PEdom.num1Dnondeg {pcds}

The domination number of Proportional Edge Proximity Catch Digraph (PE-PCD) with non-degeneracy centers - multiple interval case

Description

Returns the domination number, a minimum dominating set of PE-PCD whose vertices are the 1D data set Xp, and the domination numbers for partition intervals based on Yp when PE-PCD is constructed with vertex regions based on non-degeneracy centers.

Yp determines the end points of the intervals (i.e., partition the real line via intervalization). If there are duplicates of Yp points, only one point is retained for each duplicate value, and a warning message is printed.

PE proximity regions are defined with respect to the intervals based on Yp points with expansion parameter r \ge 1 and vertex regions in each interval are based on the centrality parameter c which is one of the 2 values of c (i.e., c \in \{(r-1)/r,1/r\}) that renders the asymptotic distribution of domination number to be non-degenerate for a given value of r in (1,2) and c is center of mass for r=2. These values are called non-degeneracy centrality parameters and the corresponding centers are called nondegeneracy centers.

Usage

PEdom.num1Dnondeg(Xp, Yp, r)

Arguments

Xp

A set of 1D points which constitute the vertices of the PE-PCD.

Yp

A set of 1D points which constitute the end points of the intervals which partition the real line.

r

A positive real number which serves as the expansion parameter in PE proximity region; must be in (1,2] here.

Value

A list with three elements

dom.num

Domination number of PE-PCD with vertex set Xp and expansion parameter r in (1,2] and centrality parameter c \in \{(r-1)/r,1/r\}.

mds

A minimum dominating set of the PE-PCD.

ind.mds

The data indices of the minimum dominating set of the PE-PCD whose vertices are Xp points.

int.dom.nums

Domination numbers of the PE-PCD components for the partition intervals.

Author(s)

Elvan Ceyhan

See Also

PEdom.num.nondeg

Examples

a<-0; b<-10
r<-1.5

#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-15; ny<-4;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;

set.seed(1)
Xp<-runif(nx,a,b)
Yp<-runif(ny,a,b)

PEdom.num1Dnondeg(Xp,Yp,r)
PEdom.num1Dnondeg(Xp,Yp,r=1.25)