pcds (version 0.1.1)

funsPG2PE1D: The functions for probability of domination number=2 for Proportional Edge Proximity Catch Digraphs (PE-PCDs) - middle interval case

Description

The function PG2PE1D and its auxiliary functions.

Returns \(P(\gamma=2)\) for PE-PCD whose vertices are a uniform data set of size n in a finite interval (a,b)where \(\gamma\) stans for the domination number.

The PE proximity region \(NPE(x,r,c)\) is defined with respect to (a,b) with centrality parameter c in \((0,1)\) and expansion parameter \(r \ge 1\).

To compute the probability \(P(\gamma=2)\) for PE-PCD in the 1D case, we partition the domain \((r,c)=(1,\infty) \times (0,1)\), and compute the probability for each partition set. The sample size (i.e. number of vertices or data points) is a positive integer, n.

Usage

PG2AI(r, c, n)
PG2AII(r, c, n)
PG2AIII(r, c, n)
PG2AIV(r, c, n)
PG2A(r, c, n)
PG2Asym(r, c, n)
PG2BIII(r, c, n)
PG2B(r, c, n)
PG2Bsym(r, c, n)
PG2CIV(r, c, n)
PG2C(r, c, n)
PG2Csym(r, c, n)
PG2PE1D(r, c, n)

Arguments

r

A positive real number which serves as the expansion parameter in PE proximity region; must be \(\ge 1\).

c

A positive real number in \((0,1)\) parameterizing the center inside \(int=(a,b)\). For the interval, \(int=(a,b)\), the parameterized center is \(M_c=a+c(b-a)\).

n

A positive integer representing the size of the uniform data set.

Value

P(domination number=2) for PE-PCD whose vertices are a uniform data set of size n in a finite interval \((a,b)\)

Auxiliary Functions for <code>PG2PE1D</code>

The auxiliary functions are PG2AI, PG2AII, PG2AIII, PG2AIV, PG2A, PG2Asym, PG2BIII, PG2B, PG2B, PG2Bsym, PG2CIV, PG2C, and PG2Csym, each corresponding to a partition of the domain of r and c. In particular, the domain partition is handled in 3 cases as

CASE A: c in \(((3-\sqrt{5})/2, 1/2)\)

CASE B: c in \((1/4,(3-\sqrt{5})/2)\) and

CASE C: c in \((0,1/4)\).

Case A - <code>c</code> in \(((3-\sqrt{5})/2, 1/2)\)

In Case A, we compute \(P(\gamma=2)\) with

PG2AIV(r,c,n) if \(1 < r < (1-c)/c\);

PG2AIII(r,c,n) if \((1-c)/c< r < 1/(1-c)\);

PG2AII(r,c,n) if \(1/(1-c)< r < 1/c\);

and PG2AI(r,c,n) otherwise.

PG2A(r,c,n) combines these functions in Case A: c in \(((3-\sqrt{5})/2,1/2)\). Due to the symmetry in the PE proximity regions, we use PG2Asym(r,c,n) for c in \((1/2,(\sqrt{5}-1)/2)\) with the same auxiliary functions

PG2AIV(r,1-c,n) if \(1 < r < c/(1-c)\);

PG2AIII(r,1-c,n) if \((c/(1-c) < r < 1/c\);

PG2AII(r,1-c,n) if \(1/c < r < 1/(1-c)\);

and PG2AI(r,1-c,n) otherwise.

Case B - <code>c</code> in \((1/4,(3-\sqrt{5})/2)\)

In Case B, we compute \(P(\gamma=2)\) with

PG2AIV(r,c,n) if \(1 < r < 1/(1-c)\);

PG2BIII(r,c,n) if \(1/(1-c) < r < (1-c)/c\);

PG2AII(r,c,n) if \((1-c)/c < r < 1/c\);

and PG2AI(r,c,n) otherwise.

PG2B(r,c,n) combines these functions in Case B: c in \((1/4,(3-\sqrt{5})/2)\). Due to the symmetry in the PE proximity regions, we use PG2Bsym(r,c,n) for c in \(((\sqrt{5}-1)/2,3/4)\) with the same auxiliary functions

PG2AIV(r,1-c,n) if \( 1< r < 1/c\);

PG2BIII(r,1-c,n) if \(1/c < r < c/(1-c)\);

PG2AII(r,1-c,n) if \(c/(1-c) < r < 1/(1-c)\);

and PG2AI(r,1-c,n) otherwise.

Case C - <code>c</code> in \((0,1/4)\)

In Case C, we compute \(P(\gamma=2)\) with

PG2AIV(r,c,n) if \(1< r < 1/(1-c)\);

PG2BIII(r,c,n) if \(1/(1-c) < r < (1-\sqrt{1-4 c})/(2 c)\);

PG2CIV(r,c,n) if \((1-\sqrt{1-4 c})/(2 c) < r < (1+\sqrt{1-4 c})/(2 c)\);

PG2BIII(r,c,n) if \((1+\sqrt{1-4 c})/(2 c) < r <1/(1-c)\);

PG2AII(r,c,n) if \(1/(1-c) < r < 1/c\);

and PG2AI(r,c,n) otherwise.

PG2C(r,c,n) combines these functions in CaseE C: c in \((0,1/4)\). Due to the symmetry in the PE proximity regions, we use PG2Csym(r,c,n) for c in \((3/4,1)\) with the same auxiliary functions

PG2AIV(r,1-c,n) if \(1< r < 1/c\);

PG2BIII(r,1-c,n) if \(1/c < r < (1-\sqrt{1-4(1-c)})/(2(1-c))\);

PG2CIV(r,1-c,n) if \((1-\sqrt{1-4(1-c)})/(2(1-c)) < r < (1+\sqrt{1-4(1-c)})/(2(1-c))\);

PG2BIII(r,1-c,n) if \((1+\sqrt{1-4(1-c)})/(2(1-c)) < r < c/(1-c)\);

PG2AII(r,1-c,n) if \(c/(1-c)< r < 1/(1-c)\);

and PG2AI(r,1-c,n) otherwise.

Combining Cases A, B, and C, we get our main function PG2PE1D which computes \(P(\gamma=2)\) for any (r,c) in its domain.

See Also

PG2PEtri and PG2PE1D.asy

Examples

# NOT RUN {
#Examples for the auxiliary functions
# }
# NOT RUN {
PG2AI(2.25,.45,10)
# }
# NOT RUN {
# }
# NOT RUN {
PG2AII(2.2,.4,10)
# }
# NOT RUN {
# }
# NOT RUN {
PG2AIII(1.2,.25,10)
# }
# NOT RUN {
# }
# NOT RUN {
PG2AIV(1.2,.25,10)
# }
# NOT RUN {
# }
# NOT RUN {
PG2A(1.2,.25,10)
# }
# NOT RUN {
# }
# NOT RUN {
PG2Asym(1.2,.75,10)
# }
# NOT RUN {
# }
# NOT RUN {
PG2BIII(1.5,.45,10)
# }
# NOT RUN {
# }
# NOT RUN {
PG2B(1.5,.30,10)
# }
# NOT RUN {
# }
# NOT RUN {
PG2Bsym(1.5,.70,10)
# }
# NOT RUN {
# }
# NOT RUN {
PG2CIV(1.5,.2,10)
# }
# NOT RUN {
# }
# NOT RUN {
PG2C(1.5,.2,10)
# }
# NOT RUN {
# }
# NOT RUN {
PG2Csym(1.5,.8,10)
# }
# NOT RUN {
#Examples for the main function PG2PE1D
r<-2
c<-.5
n<-10

PG2PE1D(r,c,n)

PG2PE1D(r=1.5,c=1/1.5,n=100)

# }