The Role of Visibility in Pursuit/Evasion Games

Kehagias, Athanasios; Mitsche, Dieter; Prałat, Paweł

doi:10.3390/robotics3040371

Cited by 12 publications

(2 citation statements)

References 32 publications

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“…It is well known that the winner of a reachability game played on a given directed graph G can be established in polynomial time [3,9]. For more information regarding cops and robbers/pursuit-evasion games, as well as their connection to reachability games, we refer the reader to [3,5,8,9,11,12,17].…”

Section: Related Workmentioning

confidence: 99%

A Game of Cops and Robbers on Graphs with Periodic Edge-Connectivity

Erlebach

Spooner

2020

SOFSEM 2020: Theory and Practice of Computer Science

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This paper considers a game in which a single cop and a single robber take turns moving along the edges of a given graph G. If there exists a strategy for the cop which enables it to be positioned at the same vertex as the robber eventually, then G is called cop-win, and robber-win otherwise. We study this classical combinatorial game in a novel context, broadening the class of potential game arenas to include the edge-periodic graphs. These are graphs with an infinite lifetime comprised of discrete time steps such that each edge e is assigned a bit pattern of length l e , with a 1 in the i-th position of the pattern indicating the presence of edge e in the i-th step of each consecutive block of l e steps. Utilising the already-developed framework of reachability games, we extend existing techniques to obtain, amongst other results, an O(LCM(L) · n 3 ) upper bound on the time required to decide if a given n-vertex edge-periodic graph G τ is cop or robber win as well as compute a strategy for the winning player (here, L is the set of all edge pattern lengths l e , and LCM(L) denotes the least common multiple of the set L). Separately, turning our attention to edge-periodic cycle graphs, we give proof of a 2 · l · LCM(L) upper bound on the length required by any edge-periodic cycle to ensure that it is robber win, where l = 1 if LCM(L) ≥ 2 · max L, and l = 2 otherwise. Furthermore, we provide lower bound constructions in the form of cop-win edge-periodic cycles: one with length 1.5 · LCM(L) in the l = 1 case and one with length 3 · LCM(L) in the l = 2 case.

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Section: Related Workmentioning

confidence: 99%

A Game of Cops and Robbers on Graphs with Periodic Edge-Connectivity

Erlebach

Spooner

2020

SOFSEM 2020: Theory and Practice of Computer Science

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“…The prototypical game of this family is, of course, the classic CR game introduced in [19,21]; for an extensive recent overview of the subject see the book [1]. While the connection between graph pursuit games and game theory is a natural one, relatively little has been published on it [12,13,14,16]. In particular, we are aware of only one previous publication (by ourselves) on graph pursuit games involving selfish pursuers [14].…”

Section: Introductionmentioning

confidence: 99%

Selfish cops and active robber: Multi-player pursuit evasion on graphs

Konstantinidis¹,

Kehagias²

2019

Theoretical Computer Science

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We introduce and study the game of "Selfish Cops and Active Robber" (SCAR) which can be seen as an multiplayer variant of the "classic" two-player Cops and Robbers (CR) game. In classic CR all cops are controlled by a single player, who has no preference over which cop captures the robber. In SCAR, on the other hand, each of N − 1 cops is controlled by a separate player, and a single robber is controlled by the N -th player; and the capturing cop player receives a higher reward than the non-capturing ones. Consequently, SCAR is an N -player pursuit game on graphs, in which each cop player has an increased motive to be the one who captures the robber. The focus of our study is the existence and properties of SCAR Nash Equilibria (NE). In particular, we prove that SCAR always has one NE in deterministic positional strategies and (for N ≥ 3) another in deterministic nonpositional strategies. Furthermore, we study conditions which, at equilibrium, guarantee either capture or escape of the robber and show that (because of the antagonism between the "selfish" cop players) the robber may, in certain SCAR configurations, be captured later than he would be in classic CR, or even not captured at all. Finally we define the selfish cop number of a graph and study its connection to the classic cop number. * The authors thank Steve Alpern and Pascal Schweitzer for several useful and inspiring discussions.

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