# The indicator for `k` being an upper bound for the domination number of Central Similarity Proximity Catch Digraph (CS-PCD) by the exact algorithm - standard equilateral triangle case Returns $I($domination number of CS-PCD is less than or equal to `k`$)$ where the vertices of the CS-PCD are the data points `Xp`, that is, returns 1 if the domination number of CS-PCD is less than the prespecified value `k`, returns 0 otherwise. It also provides the vertices (i.e., data points) in a dominating set of size `k` of CS-PCD. 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. Loops are allowed in the digraph. It takes a long time for large number of vertices (i.e., large number of row numbers). 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.numCSup.bnd.std.tri(Xp, k, t, M = c(1, 1, 1)) ``` ## Arguments - `Xp`: A set of 2D points which constitute the vertices of CS-PCD. - `k`: A positive integer representing an upper bound for the domination number of 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 A `list` with two elements - **domUB**: The upper bound `k` (to be checked) for the domination number of CS-PCD. It is prespecified as `k` in the function arguments. - **Idom.num.up.bnd**: The indicator for the upper bound for domination number of CS-PCD being the specified value `k` or not. It returns 1 if the upper bound is `k`, and 0 otherwise. - **ind.domset**: The vertices (i.e., data points) in the dominating set of size `k` if it exists, otherwise it is `NULL`. ## 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 Idom.numCSup.bnd.std.tri(Xp,1,t,M) for (k in 1:n) print(c(k,Idom.numCSup.bnd.std.tri(Xp,k,t,M)$Idom.num.up.bnd)) print(c(k,Idom.numCSup.bnd.std.tri(Xp,k,t,M)$domUB)) ``` ## References if(!exists(".Rdpack.currefs")) .Rdpack.currefs \<-new.env();Rdpack::insert_all_ref(.Rdpack.currefs) ## See Also `Idom.numCSup.bnd.tri`, `Idom.num.up.bnd`, `Idom.numASup.bnd.tri`, and `dom.num.exact` ## Author(s) Elvan Ceyhan

Idom.numCSup.bnd.std.tri function