# Centers for non-degenerate asymptotic distribution of domination number of Proportional Edge Proximity Catch Digraphs (PE-PCDs) Returns the centers which yield nondegenerate asymptotic distribution for the domination number of PE-PCD for uniform data in a triangle, `tri`$=T(v_1,v_2,v_3)$. PE proximity region is defined with respect to the triangle `tri` with expansion parameter `r` in $(1,1.5]$. Vertex regions are defined with the centers that are output of this function. Centers are stacked row-wise with row number is corresponding to the vertex row number in `tri` (see the examples for an illustration). The center labels 1,2,3 correspond to the vertices $M_1$, $M_2$, and $M_3$ (which are the three centers for `r` in $(1,1.5)$ which becomes center of mass for $r=1.5$.). 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 center.nondegPE(tri, r) ``` ## Arguments - `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 in $(1,1.5]$ for this function. ## Returns The centers (stacked row-wise) which give nondegenerate asymptotic distribution for the domination number of PE-PCD for uniform data in a triangle, `tri`. ## Examples ```r A<-c(1,1); B<-c(2,0); C<-c(1.5,2); Tr<-rbind(A,B,C); r<-1.35 Ms<-center.nondegPE(Tr,r) Ms Xlim<-range(Tr[,1]) Ylim<-range(Tr[,2]) xd<-Xlim[2]-Xlim[1] yd<-Ylim[2]-Ylim[1] plot(Tr,pch=".",xlab="",ylab="", main="Centers of nondegeneracy\n for the PE-PCD in a triangle", axes=TRUE,xlim=Xlim+xd*c(-.05,.05),ylim=Ylim+yd*c(-.05,.05)) polygon(Tr) points(Ms,pch=".",col=1) polygon(Ms,lty = 2) xc<-Tr[,1]+c(-.02,.02,.02) yc<-Tr[,2]+c(.02,.02,.03) txt.str<-c("A","B","C") text(xc,yc,txt.str) xc<-Ms[,1]+c(-.04,.04,.03) yc<-Ms[,2]+c(.02,.02,.05) txt.str<-c("M1","M2","M3") text(xc,yc,txt.str) ``` ## References if(!exists(".Rdpack.currefs")) .Rdpack.currefs \<-new.env();Rdpack::insert_all_ref(.Rdpack.currefs) ## Author(s) Elvan Ceyhan

center.nondegPE function