# The domination number of Proportional Edge Proximity Catch Digraph (PE-PCD) - multiple triangle case Returns the domination number, indices of a minimum dominating set of PE-PCD whose vertices are the data points in `Xp` in the multiple triangle case and domination numbers for the Delaunay triangles based on `Yp` points. PE proximity regions are defined with respect to the Delaunay triangles based on `Yp` points with expansion parameter $r \ge 1$ and vertex regions in each triangle are based on the center $M=(\alpha,\beta,\gamma)$ in barycentric coordinates in the interior of each Delaunay triangle or based on circumcenter of each Delaunay triangle (default for $M=(1,1,1)$ which is the center of mass of the triangle). Each Delaunay triangle is first converted to an (nonscaled) basic triangle so that `M` will be the same type of center for each Delaunay triangle (this conversion is not necessary when `M` is $CM$). Convex hull of `Yp` is partitioned by the Delaunay triangles based on `Yp` points (i.e., multiple triangles are the set of these Delaunay triangles whose union constitutes the convex hull of `Yp` points). Loops are allowed for the domination number. See (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) ) for more on the domination number of PE-PCDs. Also, see (if(!exists(".Rdpack.currefs")) .Rdpack.currefs \<-new.env();Rdpack::insert_citeOnly(keys="okabe:2000,ceyhan:comp-geo-2010,sinclair:2016;textual",package="pcds",cached_env=.Rdpack.currefs) ) for more on Delaunay triangulation and the corresponding algorithm. ```r PEdom.num(Xp, Yp, r, M = c(1, 1, 1)) ``` ## Arguments - `Xp`: A set of 2D points which constitute the vertices of the PE-PCD. - `Yp`: A set of 2D points which constitute the vertices of the Delaunay triangles. - `r`: A positive real number which serves as the expansion parameter in PE proximity region; must be $\ge 1$. - `M`: A 3D point in barycentric coordinates which serves as a center in the interior of each Delaunay triangle or circumcenter of each Delaunay triangle (for this, argument should be set as `M="CC"`), default for $M=(1,1,1)$ which is the center of mass of each triangle. ## Returns A `list` with three elements - **dom.num**: Domination number of the PE-PCD whose vertices are `Xp` points. PE proximity regions are constructed with respect to the Delaunay triangles based on the `Yp` points with expansion parameter $r \ge 1$. #- **mds**: A minimum dominating set of the PE-PCD whose vertices are `Xp` points - **ind.mds**: The vector of data indices of the minimum dominating set of the PE-PCD whose vertices are `Xp` points. - **tri.dom.nums**: The vector of domination numbers of the PE-PCD components for the Delaunay triangles. ## Examples ```r #nx is number of X points (target) and ny is number of Y points (nontarget) nx<-20; ny<-5; #try also nx<-40; ny<-10 or nx<-1000; ny<-10; set.seed(1) Xp<-cbind(runif(nx,0,1),runif(nx,0,1)) Yp<-cbind(runif(ny,0,.25), runif(ny,0,.25))+cbind(c(0,0,0.5,1,1),c(0,1,.5,0,1)) #try also Yp<-cbind(runif(ny,0,1),runif(ny,0,1)) M<-c(1,1,1) #try also M<-c(1,2,3) r<-1.5 #try also r<-2 PEdom.num(Xp,Yp,r,M) ``` ## References if(!exists(".Rdpack.currefs")) .Rdpack.currefs \<-new.env();Rdpack::insert_all_ref(.Rdpack.currefs) ## See Also `PEdom.num.tri`, `PEdom.num.tetra`, `dom.num.exact`, and `dom.num.greedy` ## Author(s) Elvan Ceyhan

PEdom.num function