Ikeda et al. Pacific Journal of Mathematics for Industry
ORIGINAL Open Access
A property of random walks on a cycle graph
, Yasunari Fukai
and Yoshihiro Mizoguchi
We analyze the Hunter vs. Rabbit game on a graph, which is a model of communication in adhoc mobile networks.
Let G be a cycle graph with N nodes. The hunter can move from a vertex to a vertex along an edge. The rabbit can
jump from any vertex to any vertex on the graph. We formalize the game using the random walk framework. The
strategy of the rabbit is formalized using a one dimensional random walk over Z. We classify strategies using the
) of their Fourier transformation. We investigate lower bounds and upper bounds of the probability
that the hunter catches the rabbit. We found a constant lower bound if β ∈ (0, 1) which does not depend on the size
N of the graph. We show the order is equivalent to O(1/ log N) if β = 1andalowerbound is1/N
if β ∈ (1, 2].
These results help us to choose the parameter β of a rabbit strategy according to the size N of the given graph. We
introduce a formalization of strategies using a random walk, theoretical estimation of bounds of a probability that the
hunter catches the rabbit, and also show computing simulation results.
Keywords: Graph theory; Random walk; Combinatorial probability; Adhoc network
We consider a game played by two players: the hunter and
the rabbit. This game is described using a graph G(V , E)
where V is a set of vertices and E is a set of edges. Both
players may use a randomized strategy. The hunter can
move from vertex to vertex along edges. The rabbit can
move to any vertex at once. The hunter’s purpose is to
hand, the rabbit considers a strategy that maximizes the
time until the hunter catch the rabbit. If the hunter moves
to a vertex that the rabbit is at, the game finishes and we
say that the hunter catches the rabbit.
ing transmission procedures in mobile adhoc networks
[5,6]. This model helps to send an electronic messages effi-
ciently using mobile phones. The expected value of time
until the hunter catches the rabbit is equal to the expected
time until the recipient receives the mail. One of our goals
is to improve these procedures.
We introduce some games resembling the Hunter vs.
Rabbit game. The first one is the Princess vs. Mon-
ster game. In this game, the Monster tries to catch the
Princess in area D. The difference between the Hunter
Graduate School of Mathematics, Kyushu University, 744, Motooka, Nishi-ku,
Fukuoka 819-0395, Japan
Full list of author information is available at the end of the article
vs. Rabbit game is that the Monster catches the Princess
if the distance between the two players is smaller than
a chosen value. Also the Monster moves at a constant
speed whereas the Princess can move at any speed. This
game is played on a cycle graph as introduced by Isaacs
. The Princess vs. Monster game has been investi-
gated by Alpern , Zelikin , and so on. Gal analyzed
the Princess-Monster game on a convex multidimensional
The next one is the Deterministic pursuit-evasion game.
In this game we consider a runaway hide dark spot, for
example a tunnel. Parsons innovated the search number
of a graph [16,17]. The search number of a graph is the
least number of people that are required to catch a run-
away hiding dark spot moving at any speed. LaPaugh 
showed that if the runaway is known not to be in edge
e at any point of time, then the runaway can not enter
edge e without being caught in the remainder of the game.
Meggido showed that the computation time of the search
number of a graph is NP-hard . If an edge can be
cleared without moving along it, but it suffices to ‘look
into’ an edge from a vertex, then the minimum number
of guards needed to catch the fugitive is called the node
search number of graph . The pursuit evasion prob-
lem in the plane was introduced by Suzuki and Yamashita
. They gave necessary and sufficient conditions for
a simple polygon to be searchable by a single pursuer.
© 2015 Ikeda et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons
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