11 research outputs found
Chasing robbers on random geometric graphs---an alternative approach
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 is called the cop number of . We
focus on , a random geometric graph in which vertices are
chosen uniformly at random and independently from , and two vertices
are adjacent if the Euclidean distance between them is at most . The main
result is that if then the cop number is
with probability that tends to as tends to infinity. The case was
proved earlier and independently in \cite{bdfm}, using a different approach.
Our method provides a tight upper bound for the number of rounds
needed to catch the robber.Comment: 6 page
Visibility Graphs, Dismantlability, and the Cops and Robbers Game
We study versions of cop and robber pursuit-evasion games on the visibility
graphs of polygons, and inside polygons with straight and curved sides. Each
player has full information about the other player's location, players take
turns, and the robber is captured when the cop arrives at the same point as the
robber. In visibility graphs we show the cop can always win because visibility
graphs are dismantlable, which is interesting as one of the few results
relating visibility graphs to other known graph classes. We extend this to show
that the cop wins games in which players move along straight line segments
inside any polygon and, more generally, inside any simply connected planar
region with a reasonable boundary. Essentially, our problem is a type of
pursuit-evasion using the link metric rather than the Euclidean metric, and our
result provides an interesting class of infinite cop-win graphs.Comment: 23 page
Chasing robbers on percolated random geometric graphs
In this paper, we study the vertex pursuit game of \emph{Cops and Robbers}, in which cops try to capture a robber on the vertices of a graph. The minimum number of cops required to win on a given graph is called the cop number of .  We focus on \G(n,r,p), a percolated random geometric graph in which vertices are chosen uniformly at random and independently from , and two vertices are adjacent with probability if the Euclidean distance between them is at most . We present asymptotic results for the game of Cops and Robber played on \G(n,r,p) for a wide range of and
Straight Line Movement in Morphing and Pursuit Evasion
Piece-wise linear structures are widely used to define problems and to represent simplified
solutions in computational geometry. A piece-wise linear structure consists of straight-line
or linear pieces connected together in a continuous geometric environment like 2D or 3D
Euclidean spaces. In this thesis two different problems both with the approach of finding
piece-wise linear solutions in 2D space are defined and studied: straight-line pursuit evasion
and straight-line morphing.
Straight-line pursuit evasion is a geometric version of the famous cops and robbers game
that is defined in this thesis for the first time. The game is played in a simply connected
region in 2D. It is a full information game where the players take turns. The copâs goal
is to catch the robber. In a turn, each player may move any distance along a straight
line as long as the line segment connecting their current location to the new location is
not blocked by the regionâs boundary. We first prove that the cop can always win the
game when the players move on the visibility graph of a simple polygon. We prove this by
showing that the visibility graph of a simple polygon is âdismantlableâ (the known class of
cop-win graphs). Polygon visibility graphs are also shown to be 2-dismantlable. Two other
settings of the game are also studied in this thesis: when the players are free to move on
the infinitely many points inside a simple polygon, and inside a splinegon. In both cases
we show that the cop can always win the game. For the case of polygons, the proposed cop
strategy gives an asymptotically tight linear bound on the number of steps the cop needs
to catch the robber. For the case of splinegons, the cop may need a quadratic number of
steps with the proposed strategy, while our best lower bound is linear.
Straight-line morphing is a type of morphing first defined in this thesis that provides a
nice and smooth transformation between straight-line graph drawings in 2D. In straight-
line morphing, each vertex of the graph moves forward along the line segment connecting
its initial position to its final position. The vertex trajectories in straight-line morphing
are very simple, but because the speed of each vertex may vary, straight-line morphs are
more general than the commonly used âlinear morphsâ where each vertex moves at uniform
speed. We explore the problem of whether an initial planar straight-line drawing of a graph
can be morphed to a final straight-line drawing of the graph using a straight-line morph
that preserves planarity at all times. We prove that this problem is NP-hard even for
the special case where the graph drawing consists of disjoint segments. We then look at
some restricted versions of the straight-line morphing: when only one vertex moves at a
time, when the vertices move one by one to their final positions uninterruptedly, and when
the edges morph one by one to their final configurations in the case of disjoint segments.
Some of the variations are shown to be still NP-complete while some others are solvable
in polynomial time. We conjecture that the class of planar straight-line morphs is as
powerful as the class of planar piece-wise linear straight-line morphs. We also explore
a simpler problem where for each edge the quadrilateral formed by its initial and final
positions together with the trajectories of its two vertices is convex. There is a necessary
condition for this case that we conjecture is also sufficient for paths and cycles
Le jeu de policiers-voleur sur différentes classes de graphes
RĂ©alisĂ© avec le support financier du Conseil de recherches en sciences naturelles et en gĂ©nie du Canada (CRSNG) et du Fonds de Recherche du QuĂ©bec â Nature et technologies (FRQNT).Ce mĂ©moire Ă©tudie le jeu de policiers-voleur et contient trois articles, chacun portant sur une classe de graphes spĂ©cifique.
Dans le premier chapitre, la notation et les définitions de base de la théorie de graphe qui nous serons utiles sont introduites. Bien que chaque article comporte une introduction citant les concepts et résultats pertinents, le premier chapitre de ce mémoire contient aussi une introduction générale au jeu de policiers-voleur et présente certains des résultats majeurs sur ce jeu.
Le deuxiĂšme chapitre contient lâarticle Ă©crit avec Seyyed Aliasghar Hosseini et Peter Bradshaw portant sur le jeu de policiers-voleurs sur les graphes de Cayley abĂ©liens. Nous amĂ©liorons la borne supĂ©rieure sur le cop number de ces graphes en raffinant les mĂ©thodes utilisĂ©es prĂ©cĂ©demment par Hamidoune, Frankl et Bradshaw.
Le troisiĂšme chapitre prĂ©sente lâarticle concernant le cop number des graphes 2K2-libres. Plus prĂ©cisĂ©ment, il est prouvĂ© que 2 policiers peuvent toujours capturer le voleur sur ces graphes, prouvant ainsi la conjecture de Sivaraman et Testa.
Finalement, le quatriĂšme chapitre est lâarticle Ă©crit avec Samuel Yvon et porte sur les graphes qui ont cop number 4. Nous montrons que tous ces graphes ont au moins 19 sommets. En dâautres mots, 3 policiers peuvent toujours capturer le voleur sur tout graphe avec au plus 18 sommets, ce qui rĂ©pond par la nĂ©gative Ă une question de Andreae formulĂ©e en 1986. Un pan important de la preuve est faite par ordinateur; ce mĂ©moire contient donc une annexe comprenant le code utilisĂ©.This thesis studies the game of cops and robbers and consists of three articles, each considering a specific class of graphs.
In the first chapter, notation and basic definitions of graph theory are introduced. Al- though each article has an introduction citing the relevant concepts and results, the first chapter of this thesis also contains a general introduction to the game of cops and robbers and presents some of its major results.
The second chapter contains the paper written with Seyyed Aliasghar Hosseini and Peter Bradshaw on the game of cops and robbers on abelian Cayley graphs. We improve the upper bound on the cop number of these graphs by refining the methods used previously by Hamidoune, Frankl and Bradshaw.
The third chapter presents the paper concerning the cop number of 2K2-free graphs. More precisely, it is proved that 2 cops can always catch the robber on these graphs, proving a conjecture of Sivaraman and Testa.
Finally, the fourth chapter is the paper written with Samuel Yvon which deals with graphs of cop number 4. We show that such graphs have at least 19 vertices. In other words, 3 cops can always catch the robber on any graph with at most 18 vertices, which answers in the negative a question by Andreae from 1986. An important part of the proof is by computer; this thesis thus has an appendix containing the code used
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 â R 2, and r â R +, the vertex set of the geometric graph G(x1,..., xn; r) is the graph on these n points, with xi, xj adjacent when âxi â xj â †r. We prove that c(G) †9 for any connected geometric graph G in R 2 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 G(n, r) denote the probability space of geometric graphs with n vertices chosen uniformly and independently from [0, 1] 2. For G â G(n, r), we show that with high probability (whp), if r â„ C1(log n/n) 1 4, then c(G) †2, and if r â„ C1(log n/n) 1 5, then c(G) = 1 where C1, C2 are absolute constants. Finally, we provide a lower bound near the connectivity regime of G(n, r): if r †1 2 (log2 n/n) 1 2 then c(G)> 1 whp.