Number Of Cops Research Articles

A Note on the Cops and Robber Game on Graphs Embedded in Non-Orientable Surfaces

The Cops and Robber game is played on undirected finite graphs. A number of cops and one robber are positioned on vertices and take turns in sliding along edges. The cops win if they can catch the robber. The minimum number of cops needed to win on a graph is called its cop number. It is known that the cop number of a graph embedded on a surface $X$ of genus $g$ is at most $3g/2 + 3$, if $X$ is orientable (Schroeder 2004), and at most $2g+1$, otherwise (Nowakowski & Schroeder 1997). We improve the bounds for non-orientable surfaces by reduction to the orientable case using covering spaces. As corollaries, using Schroeder's results, we obtain the following: the maximum cop number of graphs embeddable in the projective plane is 3; the cop number of graphs embeddable in the Klein Bottle is at most 4, and an upper bound is $3g/2 + 3/2$ for all other $g$.

Cops and Robbers on Graphs Based on Designs

AbstractWe investigate the cop number of graphs based on combinatorial designs. Incidence graphs, point graphs, and block intersection graphs are studied, with an emphasis on finding families of graphs with large cop number. We generalize known results on Meyniel extremal families by supplying bounds on the incidence graphs of transversal designs, certain G‐designs, and BIBDs with Families of graphs with diameter 2, C4‐free, and with unbounded chromatic number are described with the conjectured asymptotically maximum cop number.

Cops and Robber Game with a Fast Robber on Expander Graphs and Random Graphs

We consider a variant of the Cops and Robber game, in which the robber has unbounded speed, i.e., can take any path from her vertex in her turn, but she is not allowed to pass through a vertex occupied by a cop. Let \({c_{\infty}(G)}\) denote the number of cops needed to capture the robber in a graph G in this variant. We characterize graphs G with c ∞(G) = 1, and give an \({O( \mid V(G)\mid^2)}\) algorithm for their detection. We prove a lower bound for c ∞ of expander graphs, and use it to prove three things. The first is that if \({np \geq 4.2 {\rm log}n}\) then the random graph \({G= \mathcal{G}(n, p)}\) asymptotically almost surely has \({\eta_{1}/p \leq \eta_{2}{\rm log}(np)/p}\) , for suitable positive constants \({\eta_{1}}\) and \({\eta_{2}}\) . The second is that a fixed-degree random regular graph G with n vertices asymptotically almost surely has \({c_{\infty}(G) = \Theta(n)}\) . The third is that if G is a Cartesian product of m paths, then \({n/4km^2 \leq c_{\infty}(G) \leq n/k}\) , where \({n = \mid V(G)\mid}\) and k is the number of vertices of the longest path.

Cops and Robbers on Geometric Graphs

Cops and robbers is a turn-based pursuit game played on a graph G. One robber is pursued by a set of cops. In each round, these agents move between vertices along the edges of the graph. The cop number c(G) denotes the minimum number of cops required to catch the robber in finite time. We study the cop number of geometric graphs. For points x1, . . ., xn ∈ ℝ2, and r ∈ ℝ+, the vertex set of the geometric graph G(x1, . . ., xn; r) is the graph on these n points, with xi, xj adjacent when ∥xi − xj∥ ≤ r. We prove that c(G) ≤ 9 for any connected geometric graph G in ℝ2 and we give an example of a connected geometric graph with c(G) = 3. We improve on our upper bound for random geometric graphs that are sufficiently dense. Let (n,r) denote the probability space of geometric graphs with n vertices chosen uniformly and independently from [0,1]2. For G ∈ (n,r), we show that with high probability (w.h.p.), if r ≥ K1 (log n/n)1/4 then c(G) ≤ 2, and if r ≥ K2(log n/n)1/5 then c(G) = 1, where K1, K2 > 0 are absolute constants. Finally, we provide a lower bound near the connectivity regime of (n,r): if r ≤ K3 log n/ then c(G) > 1 w.h.p., where K3 > 0 is an absolute constant.

Meyniel’s conjecture on the cop number: A survey

Meyniel’s conjecture is one of the deepest open problems on the cop number of a graph. It states that for a connected graph G of order n, c(G) = O( √ n). While largely ignored for over 20 years, the conjecture is receiving increasing attention. We survey the origins of and recent developments towards the solution of the conjecture. We present some new results on Meyniel extremal families containing graphs of order n satisfying c(G) ≥ d√n, where d is a constant.

On Meyniel's conjecture of the cop number

AbstractMeyniel conjectured that the cop number c(G) of any connected graph G on n vertices is at most for some constant C. In this article, we prove Meyniel's conjecture in special cases that G has diameter 2 or G is a bipartite graph of diameter 3. For general connected graphs, we prove , improving the best previously known upper‐bound O(n/ lnn) due to Chiniforooshan.

Association between thyroid hormone levels, the number of circulating osteoprogenitor cells, and bone mineral density in euthyroid postmenopausal women

In postmenopausal women, an association between reduced bone mineral density (BMD) and increased number of circulating osteoprogenitor cells (COPs) has been found. Although an increased thyroid function is associated with BMD, thyroid hormones stimulate osteoblast function in vitro. We investigated whether thyroid hormones within the reference range were correlated with the number of COPs and stimulate mineralization in vitro. The number of COPs, defined as CD34+/alkaline phosphatase (AP)+ or CD34+/osteocalcin (OCN)+ cells, was quantified by fluorescence-activated cell sorting (FACS) analysis in 150 euthyroid postmenopausal women. Participants underwent measurement of serum free thyroxine (FT4), thyroid-stimulating hormone levels, and femur BMD. CD34+ cells were isolated from healthy volunteers irrespective of AP or OCN expression, and the effect of triiodothyronine (0.5-10 pmol/L)) on their ability to form mineralized nodules in vitro was studied. The number of COPs was highest among women with high-normal FT4 levels (>1.09 ng/dL). The FT4 levels were correlated positively with circulating log-CD34+/AP+ (r = 0.32, P < .001) and log-CD34/OCN+ cells (r = 0.36, P < .001) and inversely with total femur BMD (r = −0.17, P = .036) but not with femoral neck BMD. In a multivariate analysis, the FT4 levels were positively correlated with the number of COPs, independent of age and BMD. The ability of CD34+ cells to form mineralized nodules increased after exposure from low up to high-normal triiodothyronine concentrations (P for trend = .003). Among euthyroid postmenopausal women, high-normal FT4 levels are correlated with an increased number of circulating immature osteoprogenitor cells and a very mild BMD reduction. Exposure of CD34+ cells to physiological triiodothyronine concentrations stimulates mineralization in vitro.

Lower Bounds for the Cop Number when the Robber is Fast

We consider a variant of the Cops and Robbers game where the robber can movetedges at a time, and show that in this variant, the cop number of ad-regular graph with girth larger than 2t+2 is Ω(dt). By the known upper bounds on the order of cages, this implies that the cop number of a connectedn-vertex graph can be as large as Ω(n2/3) ift≥ 2, and Ω(n4/5) ift≥ 4. This improves the Ω($n^{\frac{t-3}{t-2}}$) lower bound of Frieze, Krivelevich and Loh (Variations on cops and robbers,J. Graph Theory, to appear) when 2 ≤t≤ 6. We also conjecture a general upper boundO(nt/t+1) for the cop number in this variant, generalizing Meyniel's conjecture.

Variations on cops and robbers

AbstractWe consider several variants of the classical Cops and Robbers game. We treat the version where the robber can move R≥1 edges at a time, establishing a general upper bound of , where α = 1 + 1/R, thus generalizing the best known upper bound for the classical case R = 1 due to Lu and Peng, and Scott and Sudakov. We also show that in this case, the cop number of an n‐vertex graph can be as large as n1 − 1/(R − 2) for finite R≥5, but linear in n if R is infinite. For R = 1, we study the directed graph version of the problem, and show that the cop number of any strongly connected digraph on n vertices is O(n(loglogn)2/logn). Our approach is based on expansion. © 2011 Wiley Periodicals, Inc. J Graph Theory.

On a Generalization of Meyniel's Conjecture on the Cops and Robbers Game

We consider a variant of the Cops and Robbers game where the robber can move $s$ edges at a time, and show that in this variant, the cop number of a connected graph on $n$ vertices can be as large as $\Omega(n^\frac{s}{s+1})$. This improves the $\Omega(n^{\frac{s-3}{s-2}})$ lower bound of Frieze et al., and extends the result of the second author, which establishes the above bound for $s=2,4$.

Searching for a Visible, Lazy Fugitive

Graph searching problems are described as games played on graphs, between a set of cops and a fugitive. Variants of the game restrict the abilities of the cops and the fugitive and the corresponding search numbers (the least number of cops that have a winning strategy) are related to several well-known parameters in graph theory. We study the case where the fugitive is visible (the cops’ strategy can take into account his current position) and lazy (he moves only when the cops move to his position). Our results are stated and proven in a general setting where the fugitive’s speed (i.e., the lengths of paths he can move along) can be unbounded or bounded by some constant. We give a min-max characterization of the corresponding parameters, which we show to be computable in polynomial time for fugitives with unbounded speed and speed at most 3 and to be nondeterministic polynomial-time complete (NP-complete) for all other finite speeds. This is in contrast to the other standard versions of the game, where the parameters corresponding to fugitives with unbounded speed are NP-complete. Several consequences of our results are also discussed.

A Bound for the Cops and Robbers Problem

‡ Abstract. In this short paper we study the game of cops and robbers, which is played on the vertices of some fixed graph G. Cops and a robber are allowed to move along the edges of G, and the goal of cops is to capture the robber. The cop number cðGÞ of G is the minimum number of cops required to win the game. Meyniel conjectured a long time ago that Oð ffiffiffi p Þ cops are enough for any connected G on n vertices. Improving several previous results, we prove that the cop number of an n-vertex graph is at most n2 −ð1þoð1ÞÞ ffiffiffiffiffiffiffiffi log n p .A similar result independently and slightly before us was also obtained by Lu and Peng.

Large classes of infinite k‐cop‐win graphs

AbstractWhile finite cop‐win finite graphs possess a good structural characterization, none is known for infinite cop‐win graphs. As evidence that such a characterization might not exist, we provide as large as possible classes of infinite graphs with finite cop number. More precisely, for each infinite cardinal κ and each positive integer k, we construct 2κ non‐isomorphic k‐cop‐win graphs satisfying additional properties such as vertex‐transitivity, or having universal endomorphism monoid and automorphism group. © 2010 Wiley Periodicals, Inc. J Graph Theory 65: 334–342, 2010

Chasing robbers on random graphs: Zigzag theorem

In this paper, we study the vertex pursuit game of Cops and Robbers where 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 the cop number of G. We present asymptotic results for the game of Cops and Robber played on a random graph G(n,p) for a wide range of p = p(n). It has been shown that the cop number as a function of an average degree forms an intriguing zigzag shape. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 2010

Cops and Robbers from a distance

Cops and Robbers is a pursuit and evasion game played on graphs that has received much attention. We consider an extension of Cops and Robbers, distance k Cops and Robbers, where the cops win if at least one of them is of distance at most k from the robber in G . The cop number of a graph G is the minimum number of cops needed to capture the robber in G . The distance k analogue of the cop number, written c k ( G ) , equals the minimum number of cops needed to win at a given distance k . We study the parameter c k from algorithmic, structural, and probabilistic perspectives. We supply a classification result for graphs with bounded c k ( G ) values and develop an O ( n 2 s + 3 ) algorithm for determining if c k ( G ) ≤ s for s fixed. We prove that if s is not fixed, then computing c k ( G ) is NP-hard. Upper and lower bounds are found for c k ( G ) in terms of the order of G . We prove that ( n k ) 1 / 2 + o ( 1 ) ≤ c k ( n ) = O ( n log ( 2 n k + 1 ) log ( k + 2 ) k + 1 ) , where c k ( n ) is the maximum of c k ( G ) over all n -vertex connected graphs. The parameter c k ( G ) is investigated asymptotically in random graphs G ( n , p ) for a wide range of p = p ( n ) . For each k ≥ 0 , it is shown that c k ( G ) as a function of the average degree d ( n ) = p n forms an intriguing zigzag shape.

Parameterized pursuit-evasion games

We study the parameterized complexity of four variants of pursuit-evasion on graphs: Seeded Pursuit Evasion, Short Seeded Pursuit Evasion, Directed Pursuit Evasion and Short Directed Pursuit Evasion. Both Seeded Pursuit Evasion and Short Seeded Pursuit Evasion are played on undirected graphs with given starting positions for both the cops and the robber. Directed Pursuit Evasion and its short variant are played on directed graphs, with the players free to choose their starting positions. We show for Seeded Pursuit Evasion and Directed Pursuit Evasion that finding a winning strategy for the cops is AW[*]-hard when we parameterize by the number of cops. Further, we show that the short ( k -move) variants of these problems ( Short Seeded Pursuit Evasion and Short Directed Pursuit Evasion) are AW[*]-complete when we parameterize by both the number of cops and turns.

Cop–robber guarding game with cycle robber-region

A cop–robber guarding game is played by the robber-player and the cop-player on a graph G with a partition R and C of the vertex set. The robber-player starts the game by placing a robber (her pawn) on a vertex in R , followed by the cop-player who places a set of cops (her pawns) on some vertices in C . The two players take turns in moving their pawns to adjacent vertices in G . The cop-player moves the cops within C to prevent the robber-player from moving the robber to any vertex in C . The robber-player wins if it gets a turn to move the robber onto a vertex in C on which no cop situates, and the cop-player wins otherwise. The problem is to find the minimum number of cops that admits a winning strategy to the cop-player. It has been shown that the problem is polynomially solvable if R induces a path, whereas it is NP-complete if R induces a tree. In this paper, we show that the problem remains NP-complete even if R induces a 3-star and that the problem is polynomially solvable if R induces a cycle.

How to Guard a Graph?

We initiate the study of the algorithmic foundations of games in which a set of cops has to guard a region in a graph (or digraph) against a robber. The robber and the cops are placed on vertices of the graph; they take turns in moving to adjacent vertices (or staying). The goal of the robber is to enter the guarded region at a vertex with no cop on it. The problem is to find the minimum number of cops needed to prevent the robber from entering the guarded region. The problem is highly non-trivial even if the robber’s or the cops’ regions are restricted to very simple graphs. The computational complexity of the problem depends heavily on the chosen restriction. In particular, if the robber’s region is only a path, then the problem can be solved in polynomial time. When the robber moves in a tree (or even in a star), then the decision version of the problem is NP-complete. Furthermore, if the robber is moving in a directed acyclic graph, the problem becomes PSPACE-complete.

Pursuing a fast robber on a graph

The Cops and Robbers game as originally defined independently by Quilliot and by Nowakowski and Winkler in the 1980s has been much studied, but very few results pertain to the algorithmic and complexity aspects of it. In this paper we prove that computing the minimum number of cops that are guaranteed to catch a robber on a given graph is NP-hard and that the parameterized version of the problem is W[2]-hard; the proof extends to the case where the robber moves s time faster than the cops. We show that on split graphs, the problem is polynomially solvable if s = 1 but is NP-hard if s = 2 . We further prove that on graphs of bounded cliquewidth the problem is polynomially solvable for s ≤ 2 . Finally, we show that for planar graphs the minimum number of cops is unbounded if the robber is faster than the cops.

A better bound for the cop number of general graphs

AbstractIn this note, we prove that the cop number of any n‐vertex graph G, denoted by ${{c}}({{G}})$, is at most ${{O}}\big({{{n}}\over {{\rm lg}} {{n}}}\big)$. Meyniel conjectured ${{c}}({{G}})={{O}}(\sqrt{{{n}}})$. It appears that the best previously known sublinear upper‐bound is due to Frankl, who proved ${{c}}({{G}})\leq ({{1}}+ {{o}}({{1}})){{{n}}{{\rm lg}}{{\rm lg}} {{n}}\over {{\rm lg}} {{n}}}$. © 2008 Wiley Periodicals, Inc. J Graph Theory 58: 45–48, 2008