A (finite or infinite) graph is called constructible if it may be obtained recursively from the one-point graph by repeatedly adding dominated vertices. In the finite case, the constructible graphs are precisely the cop-win graphs, but for infinite graphs the situation is not well understood. One of our aims in this paper is to give a graph that is cop-win but not constructible. This is the first known such example. We also show that every countable ordinal arises as the rank of some constructible graph, answering a question of Evron, Solomon and Stahl. In addition, we give a finite constructible graph for which there is no construction order whose associated domination map is a homomorphism, answering a question of Chastand, Laviolette and Polat. Lehner showed that every constructible graph is a weak cop win (meaning that the cop can eventually force the robber out of any finite set). Our other main aim is to investigate how this notion relates to the notion of `locally constructible' (every finite graph is contained in a finite constructible subgraph). We show that, under mild extra conditions, every locally constructible graph is a weak cop win. But we also give an example to show that, in general, a locally constructible graph need not be a weak cop win. Surprisingly, this graph may even be chosen to be locally finite. We also give some open problems.
Let $τ(G)$ and $τ_G(e)$ be the number of spanning trees of a connected graph $G$ and the number of spanning trees of $G$ containing edge $e$. The ratio $d_{G}(e)=τ_{G}(e)/τ(G)$ is called the spanning tree edge density of $e$, or simply density of $e$. The maximum density $\mbox{dep}(G)=\max\limits_{e\in E(G)}d_{G}(e)$ is called the spanning tree edge dependence of $G$, or simply dependence of $G$. Given a rational number $p/q\in (0,1)$, if there exists a graph $G$ and an edge $e\in E(G)$ such that $d_{G}(e)=p/q$, then we say the density $p/q$ is constructible. More specially, if there exists a…
We investigate various pursuit-evasion parameters on latin square graphs, including the cop number, metric dimension, and localization number. The cop number of latin square graphs is studied, and for $k$-MOLS$(n),$ bounds for the cop number are given. If $n>(k+1)^2,$ then the cop number is shown to be $k+2.$ Lower and upper bounds are provided for the metric dimension and localization number of latin square graphs. The metric dimension of back-circulant latin squares shows that the lower bound is close to tight. Recent results on covers and partial transversals of latin squares provide the up…
This paper solves a pursuit-evasion problem in which a prince must find a princess who is constrained to move on each day from one vertex of a finite graph to another. Unlike the related and much studied `Cops and Robbers Game', the prince has no knowledge of the position of the princess; he may, however, visit any single room he wishes on each day. We characterize the graphs for which the prince has a winning strategy, and determine, for each such graph, the minimum number of days the prince requires to guarantee to find the princess.
We compare two kinds of pursuit-evasion games played on graphs. In Cops and Robbers, the cops can move strategically to adjacent vertices as they please, while in a new variant, called deterministic Zombies and Survivors, the zombies (the counterpart of the cops) are required to always move towards the survivor (the counterpart of the robber). The cop number of a graph is the minimum number of cops required to catch the robber on that graph; the zombie number of a graph is the minimum number of zombies required to catch the survivor on that graph. We answer two questions from the 2016 paper of…
Cops and robbers is a game between two players, where one tries to catch the other by moving along the edges of a graph. It is well known that on a finite graph the cop has a winning strategy if and only if the graph is constructible and that finiteness is necessary for this result. We propose the notion of weakly cop-win graphs, a winning criterion for infinite graphs which could lead to a generalisation. In fact, we generalise one half of the result, that is, we prove that every constructible graph is weakly cop-win. We also show that a similar notion studied by Chastand et al. (which they a…
We examine a dynamic model for the disruption of information flow in hierarchical social networks by considering the vertex-pursuit game Seepage played in directed acyclic graphs (DAGs). In Seepage, agents attempt to block the movement of an intruder who moves downward from the source node to a sink. The minimum number of such agents required to block the intruder is called the green number. We propose a generalized stochastic model for DAGs with given expected total degree sequence. Seepage and the green number is analyzed in stochastic DAGs in both the cases of a regular and power law degree…
We study zombies and survivor, a variant of the game of cops and robber on graphs. In this variant, the single survivor plays the role of the robber and attempts to escape from the zombies that play the role of the cops. The zombies are restricted, on their turn, to always follow an edge of a shortest path towards the survivor. Let $z(G)$ be the smallest number of zombies required to catch the survivor on a graph $G$ with $n$ vertices. We show that there exist outerplanar graphs and visibility graphs of simple polygons such that $z(G) = Θ(n)$. We also show that there exist maximum-degree-$3$ o…
We consider a class of pursuit-evasion problems where an evader enters a directed acyclic graph and attempts to reach one of the terminal nodes. A pursuer enters the graph at a later time and attempts to capture the evader before it reaches a terminal node. The pursuer can only obtain information about the evader's path via sensors located at each node in the graph; the sensor measurements are either green or red (indicating whether or not the evader has passed through that node). We first show that it is NP-hard to determine whether the pursuer can enter with some nonzero delay and still be g…
It is proved that in Godel's constructible universe, for every infinite successor cardinal k, there exist graphs G and H of size and chromatic number k, for which the tensor product graph (G x H) is countably chromatic.
These are some personal notes about the pursuit game of Cops and Robbers that I made starting in 2007. More old and new problems (and some solutions) will be added in future versions of these notes.
The game of Cops and Robbers is a well known pursuit-evasion game played on graphs. It has been proved \cite{bounded_degree} that cubic graphs can have arbitrarily large cop number $c(G)$, but the known constructions show only that the set $\{c(G) \mid G \text{ cubic}\}$ is unbounded. In this paper we prove that there are arbitrarily large subcubic graphs $G$ whose cop number is at least $n^{1/2-o(1)}$ where $n=|V(G)|$. We also show that proving Meyniel's conjecture for graphs of bounded degree implies a weak Meyniel's conjecture for all graphs.
A gambler moves on the vertices $1, \ldots, n$ of a graph using the probability distribution $p_{1}, \ldots, p_{n}$. A cop pursues the gambler on the graph, only being able to move between adjacent vertices. What is the expected number of moves that the gambler can make until the cop catches them? Komarov and Winkler proved an upper bound of approximately $1.97n$ for the expected capture time on any connected $n$-vertex graph when the cop does not know the gambler's distribution. We improve this upper bound to approximately $1.95n$ by modifying the cop's pursuit algorithm.
We consider the pursuit and evasion game on finite, connected, undirected graphs known as cops and robbers. Meyniel conjectured that for every graph on n vertices a rootish number of cops can win the game. We prove that this holds up to a log(n) factor for random graphs G(n,p) if p is not very small, and this is close to be tight unless the graph is very dense. We analyze the area-defending strategy (used by Aigner in case of planar graphs) and show examples where it can not be too efficient.
In the ordinary version of the pursuit-evasion game "cops and robbers", a team of cops and a robber occupy vertices of a graph and alternately move along the graph's edges, with perfect information about each other. If a cop lands on the robber, the cops win; if the robber can evade the cops indefinitely, he wins. In the variant "lazy cops and robbers", the cops may only choose one member of their squad to make a move when it's their turn. The minimum number of cops (respectively lazy cops) required to catch the robber is called the "cop number" (resp. "lazy cop number") of G and is denoted $c…
We study a game of pursuit and evasion introduced by Seager in 2012, in which a cop searches the robber from outside the graph, using distance queries. A graph on which the cop wins is called locatable. In her original paper, Seager asked whether there exists a characterisation of the graph property of locatable graphs by either forbidden or forbidden induced subgraphs, both of which we answer in the negative. We then proceed to show that such a characterisation does exist for graphs of diameter at most 2, stating it explicitly, and note that this is not true for higher diameter. Exploring a d…
Betti numbers for cochordal zero-divisor graphs of commutative rings
School of Mathematics and Statistics, Shandong University of Technology（山东理工大学数学与统计学院）；School of Mechanical Engineering, Shandong University of Technology（山东理工大学机械工程学院）
The game of cops and robber is a turn based vertex pursuit game played on a connected graph between a team of cops and a single robber. The cops and the robber move alternately along the edges of the graph. We say the team of cops win the game if a cop and the robber are at the same vertex of the graph. The minimum number of cops required to win in each component of a graph is called the cop number of the graph. Sivaraman [Discrete Math. 342(2019), pp. 2306-2307] conjectured that for every $t\geq 5$, the cop number of a connected $P_t$-free graph is at most $t-3$, where $P_t$ denotes a path on…
We study the vertex pursuit game of \emph{Cops and Robbers}, in which cops try to capture a robber on the vertices of the graph. The minimum number of cops required to win on a given graph $G$ is called the cop number of $G$. We focus on $G_{d}(n,r)$, a random geometric graph in which $n$ vertices are chosen uniformly at random and independently from $[0,1]^d$, and two vertices are adjacent if the Euclidean distance between them is at most $r$. The main result is that if $r^{3d-1}>c_d \frac{\log n}{n}$ then the cop number is $1$ with probability that tends to $1$ as $n$ tends to infinity. The …
In a pursuit evasion game on a finite, simple, undirected, and connected graph $G$, a first player visits vertices $m_1,m_2,\ldots$ of $G$, where $m_{i+1}$ is in the closed neighborhood of $m_i$ for every $i$, and a second player probes arbitrary vertices $c_1,c_2,\ldots$ of $G$, and learns whether or not the distance between $c_{i+1}$ and $m_{i+1}$ is at most the distance between $c_i$ and $m_i$. Up to what distance $d$ can the second player determine the position of the first? For trees of bounded maximum degree and grids, we show that $d$ is bounded by a constant. We conjecture that $d=O(\l…
The game of cops and robbers is a pursuit game on graphs where a set of agents, called the cops try to get to the same position of another agent, called the robber. Cops and robbers has been studies on several classes of graphs including geometrically represented graphs. For example, it has been shown that string graphs, including geometric graphs, have cop number at most 15. On the other hand, little is known about geometric graphs of any cop number less than 15 and there is only one example of a geometric graph of cop number three that has as many as 1440 vertices. In this paper we present a…
The game of Cops and Robber is traditionally played on a finite graph. The purpose of this note is to introduce and analyze the game that is played on an arbitrary geodesic space. The game is defined in such a way that it preserves the beauty and power of discrete games played on graphs and also keeps the specialties of the pursuit-evasion games played on polyhedral complexes. It is shown that the game can be approximated by finite games of discrete type and as a consequence a min-max theorem is obtained.
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. In this work we introduce a strengthening of one of these models by considering the probability that the remaining graph is not just connected but also cop-win. A graph is cop-win if one cop can guarantee capture of a fleeing robber in the well-studied pursuit-evasion game of Cops and Robber. More precisely, for a graph $G$ with nodes that are operational independently with prob…
