops =-=[10]-=-, or make the robber invisible or drunk [17, 18]. See Chapter 8 of [11] for a non-comprehensive survey of variants of Cops and Robbers. We are interested in slowing the cops down to create a situation... |

5 |
Pra lat, Cops and Robbers playing on edges
- Dudek, Gordinowicz, et al.
(Show Context)
Citation Context ...[3, 6, 7]. A number of variants of Cops and Robbers have been studied. For example, we may allow a cop to capture the robber from a distance k, where k is a non-negative integer [8, 9], play on edges =-=[13]-=-, allow one or both players to move with different speeds [2, 14] or to teleport, allow the robber to capture the cops [10], or make the robber invisible or drunk [17, 18]. See Chapter 8 of [11] for a... |

5 |
The copnumber of a graph is bounded by b 32 genus(G)c+3, in: Categorical perspectives
- Schroeder
- 1998
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Citation Context ...hich G can be embedded without edge crossings. Graphs with genus 0 are the planar graphs, and it was shown in [1] that planar graphs have cop number at most 3. If G has genus g, then it was proved in =-=[26]-=- that c(G) ≤ b3g 2 c+3. In the same paper, it was conjectured that c(G) ≤ g + 3. We conclude the paper with a straightforward asymptotic upper bound on cL for graphs with genus g. We use a well-known ... |

3 |
Variations of Cops and Robber on the hypercube
- Offner, Okajian
(Show Context)
Citation Context ...wing the cops down to create a situation akin to chess, where at most one chess piece can move in a round. Hence, our focus in the present article is a recent variant introduced by Offner and Ojakian =-=[21]-=-, where at most one cop can move in any given round. We refer to this variant, whose formal definition appears in Section 1.1, as Lazy Cops and Robbers; the analogue of the cop number is called the la... |