# The domination number of Proportional Edge Proximity Catch Digraph (PE-PCD) - one triangle case Returns the domination number of PE-PCD whose vertices are the data points in `Xp`. PE proximity region is defined with respect to the triangle `tri` with expansion parameter $r \ge 1$ and vertex regions are constructed with 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 the circumcenter of `tri`. 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 PEdom.num.tri(Xp, tri, r, M = c(1, 1, 1)) ``` ## Arguments - `Xp`: A set of 2D points which constitute the vertices of the digraph. - `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 $(1,1,1)$, i.e., the center of mass. ## Returns A `list` with two elements - **dom.num**: Domination number of PE-PCD with vertex set = `Xp` and expansion parameter $r \ge 1$ and center `M` - **mds**: A minimum dominating set of PE-PCD with vertex set = `Xp` and expansion parameter $r \ge 1$ and center `M` - **ind.mds**: Indices of the minimum dominating set `mds` ## Examples ```r A<-c(0,0); B<-c(1,0); C<-c(1/2,sqrt(3)/2) Tr<-rbind(A,B,C) n<-10 #try also n<-20 Xp<-runif.tri(n,Tr)$g M<-as.numeric(runif.tri(1,Tr)$g) #try also M<-c(1,1,1) r<-1.4 PEdom.num.tri(Xp,Tr,r,M) IM<-inci.matPEtri(Xp,Tr,r,M) dom.num.greedy #try also dom.num.exact(IM) gr.gam<-dom.num.greedy(IM) gr.gam Xp[gr.gam$i,] PEdom.num.tri(Xp,Tr,r,M=c(.4,.4)) ``` ## References if(!exists(".Rdpack.currefs")) .Rdpack.currefs \<-new.env();Rdpack::insert_all_ref(.Rdpack.currefs) ## See Also `PEdom.num.nondeg`, `PEdom.num`, and `PEdom.num1D` ## Author(s) Elvan Ceyhan

PEdom.num.tri function