# Indicator for an upper bound for the domination number of Central Similarity Proximity Catch Digraph (CS-PCD) by the exact algorithm - one 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 triangle `tri`$=T(A,B,C)$ 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 `tri`; default is $M=(1,1,1)$ i.e., the center of mass of `tri`. Edges of `tri`, $AB$, $BC$, $AC$, are also labeled as 3, 1, and 2, respectively. Loops are allowed in the digraph. See also (if(!exists(".Rdpack.currefs")) .Rdpack.currefs \<-new.env();Rdpack::insert_citeOnly(keys="ceyhan:mcap2012;textual",package="pcds",cached_env=.Rdpack.currefs) ). Caveat: It takes a long time for large number of vertices (i.e., large number of row numbers). ```r Idom.numCSup.bnd.tri(Xp, k, tri, t, M = c(1, 1, 1)) ``` ## Arguments - `Xp`: A set of 2D points which constitute the vertices of CS-PCD. - `k`: A positive integer to be tested for an upper bound for the domination number of CS-PCDs. - `tri`: A $3 \times 2$ matrix with each row representing a vertex of the triangle. - `t`: A positive real number which serves as the expansion parameter in CS proximity region in the triangle `tri`. - `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`; default is $M=(1,1,1)$, i.e. the center of mass of `tri`. ## 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(1,1); B<-c(2,0); C<-c(1.5,2); Tr<-rbind(A,B,C); n<-10 set.seed(1) Xp<-runif.tri(n,Tr)$gen.points M<-as.numeric(runif.tri(1,Tr)$g) #try also M<-c(1.6,1.0) t<-.5 Idom.numCSup.bnd.tri(Xp,1,Tr,t,M) for (k in 1:n) print(c(k,Idom.numCSup.bnd.tri(Xp,k,Tr,t,M))) ``` ## References if(!exists(".Rdpack.currefs")) .Rdpack.currefs \<-new.env();Rdpack::insert_all_ref(.Rdpack.currefs) ## See Also `Idom.numCSup.bnd.std.tri`, `Idom.num.up.bnd`, `Idom.numASup.bnd.tri`, and `dom.num.exact` ## Author(s) Elvan Ceyhan

Idom.numCSup.bnd.tri function