1d:T797,# The domination number of Proportional Edge Proximity Catch Digraph (PE-PCD) - one tetrahedron 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 tetrahedron `th` with expansion parameter $r \ge 1$ and vertex regions are based on the center `M` which is circumcenter (`"CC"`) or center of mass (`"CM"`) of `th` with default=`"CM"`. See also (if(!exists(".Rdpack.currefs")) .Rdpack.currefs \<-new.env();Rdpack::insert_citeOnly(keys="ceyhan:Phd-thesis,ceyhan:comp-geo-2010;textual",package="pcds",cached_env=.Rdpack.currefs) ). ```r PEdom.num.tetra(Xp, th, r, M = "CM") ``` ## Arguments - `Xp`: A set of 3D points which constitute the vertices of the digraph. - `th`: A $4 \times 3$ matrix with each row representing a vertex of the tetrahedron. - `r`: A positive real number which serves as the expansion parameter in PE proximity region; must be $\ge 1$. - `M`: The center to be used in the construction of the vertex regions in the tetrahedron, `th`. Currently it only takes `"CC"` for circumcenter and `"CM"` for center of mass; default=`"CM"`. ## 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(0,0,0); B<-c(1,0,0); C<-c(1/2,sqrt(3)/2,0); D<-c(1/2,sqrt(3)/6,sqrt(6)/3) tetra<-rbind(A,B,C,D) n<-10 #try also n<-20 Xp<-runif.tetra(n,tetra)$g M<-"CM" #try also M<-"CC" r<-1.25 PEdom.num.tetra(Xp,tetra,r,M) P1<-c(.5,.5,.5) PEdom.num.tetra(P1,tetra,r,M) ``` ## References if(!exists(".Rdpack.currefs")) .Rdpack.currefs \<-new.env();Rdpack::insert_all_ref(.Rdpack.currefs) ## See Also `PEdom.num.tri` ## Author(s) Elvan Ceyhan

PEdom.num.tetra function