PEdom.num.tri {pcds}

The domination number of Proportional Edge Proximity Catch Digraph (PE-PCD) - one triangle 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 triangle tri with expansion parameter r \ge 1 and vertex regions are constructed with 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 the circumcenter of tri.

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

Usage

PEdom.num.tri(Xp, tri, r, M = c(1, 1, 1))

Arguments

Xp

A set of 2D points which constitute the vertices of the digraph.

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; 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 (1,1,1), i.e., the center of mass.

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 (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

PEdom.num.nondeg, PEdom.num, and PEdom.num1D

Examples

A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2)
Tr<-rbind(A,B,C)
n<-10  #try also n<-20
Xp<-runif.tri(n,Tr)$g

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

r<-1.4

PEdom.num.tri(Xp,Tr,r,M)
IM<-inci.matPEtri(Xp,Tr,r,M)
dom.num.greedy #try also dom.num.exact(IM)

gr.gam<-dom.num.greedy(IM)
gr.gam
Xp[gr.gam$i,]

PEdom.num.tri(Xp,Tr,r,M=c(.4,.4))