PEdom.num.tetra {pcds}

The domination number of Proportional Edge Proximity Catch Digraph (PE-PCD) - one tetrahedron case

Description

Returns the domination number of PE-PCD whose vertices are the data points in Xp.

PE proximity region is defined with respect to the tetrahedron th with expansion parameter r \ge 1 and vertex regions are based on the center M which is circumcenter ("CC") or center of mass ("CM") of th with default="CM".

See also (Ceyhan (2005, 2010)).

Usage

PEdom.num.tetra(Xp, th, r, M = "CM")

Arguments

Xp

A set of 3D points which constitute the vertices of the digraph.

th

A 4 \times 3 matrix with each row representing a vertex of the tetrahedron.

r

A positive real number which serves as the expansion parameter in PE proximity region; must be \ge 1.

M

The center to be used in the construction of the vertex regions in the tetrahedron, th. Currently it only takes "CC" for circumcenter and "CM" for center of mass; default="CM".

Value

A list with two elements

dom.num

Domination number of PE-PCD with vertex set = Xp and expansion parameter r \ge 1 and center M

mds

A minimum dominating set of PE-PCD with vertex set = Xp and expansion parameter r \ge 1 and center M

ind.mds

Indices of the minimum dominating set mds

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 (2010). “Extension of One-Dimensional Proximity Regions to Higher Dimensions.” Computational Geometry: Theory and Applications, 43(9), 721-748.

See Also

PEdom.num.tri

Examples

A<-c(0,0,0); B<-c(1,0,0); C<-c(1/2,sqrt(3)/2,0); D<-c(1/2,sqrt(3)/6,sqrt(6)/3)
tetra<-rbind(A,B,C,D)
n<-10  #try also n<-20

Xp<-runif.tetra(n,tetra)$g

M<-"CM"  #try also M<-"CC"
r<-1.25

PEdom.num.tetra(Xp,tetra,r,M)

P1<-c(.5,.5,.5)
PEdom.num.tetra(P1,tetra,r,M)