Pursuing a fast robber on a graph

AbstractThe 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

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 move t edges at a time, and show that in this variant, the cop number of a d-regular graph with girth larger than 2t+2 is Omega(d^t). By the known upper bounds on the order of cages, this implies that the cop number of a connected n-vertex graph can be as large as Omega(n^{2/3}) if t>1, and Omega(n^{4/5}) if t>3. This improves the Omega(n^{(t-3)/(t-2)}) lower bound of Frieze, Krivelevich, and Loh (Variations on Cops and Robbers, J. Graph Theory, 2011) when 1<t<7. We also conjecture a general upper bound O(n^{t/t+1}) for the cop number in this variant, generalizing Meyniel's conjecture.Comment: 5 page

Variations on Cops and Robbers

We 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 N / \alpha ^{(1-o(1))\sqrt{log_\alpha N}}, where \alpha = 1 + 1/R, thus generalizing the best known upper bound for the classical case R = 1 due to Lu and Peng. We also show that in this case, the cop number of an N-vertex graph can be as large as N^{1 - 1/(R-2)} for finite R, 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 at most O(N(log log N)^2/log N). Our approach is based on expansion.Comment: 18 page

Cops and Robbers is EXPTIME-complete

We investigate the computational complexity of deciding whether k cops can capture a robber on a graph G. In 1995, Goldstein and Reingold conjectured that the problem is EXPTIME-complete when both G and k are part of the input; we prove this conjecture.Comment: v2: updated figures and slightly clarified some minor point

Cop and robber game and hyperbolicity

In this note, we prove that all cop-win graphs G in the game in which the robber and the cop move at different speeds s and s' with s'<s, are \delta-hyperbolic with \delta=O(s^2). We also show that the dependency between \delta and s is linear if s-s'=\Omega(s) and G obeys a slightly stronger condition. This solves an open question from the paper (J. Chalopin et al., Cop and robber games when the robber can hide and ride, SIAM J. Discr. Math. 25 (2011) 333-359). Since any \delta-hyperbolic graph is cop-win for s=2r and s'=r+2\delta for any r>0, this establishes a new - game-theoretical - characterization of Gromov hyperbolicity. We also show that for weakly modular graphs the dependency between \delta and s is linear for any s'<s. Using these results, we describe a simple constant-factor approximation of the hyperbolicity \delta of a graph on n vertices in O(n^2) time when the graph is given by its distance-matrix

Statistical Model Checking for Cops and Robbers Game on Random Graph Models

Cops and robbers problem has been studied over the decades with many variants and applications in graph searching problem. In this work, we study a variant of cops and robbers problem on graphs. In this variant, there are di�erent types of cops and a minimum number of each type of cops are required to catch a robber. We studied this model over various random graph models and analyzed the properties using statistical model checking. To the best of our knowledge this variant of the cops and robber problem has not been studied yet. We have used statistical techniques to estimate the probability of robber getting caught in di�erent random graph models. We seek to compare the ease of catching robbers performing random walk on graphs, especially complex networks. In this work, we report the experiments that yields interesting empirical results. Through the experiments we have observed that it is easier to catch a robber in Barab�asi Albert model than in Erd�os-R�enyi graph model. We have also experimented with k-Regular graphs and real street networks. In our work, the model is framed as the multi-agent based system and we have implemented a statistical model checker, SMCA tool which veri�es agents based systems using statistical techniques. SMCA tool can take the model in JAVA programming language and support Probabilistic - Bounded LTL logic for property specification