Idom.setPEtri {pcds}

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

Description

Returns I(S a dominating set of PE-PCD whose vertices are the data set Xp), 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 triangle tri with the 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 the triangle tri or based on the circumcenter of tri; default is M=(1,1,1), i.e., the center of mass of tri. The triangle tri=T(A,B,C) has edges AB, BC, AC which are also labeled as edges 3, 1, and 2, respectively.

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

Usage

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

Arguments

S

A set of 2D points which is to be tested for being a dominating set for the PE-PCDs.

Xp

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

tri

A 3 \times 2 matrix with each row representing a vertex of the triangle.

r

A positive real number which serves as the expansion parameter in PE proximity region constructed in the triangle tri; 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 triangle tri or the circumcenter of tri which may be entered as "CC" as well; default is M=(1,1,1), i.e., the center of mass of tri.

Value

I(S a dominating set of PE-PCD), that is, returns 1 if S is a dominating set of PE-PCD whose vertices are the data points in Xp; and returns 0 otherwise, where PE proximity region is constructed in the triangle tri.

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.setPEstd.tri, IarcPEset2pnt.tri, Idom.setCStri, and Idom.setAStri

Examples

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(1.6,1.0)

r<-1.5

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

S<-rbind(Xp[1,],Xp[2,],Xp[3,],Xp[5,])
Idom.setPEtri(S,Xp,Tr,r,M)

S<-rbind(c(.1,.1),c(.3,.4),c(.5,.3))
Idom.setPEtri(S,Xp,Tr,r,M)