Results 1 
6 of
6
Cops and robber game with a fast robber
, 2011
"... I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ii Graph searching problems are described as games played ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
(Show Context)
I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ii Graph searching problems are described as games played on graphs, between a set of searchers and a fugitive. Variants of the game restrict the abilities of the searchers and the fugitive and the corresponding search number (the least number of searchers that have a winning strategy) is related to several wellknown parameters in graph theory. One popular variant is called the Cops and Robber game, where the searchers (cops) and the fugitive (robber) move in rounds, and in each round they move to an adjacent vertex. This game, defined in late 1970’s, has been studied intensively. The most famous open problem is Meyniel’s conjecture, which states that the cop number (the minimum number of cops that can always capture the robber) of a connected graph on n vertices is O( n). We consider a version of the Cops and Robber game, where the robber is faster than the cops, but is not allowed to jump over the cops. This version was first studied in 2008. We show that when the robber has speed s, the cop number of a connected nvertex graph can be as large as Ω(ns/s+1). This improves the Ω(n s−3 s−2) lower bound of Frieze, Krivelevich, and Loh (Variations on Cops and Robbers, J. Graph Theory, to appear). We also conjecture a general upper bound O(ns/s+1) for the cop number, generalizing Meyniel’s conjecture. Then we focus on the version where the robber is infinitely fast, but is again not allowed to jump over the cops. We give a mathematical characterization for graphs with cop number one. For a graph with treewidth tw and maximum degree ∆, we prove the cop number is between tw+1 ∆+1 and tw + 1. Using this we show that the cop number of the
Cops and robbers from a distance
 Theor. Comput. Sci
"... Abstract. 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 ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
(Show Context)
Abstract. 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 ck(G), equals the minimum number of cops needed to win at a given distance k. We study the parameter ck from algorithmic, structural, and probabilistic perspectives. We supply a classification result for graphs with bounded ck(G) values and develop an O(n2s+3) algorithm for determining if ck(G) ≤ s for s fixed. We prove that if s is not fixed, then computing ck(G) is NPhard. Upper and lower bounds are found for ck(G) in terms of the order of G. We prove that n
Cops and Robber Game with a Fast Robber on Interval, Chordal, and Planar Graphs
"... ar ..."
(Show Context)
A NOTE ON BOUNDS FOR THE COP NUMBER USING TREE DECOMPOSITIONS
"... Abstract. In this short note, we supply a new upper bound on the cop number in terms of tree decompositions. Our results in some cases extend a previously derived bound on the cop number using treewidth. 1. ..."
Abstract
 Add to MetaCart
(Show Context)
Abstract. In this short note, we supply a new upper bound on the cop number in terms of tree decompositions. Our results in some cases extend a previously derived bound on the cop number using treewidth. 1.
Zerovisibility cops & robber and the pathwidth of a graph
"... We examine the zerovisibility cops and robber graph searching model, which differs from the classical cops & robber game in one way: the robber is invisible. We show that this model is not monotonic. We also provide bounds on both the zerovisibility copnumber and monotonic zerovisibility copn ..."
Abstract
 Add to MetaCart
(Show Context)
We examine the zerovisibility cops and robber graph searching model, which differs from the classical cops & robber game in one way: the robber is invisible. We show that this model is not monotonic. We also provide bounds on both the zerovisibility copnumber and monotonic zerovisibility copnumber in terms of the pathwidth. 1