IndASdomUBtri: Indicator for an upper bound for the domination number of Arc Slice Proximity Catch Digraph (AS-PCD) by the exact algorithm - one triangle case

Description

Returns \(I(\)domination number of AS-PCD whose vertices are the data points Xp is less than or equal to k\()\), that is, returns 1 if the domination number of AS-PCD is less than the prespecified value k, returns 0 otherwise. It also provides the vertices (i.e., data points) in a dominating set of size k of AS-PCD.

AS proximity regions are constructed with respect to the triangle tri and vertex regions are based on the center M="CC" for circumcenter of tri; or \(M=(m_1,m_2)\) in Cartesian coordinates or \(M=(\alpha,\beta,\gamma)\) in barycentric coordinates in the interior of the triangle tri; default is M="CC" i.e., circumcenter of tri.

The vertices of triangle, tri, are labeled as \(1,2,3\) according to the row number the vertex is recorded in tri. Loops are allowed in the digraph. It takes a long time for large number of vertices (i.e., large number of row numbers).

Usage

IndASdomUBtri(Xp, k, tri, M = "CC")

Value

A list with the elements

domUB: The suggested upper bound (to be checked) for the domination number of AS-PCD. It is prespecified as k in the function arguments.
IndUBdom: The indicator for the upper bound for domination number of AS-PCD being the specified value k or not. It returns 1 if the upper bound is k, and 0 otherwise.
ind.domset: The vertices (i.e., data points) in the dominating set of size k if it exists, otherwise it yields NULL.

Arguments

Xp: A set of 2D points which constitute the vertices of the AS-PCD.
k: A positive integer to be tested for an upper bound for the domination number of AS-PCDs.
tri: Three 2D points, stacked row-wise, each row representing a vertex of the triangle.
M: The center of the triangle. "CC" stands for circumcenter of the triangle tri or a 2D point in Cartesian coordinates or a 3D point in barycentric coordinates which serves as a center in the interior of tri; default is M="CC" i.e., the circumcenter of tri.

Author

Elvan Ceyhan

Examples

Run this code

if (FALSE) {
A<-c(1,1); B<-c(2,0); C<-c(1.5,2);

Tr<-rbind(A,B,C);
n<-10

set.seed(1)
Xp<-runif.tri(n,Tr)$gen.points

M<-as.numeric(runif.tri(1,Tr)$g)  #try also M<-c(1.6,1.2)

IndASdomUBtri(Xp,1,Tr)

for (k in 1:n)
  print(c(k,IndASdomUBtri(Xp,k,Tr,M)))

IndASdomUBtri(Xp,k=4,Tr,M)

P<-c(.4,.2)
IndASdomUBtri(P,1,Tr,M)

IndASdomUBtri(rbind(Xp,Xp),k=2,Tr,M)
}