A Location Game On Disjoint Circles

general notion formally although we lay out complete specifications of the
strategies we report. We will use uppercase letters S, T , X , Y to denote
pure strategies. Strategy X is called a winning strategy (a tying strategy)
for player p if no matter what player q does, by using X player p guarantees
that Sp > Sq (Sp ≥ Sq). Throughout the paper we will use the standard
notation m | n, to denote the fact that m divides n and m - n, to denote its
negation.

2.1 Some definitions and existing results

We first develop some concepts and notations. Let C be a circle and let
P ⊆ C be a finite set of points on the circle. Then an arc (x, y) ⊆ C
such that {x, y} ⊆ P and (x, y) ∩ P = 0 is called an interval. Now let
PR and PG such that PR ∪ PG = P be sets of red and green points of P ,
respectively. Then an arc (x, y) ⊆ C such that {x, y} ⊆ PR ({x, y} ⊆ PG)
and and (x, y) ∩ PR = 0 ((x, y) ∩ PG = 0) is called a red (green) interval.
An interval that is neither red nor green is called a bichromatic interval
and an interval which is not bichromatic shall be at times referred to as
a monochromatic interval in general. We will use r^C (g^C) to denote the
number of red (green) points placed on the circle C. We will also use I^R(C)
(I^G(C)) to denote the number of red (green) intervals on the circle C.

Given a circle C and a point x ∈ C , an antipode of x is the point y ∈ C
such that d(x, y) = 1/2. The pair of points {x, y} is called a pair of antipodes
of C. Let m be a positive natural number. Then the set of key positions ⁵
on C determined by point x and m is the set

κ(C, x, m) = {p ∈ C : a(p, x) = l/m, where l ∈ {0, . . . , m - 1}}.

By the set of key positions determined by m we mean a set of key positions
⁵We use a term key position here for what was called a key point in the paper Ahn
et al. [2004]. We found the name key position somewhat more apropriate in our context.

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