相关论文: Notes on Cops and Robber game on graphs

We consider a pursuit-evasion game that describes the process of extinguishing a fire burning on the nodes of an undirected graph. We denote the minimum number of firefighters required by ffn(G) and provide almost sharp bounds to this graph…

Achievement systems have been actively adopted in gaming platforms to maintain players' interests. Among them, trophies in PlayStation games are one of the most successful achievement systems. While the importance of trophy design has been…

Pursuit-evasion games are ubiquitous in nature and in an artificial world. In nature, pursuer(s) and evader(s) are intelligent agents that can learn from experience, and dynamics (i.e., Newtonian or Lagrangian) is vital for the pursuer and…

We investigate the Dots and Boxes game, also known as ``Strings and Coins,'' for certain specific families of graphs. These include complete graphs, wheel graphs, and friendship graphs.

The cop throttling number of a graph, introduced in 2018 by Breen et al., optimizes the balance between the number of cops used and the number of rounds required to catch the robber in a game of Cops and Robbers. In 2019, Cox and Sanaei…

Introduced by Harris, Insko, Prieto Langarica, Stoisavljevic, and Sullivan, the \emph{tipsy cop and drunken robber} is a variant of the cop and robber game on graphs in which the robber simply moves randomly along the graph, while the cop…

We start with the well-known game below: Two players hold a sheet of paper to their forehead on which a positive integer is written. The numbers are consecutive and each player can only see the number of the other one. In each time step,…

In the $(s,d)$-spy game over a graph, introduced by Cohen et al. in 2016, one spy and $k$ guards occupy vertices of a graph and, at each turn, each guard may move along one edge and the spy may move along at most $s$ edges. The guards win…

The localization game is a two player combinatorial game played on a graph $G=(V,E)$. The cops choose a set of vertices $S_1 \subseteq V$ with $|S_1|=k$. The robber then chooses a vertex $v \in V$ whose location is hidden from the cops, but…

In the game of \emph{cops and robbers} on a graph $G = (V,E)$, $k$ cops try to catch a robber. On the cop turn, each cop may move to a neighboring vertex or remain in place. On the robber's turn, he moves similarly. The cops win if there is…

A \emph{periodic graph} ${\cal G}=(G_0, G_1, G_2, \dots)$ with period $p$ is an infinite periodic sequence of graphs $G_i = G_{i + p} = (V,E_i)$, where $i \geq 0$. The graph $G=(V,\cup_i E_i)$ is called the footprint of ${\cal G}$.…

"Zombies and Survivor" is a variant of the well-studied game of "Cops and Robber" where the zombies (cops) can only move closer to the survivor (robber). We consider the deterministic version of the game where a zombie can choose their path…

We study a pursuit-evasion problem which can be viewed as an extension of the keep-away game. In the game, pursuer(s) will attempt to intersect or catch the evader, while the evader can visit a fixed set of locations, which we denote as the…

We present two efficient algorithms that compute the optimal strategy for cop in the game of Cop v.s. Gambler where the gambler's strategy is not optimal but known to the cop. The first algorithm is analogous to Bellman-Ford algorithm for…

We study the localization game on dense random graphs. In this game, a {\em cop} $x$ tries to locate a {\em robber} $y$ by asking for the graph distance of $y$ from every vertex in a sequence of sets $W_1,W_2,\ldots,W_\ell$. We prove high…

We consider a Cops-and-Robber game played on the subsets of an $n$-set. The robber starts at the full set; the cops start at the empty set. On each turn, the robber moves down one level by discarding an element, and each cop moves up one…

We introduce and analyze the Walker-Breaker game, a variant of Maker-Breaker games where Maker is constrained to choose edges of a walk or path in a given graph G, with the goal of visiting as many vertices of the underlying graph as…

This paper proposes a new perspective on the conventional planar target tracking problem. One evader and one pursuer are considered in the dynamics. In the planar tracking, pursuer has the ability to measure the position and the velocity…

The game of cops and robber has been studied for many years. Denoting $\mathsf{Forb}(4K_1)$ to be the family of all graphs that contain no induced subgraph isomorphic to $4K_1$ (e.g., with independence number less than $4$), we prove that…

Shmuel Gal and Jerome Casas have recently introduced a game theoretic model that combines search and pursuit by a predator for a prey animal. The prey (hider) can hide in a finite number of locations. The predator (searcher) can inspect any…