# The indicator for the set of points `S` being a dominating set or not for Central Similarity Proximity Catch Digraphs (CS-PCDs) - standard equilateral triangle case Returns $I($`S` a dominating set of the CS-PCD$)$ where the vertices of the CS-PCD are the data set `Xp`), that is, returns 1 if `S` is a dominating set of CS-PCD, returns 0 otherwise. CS proximity region is constructed with respect to the standard equilateral triangle $T_e=T(A,B,C)=T((0,0),(1,0),(1/2,\sqrt{3}/2))$ with expansion parameter $t>0$ and edge 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 $T_e$; default is $M=(1,1,1)$ i.e., the center of mass of $T_e$ (which is equivalent to the circumcenter of $T_e$). Edges of $T_e$, $AB$, $BC$, $AC$, are also labeled as 3, 1, and 2, respectively. See also (if(!exists(".Rdpack.currefs")) .Rdpack.currefs \<-new.env();Rdpack::insert_citeOnly(keys="ceyhan:mcap2012;textual",package="pcds",cached_env=.Rdpack.currefs) ). ```r Idom.setCSstd.tri(S, Xp, t, M = c(1, 1, 1)) ``` ## Arguments - `S`: A set of 2D points which is to be tested for being a dominating set for the CS-PCDs. - `Xp`: A set of 2D points which constitute the vertices of the CS-PCD. - `t`: A positive real number which serves as the expansion parameter in CS proximity region in the standard equilateral triangle $T_e=T((0,0),(1,0),(1/2,\sqrt{3}/2))$. - `M`: A 2D point in Cartesian coordinates or a 3D point in barycentric coordinates which serves as a center in the interior of the standard equilateral triangle $T_e$; default is $M=(1,1,1)$ i.e. the center of mass of $T_e$. ## Returns $I($`S` a dominating set of the CS-PCD$)$, that is, returns 1 if `S` is a dominating set of CS-PCD, returns 0 otherwise, where CS proximity region is constructed in the standard equilateral triangle $T_e$ ## Examples ```r A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2); Te<-rbind(A,B,C); n<-10 set.seed(1) Xp<-runif.std.tri(n)$gen.points M<-as.numeric(runif.std.tri(1)$g) #try also M<-c(.6,.2) t<-.5 S<-rbind(Xp[1,],Xp[2,]) Idom.setCSstd.tri(S,Xp,t,M) S<-rbind(Xp[1,],Xp[2,],Xp[3,],Xp[5,]) Idom.setCSstd.tri(S,Xp,t,M) ``` ## References if(!exists(".Rdpack.currefs")) .Rdpack.currefs \<-new.env();Rdpack::insert_all_ref(.Rdpack.currefs) ## See Also `Idom.setCStri` and `Idom.setPEstd.tri` ## Author(s) Elvan Ceyhan

Idom.setCSstd.tri function