Cops and robbers in graphs with large girth and Cayley graphs
Discrete Applied Mathematics
On bridged graphs and cop-win graphs
Journal of Combinatorial Theory Series A
Regular Article: On the Cop Number of a Graph
Advances in Applied Mathematics
The complexity of pursuit on a graph
Theoretical Computer Science
The art of computer programming, volume 1 (3rd ed.): fundamental algorithms
The art of computer programming, volume 1 (3rd ed.): fundamental algorithms
A better bound for the cop number of general graphs
Journal of Graph Theory
An annotated bibliography on guaranteed graph searching
Theoretical Computer Science
Pursuing a fast robber on a graph
Theoretical Computer Science
Chasing robbers on random graphs: Zigzag theorem
Random Structures & Algorithms
Variations on Cops and Robbers
Journal of Graph Theory
On Meyniel's conjecture of the cop number
Journal of Graph Theory
Hi-index | 5.23 |
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 (nk)^1^/^2^+^o^(^1^)@?c"k(n)=O(nlog(2nk+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)=pn forms an intriguing zigzag shape.