# The indicator for two points constituting a dominating set for Proportional Edge Proximity Catch Digraphs (PE-PCDs) - one triangle case Returns $I(${`p1,p2`} is a dominating set of the PE-PCD$)$ where the vertices of the PE-PCD are the 2D data set `Xp`, that is, returns 1 if {`p1,p2`} is a dominating set of PE-PCD, and returns 0 otherwise. Point, `p1`, is in the region of vertex `rv1` (default is `NULL`) and point, `p2`, is in the region of vertex `rv2` (default is `NULL`); vertices (and hence `rv1` and `rv2`) are labeled as $1,2,3$ in the order they are stacked row-wise in `tri`. PE proximity regions are defined with respect to the triangle `tri` and vertex regions are based on 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 circumcenter of `tri`; default is $M=(1,1,1)$, i.e., the center of mass of `tri`. `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 (if(!exists(".Rdpack.currefs")) .Rdpack.currefs \<-new.env();Rdpack::insert_citeOnly(keys="ceyhan:Phd-thesis,ceyhan:masa-2007,ceyhan:dom-num-NPE-Spat2011,ceyhan:mcap2012;textual",package="pcds",cached_env=.Rdpack.currefs) ). ```r Idom.num2PEtri( p1, p2, Xp, tri, r, M = c(1, 1, 1), rv1 = NULL, rv2 = NULL, ch.data.pnts = FALSE ) ``` ## Arguments - `p1, p2`: Two 2D points to be tested for constituting a dominating set of the PE-PCD. - `Xp`: A set of 2D points which constitutes 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; 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`. - `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 `tri` (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`). ## Returns $I(${`p1,p2`} is a dominating set of the PE-PCD$)$ where the vertices of the PE-PCD are the 2D data set `Xp`, that is, returns 1 if {`p1,p2`} is a dominating set of PE-PCD, and returns 0 otherwise. ## Examples ```r A<-c(1,1); B<-c(2,0); C<-c(1.5,2); Tr<-rbind(A,B,C); n<-10 #try also n<-20 set.seed(1) Xp<-runif.tri(n,Tr)$g M<-as.numeric(runif.tri(1,Tr)$g) #try also M<-c(1.6,1.0) r<-1.5 #try also r<-2 Idom.num2PEtri(Xp[1,],Xp[2,],Xp,Tr,r,M) ind.gam2<-vector() for (i in 1:(n-1)) for (j in (i+1):n) {if (Idom.num2PEtri(Xp[i,],Xp[j,],Xp,Tr,r,M)==1) ind.gam2<-rbind(ind.gam2,c(i,j))} ind.gam2 #or try rv1<-rel.vert.tri(Xp[1,],Tr,M)$rv; rv2<-rel.vert.tri(Xp[2,],Tr,M)$rv Idom.num2PEtri(Xp[1,],Xp[2,],Xp,Tr,r,M,rv1,rv2) Idom.num2PEtri(Xp[1,],c(1,2),Xp,Tr,r,M,ch.data.pnts = FALSE) #gives an error message if ch.data.pnts = TRUE #since not both points, p1 and p2, are data points in Xp ``` ## References if(!exists(".Rdpack.currefs")) .Rdpack.currefs \<-new.env();Rdpack::insert_all_ref(.Rdpack.currefs) ## See Also `Idom.num2PEbasic.tri`, `Idom.num2AStri`, and `Idom.num2PEtetra` ## Author(s) Elvan Ceyhan

Idom.num2PEtri function