Idom.numCSup.bnd.std.tri {pcds}

The indicator for k being an upper bound for the domination number of Central Similarity Proximity Catch Digraph (CS-PCD) by the exact algorithm - standard equilateral triangle case

Description

Returns I(domination number of CS-PCD is less than or equal to k) where the vertices of the CS-PCD are the data points Xp, that is, returns 1 if the domination number of CS-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 CS-PCD.

CS proximity region is constructed with respect to the standard equilateral triangle T_e=T(A,B,C)=T((0,0),(1,0),(1/2,\sqrt{3}/2)) with expansion parameter t>0 and edge regions are based on the center M=(m_1,m_2) in Cartesian coordinates or M=(\alpha,\beta,\gamma) in barycentric coordinates in the interior of T_e; default is M=(1,1,1) i.e., the center of mass of T_e (which is equivalent to the circumcenter of T_e).

Edges of T_e, AB, BC, AC, are also labeled as 3, 1, and 2, respectively. Loops are allowed in the digraph. It takes a long time for large number of vertices (i.e., large number of row numbers).

See also (Ceyhan (2012)).

Usage

Idom.numCSup.bnd.std.tri(Xp, k, t, M = c(1, 1, 1))

Arguments

Xp

A set of 2D points which constitute the vertices of CS-PCD.

k

A positive integer representing an upper bound for the domination number of CS-PCD.

t

A positive real number which serves as the expansion parameter in CS proximity region in the standard equilateral triangle T_e=T((0,0),(1,0),(1/2,\sqrt{3}/2)).

M

A 2D point in Cartesian coordinates or a 3D point in barycentric coordinates which serves as a center in the interior of the standard equilateral triangle T_e; default is M=(1,1,1) i.e. the center of mass of T_e.

Value

A list with two elements

domUB

The upper bound k (to be checked) for the domination number of CS-PCD. It is prespecified as k in the function arguments.

Idom.num.up.bnd

The indicator for the upper bound for domination number of CS-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 is NULL.

Author(s)

Elvan Ceyhan

References

Ceyhan E (2012). “An investigation of new graph invariants related to the domination number of random proximity catch digraphs.” Methodology and Computing in Applied Probability, 14(2), 299-334.

See Also

Idom.numCSup.bnd.tri, Idom.num.up.bnd, Idom.numASup.bnd.tri, and dom.num.exact

Examples

A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2);
Te<-rbind(A,B,C);
n<-10

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

M<-as.numeric(runif.std.tri(1)$g)  #try also M<-c(.6,.2)

t<-.5

Idom.numCSup.bnd.std.tri(Xp,1,t,M)

for (k in 1:n)
  print(c(k,Idom.numCSup.bnd.std.tri(Xp,k,t,M)$Idom.num.up.bnd))
  print(c(k,Idom.numCSup.bnd.std.tri(Xp,k,t,M)$domUB))