PEdomMTnd: The domination number of Proportional Edge Proximity Catch Digraph (PE-PCD) with non-degeneracy centers - multiple triangle case

Description

Returns the domination number and a minimum dominating set of PE-PCD whose vertices are the data points in Xp in the multiple triangle case and the Delaunay triangles based on Yp points.

PE proximity regions are defined with respect to the Delaunay triangles based on Yp points with expansion parameter $r \geq 1$ and vertex regions in each triangle are based on the center $M = (α, β, γ)$ in barycentric coordinates in the interior of each Delaunay triangle or based on circumcenter of each Delaunay triangle (default for $M = (1, 1, 1)$ which is the center $M$ where M is one of the 3 centers that renders the asymptotic distribution of domination number to be non-degenerate for a given value of r in $(1, 1.5)$ and M is center of mass for $r = 1.5$ .

Convex hull of Yp is partitioned by the Delaunay triangles based on Yp points (i.e., multiple triangles are the set of these Delaunay triangles whose union constitutes the convex hull of Yp points). Loops are allowed for the domination number.

See (ceyhan:Phd-thesis,ceyhan:masa-2007,ceyhan:dom-num-NPE-Spat2011,ceyhan:mcap2012;textualpcds) more on the domination number of PE-PCDs. Also see (okabe:2000,ceyhan:comp-geo-2010,sinclair:2016;textualpcds) for more on Delaunay triangulation and the corresponding algorithm.

Usage

PEdomMTnd(Xp, Yp, r)

Value

A list with two elements

dom.num: Domination number of the PE-PCD whose vertices are Xp points. PE proximity regions are constructed with respect to the Delaunay triangles based on the Yp points with expansion parameter $r \geq 1$ .
mds: A minimum dominating set of the PE-PCD whose vertices are Xp points

Arguments

Xp: A set of 2D points which constitute the vertices of the PE-PCD.
Yp: A set of 2D points which constitute the vertices of the Delaunay triangles.
r: A positive real number which serves as the expansion parameter in PE proximity region; must be $\geq 1$ .

Author

Elvan Ceyhan

References

Examples

Run this code

#nx is number of X points (target) and ny is number of Y points (nontarget)
nx<-20; ny<-4;  #try also nx<-40; ny<-10 or nx<-1000; ny<-10;

r<-1.5  #try also r<-2

set.seed(1)
Xp<-cbind(runif(nx,0,1),runif(nx,0,1))
Yp<-cbind(runif(ny,0,1),runif(ny,0,1))

PEdomMTnd(Xp,Yp,r)

PEdomMTnd(Xp,Yp,r=1.4)

r<-1.5  #try also  #r<-2
PEdomMTnd(Xp,Yp,r)  #this may be different due to random selection of the center for r in (1,1.5)

PEdomMTnd(Xp,Yp[1:3,],r)

PEdomMTnd(Xp,rbind(Yp,Yp),r)

dat.fr<-data.frame(a=Xp)
PEdomMTnd(dat.fr,Yp,r)

dat.fr<-data.frame(a=Yp)
PEdomMTnd(Xp,dat.fr,r)

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