Idom.setPEstd.tri {pcds}

The indicator for the set of points S being a dominating set or not for Proportional Edge Proximity Catch Digraphs (PE-PCDs) - standard equilateral triangle case

Description

Returns I(S a dominating set of PE-PCD whose vertices are the data points Xp) for S in the standard equilateral triangle, that is, returns 1 if S is a dominating set of PE-PCD, and returns 0 otherwise.

PE 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 r \ge 1 and vertex 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 also equivalent to the circumcenter of T_e). Vertices of T_e are also labeled as 1, 2, and 3, respectively.

See also (Ceyhan (2005); Ceyhan and Priebe (2007); Ceyhan (2011, 2012)).

Usage

Idom.setPEstd.tri(S, Xp, r, M = c(1, 1, 1))

Arguments

S

A set of 2D points whose PE proximity regions are considered.

Xp

A set of 2D points which constitutes the vertices of the PE-PCD.

r

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

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

I(S a dominating set of PE-PCD) for S in the standard equilateral triangle, that is, returns 1 if S is a dominating set of PE-PCD, and returns 0 otherwise, where PE proximity region is constructed in the standard equilateral triangle T_e.

Author(s)

Elvan Ceyhan

References

Ceyhan E (2005). An Investigation of Proximity Catch Digraphs in Delaunay Tessellations, also available as technical monograph titled Proximity Catch Digraphs: Auxiliary Tools, Properties, and Applications. Ph.D. thesis, The Johns Hopkins University, Baltimore, MD, 21218.

Ceyhan E (2011). “Spatial Clustering Tests Based on Domination Number of a New Random Digraph Family.” Communications in Statistics - Theory and Methods, 40(8), 1363-1395.

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.

Ceyhan E, Priebe CE (2007). “On the Distribution of the Domination Number of a New Family of Parametrized Random Digraphs.” Model Assisted Statistics and Applications, 1(4), 231-255.

See Also

Idom.setPEtri and Idom.setCSstd.tri

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)

r<-1.5

S<-rbind(Xp[1,],Xp[2,])
Idom.setPEstd.tri(S,Xp,r,M)

S<-rbind(Xp[1,],Xp[2,],Xp[3,],Xp[5,],c(.2,.5))
Idom.setPEstd.tri(S,Xp[3,],r,M)