Idom.num2ASbasic.tri {pcds}

The indicator for two points being a dominating set for Arc Slice Proximity Catch Digraphs (AS-PCDs) - standard basic triangle case

Description

Returns I({p1,p2} is a dominating set of AS-PCD) where vertices of AS-PCD are the 2D data set Xp), that is, returns 1 if {p1,p2} is a dominating set of AS-PCD, returns 0 otherwise.

AS proximity regions are defined with respect to the standard basic triangle T_b=T(c(0,0),c(1,0),c(c1,c2)), In the standard basic triangle, T_b, c_1 is in [0,1/2], c_2>0 and (1-c_1)^2+c_2^2 \le 1.

Any given triangle can be mapped to the standard basic triangle by a combination of rigid body motions (i.e., translation, rotation and reflection) and scaling, preserving uniformity of the points in the original triangle. Hence standard basic triangle is useful for simulation studies under the uniformity hypothesis.

Point, p1, is in the vertex region of vertex rv1 (default is NULL) and point, p2, is in the vertex region of vertex rv2 (default is NULL); vertices are labeled as 1,2,3 in the order they are stacked row-wise.

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 standard basic triangle T_b or based on circumcenter of T_b; default is M="CC", i.e., circumcenter of T_b.

ch.data.pnts is for checking whether points p1 and p2 are data points in Xp or not (default is FALSE), so by default this function checks whether the points p1 and p2 would be a dominating set if they actually were in the data set.

See also (Ceyhan (2005, 2010)).

Usage

Idom.num2ASbasic.tri(
  p1,
  p2,
  Xp,
  c1,
  c2,
  M = "CC",
  rv1 = NULL,
  rv2 = NULL,
  ch.data.pnts = FALSE
)

Arguments

p1, p2

Two 2D points to be tested for constituting a dominating set of the AS-PCD.

Xp

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

c1, c2

Positive real numbers which constitute the vertex of the standard basic triangle adjacent to the shorter edges; c_1 must be in [0,1/2], c_2>0 and (1-c_1)^2+c_2^2 \le 1.

M

The center of the triangle. "CC" stands for circumcenter of the triangle T_b or a 2D point in Cartesian coordinates or a 3D point in barycentric coordinates which serves as a center in the interior of the triangle T_b; default is M="CC" i.e., the circumcenter of T_b.

rv1, rv2

The indices of the vertices whose regions contains p1 and p2, respectively. They take the vertex labels as 1,2,3 as in the row order of the vertices in T_b (default is NULL for both).

ch.data.pnts

A logical argument for checking whether points p1 and p2 are data points in Xp or not (default is FALSE).

Value

I({p1,p2} is a dominating set of the AS-PCD) where the vertices of AS-PCD are the 2D data set Xp), that is, returns 1 if {p1,p2} is a dominating set of AS-PCD, returns 0 otherwise

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.

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.

See Also

Idom.num2AStri

Examples

c1<-.4; c2<-.6;
A<-c(0,0); B<-c(1,0); C<-c(c1,c2);
Tb<-rbind(A,B,C)
n<-10

set.seed(1)
Xp<-runif.basic.tri(n,c1,c2)$g

M<-as.numeric(runif.basic.tri(1,c1,c2)$g)  #try also M<-c(.6,.2)

Idom.num2ASbasic.tri(Xp[1,],Xp[2,],Xp,c1,c2,M)
Idom.num2ASbasic.tri(Xp[1,],Xp[1,],Xp,c1,c2,M)  #one point can not a dominating set of size two

Idom.num2ASbasic.tri(c(.2,.4),c(.2,.5),rbind(c(.2,.4),c(.2,.5)),c1,c2,M)

ind.gam2<-vector()
for (i in 1:(n-1))
  for (j in (i+1):n)
  {if (Idom.num2ASbasic.tri(Xp[i,],Xp[j,],Xp,c1,c2,M)==1)
   ind.gam2<-rbind(ind.gam2,c(i,j))}
ind.gam2

#or try
rv1<-rel.vert.basic.triCC(Xp[1,],c1,c2)$rv
rv2<-rel.vert.basic.triCC(Xp[2,],c1,c2)$rv
Idom.num2ASbasic.tri(Xp[1,],Xp[2,],Xp,c1,c2,M,rv1,rv2)
Idom.num2ASbasic.tri(c(.2,.4),Xp[2,],Xp,c1,c2,M,rv1,rv2)

#or try
rv1<-rel.vert.basic.triCC(Xp[1,],c1,c2)$rv
Idom.num2ASbasic.tri(Xp[1,],Xp[2,],Xp,c1,c2,M,rv1)

#or try
Rv2<-rel.vert.basic.triCC(Xp[2,],c1,c2)$rv
Idom.num2ASbasic.tri(Xp[1,],Xp[2,],Xp,c1,c2,M,rv2=Rv2)

Idom.num2ASbasic.tri(c(.3,.2),c(.35,.25),Xp,c1,c2,M)