# The functions for probability of domination number $= 2$ for Proportional Edge Proximity Catch Digraphs (PE-PCDs) - middle interval case The function `Pdom.num2PE1D` 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 $\gamma$ stands for the domination number. The PE proximity region $N_{PE}(x,r,c)$ 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`. ```r Pdom.num2AI(r, c, n) Pdom.num2AII(r, c, n) Pdom.num2AIII(r, c, n) Pdom.num2AIV(r, c, n) Pdom.num2A(r, c, n) Pdom.num2Asym(r, c, n) Pdom.num2BIII(r, c, n) Pdom.num2B(r, c, n) Pdom.num2Bsym(r, c, n) Pdom.num2CIV(r, c, n) Pdom.num2C(r, c, n) Pdom.num2Csym(r, c, n) Pdom.num2PE1D(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, $(a,b)$, the parameterized center is $M_c=a+c(b-a)$. - `n`: A positive integer representing the size of the uniform data set. ## Returns $P($domination number$\le 1)$ for PE-PCD whose vertices are a uniform data set of size `n` in a finite interval $(a,b)$ ## Auxiliary Functions for `Pdom.num2PE1D` The auxiliary functions are `Pdom.num2AI, Pdom.num2AII, Pdom.num2AIII, Pdom.num2AIV, Pdom.num2A,Pdom.num2Asym, Pdom.num2BIII, Pdom.num2B, Pdom.num2B, Pdom.num2Bsym,Pdom.num2CIV, Pdom.num2C`, and `Pdom.num2Csym`, 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 - $c \in ((3-\sqrt{5})/2, 1/2)$ In Case A, we compute $P(\gamma=2)$ with `Pdom.num2AIV(r,c,n)` if $1 < r < (1-c)/c$; `Pdom.num2AIII(r,c,n)` if $(1-c)/c< r < 1/(1-c)$; `Pdom.num2AII(r,c,n)` if $1/(1-c)< r < 1/c$; and `Pdom.num2AI(r,c,n)` otherwise. `Pdom.num2A(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 `Pdom.num2Asym(r,c,n)` for $c$ in $(1/2,(\sqrt{5}-1)/2)$ with the same auxiliary functions `Pdom.num2AIV(r,1-c,n)` if $1 < r < c/(1-c)$; `Pdom.num2AIII(r,1-c,n)` if $(c/(1-c) < r < 1/c$; `Pdom.num2AII(r,1-c,n)` if $1/c < r < 1/(1-c)$; and `Pdom.num2AI(r,1-c,n)` otherwise. ## Case B - $c \in (1/4,(3-\sqrt{5})/2)$ In Case B, we compute $P(\gamma=2)$ with `Pdom.num2AIV(r,c,n)` if $1 < r < 1/(1-c)$; `Pdom.num2BIII(r,c,n)` if $1/(1-c) < r < (1-c)/c$; `Pdom.num2AII(r,c,n)` if $(1-c)/c < r < 1/c$; and `Pdom.num2AI(r,c,n)` otherwise. `Pdom.num2B(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 `Pdom.num2Bsym(r,c,n)` for `c` in $((\sqrt{5}-1)/2,3/4)$ with the same auxiliary functions `Pdom.num2AIV(r,1-c,n)` if $ 1< r < 1/c$; `Pdom.num2BIII(r,1-c,n)` if $1/c < r < c/(1-c)$; `Pdom.num2AII(r,1-c,n)` if $c/(1-c) < r < 1/(1-c)$; and `Pdom.num2AI(r,1-c,n)` otherwise. ## Case C - $c \in (0,1/4)$ In Case C, we compute $P(\gamma=2)$ with `Pdom.num2AIV(r,c,n)` if $1< r < 1/(1-c)$; `Pdom.num2BIII(r,c,n)` if $1/(1-c) < r < (1-\sqrt{1-4 c})/(2 c)$; `Pdom.num2CIV(r,c,n)` if $(1-\sqrt{1-4 c})/(2 c) < r < (1+\sqrt{1-4 c})/(2 c)$; `Pdom.num2BIII(r,c,n)` if $(1+\sqrt{1-4 c})/(2 c) < r <1/(1-c)$; `Pdom.num2AII(r,c,n)` if $1/(1-c) < r < 1/c$; and `Pdom.num2AI(r,c,n)` otherwise. `Pdom.num2C(r,c,n)` combines these functions in Case C: $c \in (0,1/4)$. Due to the symmetry in the PE proximity regions, we use `Pdom.num2Csym(r,c,n)` for $c \in (3/4,1)$ with the same auxiliary functions `Pdom.num2AIV(r,1-c,n)` if $1< r < 1/c$; `Pdom.num2BIII(r,1-c,n)` if $1/c < r < (1-\sqrt{1-4(1-c)})/(2(1-c))$; `Pdom.num2CIV(r,1-c,n)` if $(1-\sqrt{1-4(1-c)})/(2(1-c)) < r < (1+\sqrt{1-4(1-c)})/(2(1-c))$; `Pdom.num2BIII(r,1-c,n)` if $(1+\sqrt{1-4(1-c)})/(2(1-c)) < r < c/(1-c)$; `Pdom.num2AII(r,1-c,n)` if $c/(1-c)< r < 1/(1-c)$; and `Pdom.num2AI(r,1-c,n)` otherwise. Combining Cases A, B, and C, we get our main function `Pdom.num2PE1D` which computes $P(\gamma=2)$ for any (`r,c`) in its domain. ## Examples ```r #Examples for the main function Pdom.num2PE1D r<-2 c<-.5 Pdom.num2PE1D(r,c,n=10) Pdom.num2PE1D(r=1.5,c=1/1.5,n=100) ``` ## See Also `Pdom.num2PEtri` and `Pdom.num2PE1Dasy` ## Author(s) Elvan Ceyhan

funsPDomNum2PE1D function