# The function `gammakCSstd.tri` is for $k$ ($k=2,3,4,5$) points constituting a dominating set for Central Similarity Proximity Catch Digraphs (CS-PCDs) - standard equilateral triangle case Four indicator functions: `Idom.num2CSstd.tri`, `Idom.num3CSstd.tri`, `Idom.num4CSstd.tri`, `Idom.num5CSstd.tri` and `Idom.num6CSstd.tri`. The function `gammakCSstd.tri` returns I({`p1`,...,`pk`} is a dominating set of the CS-PCD) where vertices of CS-PCD are the 2D data set `Xp`, that is, returns 1 if {`p1`,...,`pk`} is a dominating set of CS-PCD, returns 0 otherwise for $k=2,3,4,5,6$. CS proximity region is constructed with respect to $T_e=T(A,B,C)=T((0,0),(1,0),(1/2,\sqrt{3}/2))$ with expansion parameter $t>0$ and edge regions are based on center of mass $CM=(1/2,\sqrt{3}/6)$. `ch.data.pnts` is for checking whether points `p1`,...,`pk` are data points in `Xp` or not (default is `FALSE`), so by default this function checks whether the points `p1`,...,`pk` would be a dominating set if they actually were in the data set. See also (if(!exists(".Rdpack.currefs")) .Rdpack.currefs \<-new.env();Rdpack::insert_citeOnly(keys="ceyhan:Phd-thesis,ceyhan:comp-geo-2010,ceyhan:mcap2012;textual",package="pcds",cached_env=.Rdpack.currefs) ). ```r Idom.num2CSstd.tri(p1, p2, Xp, t, ch.data.pnts = FALSE) Idom.num3CSstd.tri(p1, p2, p3, Xp, t, ch.data.pnts = FALSE) Idom.num4CSstd.tri(p1, p2, p3, p4, Xp, t, ch.data.pnts = FALSE) Idom.num5CSstd.tri(p1, p2, p3, p4, p5, Xp, t, ch.data.pnts = FALSE) Idom.num6CSstd.tri(p1, p2, p3, p4, p5, p6, Xp, t, ch.data.pnts = FALSE) ``` ## Arguments - `p1, p2, p3, p4, p5, p6`: The points {$p1,\ldots,pk$} are $k$ 2D points (for $k=2,3,4,5,6$) to be tested for constituting a dominating set of the CS-PCD. - `Xp`: A set of 2D points which constitutes the vertices of the CS-PCD. - `t`: A positive real number which serves as the expansion parameter in CS proximity region. - `ch.data.pnts`: A logical argument for checking whether points {$p1,\ldots,pk$} are data points in `Xp` or not (default is `FALSE`). ## Returns The function `gammakCSstd.tri` returns {`p1`,...,`pk`} is a dominating set of the CS-PCD) where vertices of the CS-PCD are the 2D data set `Xp`), that is, returns 1 if {`p1`,...,`pk`} is a dominating set of CS-PCD, returns 0 otherwise. ## Examples ```r set.seed(123) #Examples for Idom.num2CSstd.tri t<-1.5 n<-10 #try also 10, 20 (it may take longer for larger n) set.seed(1) Xp<-runif.std.tri(n)$gen.points Idom.num2CSstd.tri(Xp[1,],Xp[2,],Xp,t) Idom.num2CSstd.tri(c(.2,.2),Xp[2,],Xp,t) ind.gam2<-vector() for (i in 1:(n-1)) for (j in (i+1):n) {if (Idom.num2CSstd.tri(Xp[i,],Xp[j,],Xp,t)==1) ind.gam2<-rbind(ind.gam2,c(i,j))} ind.gam2 #Examples for Idom.num3CSstd.tri t<-1.5 n<-10 #try also 10, 20 (it may take longer for larger n) set.seed(1) Xp<-runif.std.tri(n)$gen.points Idom.num3CSstd.tri(Xp[1,],Xp[2,],Xp[3,],Xp,t) ind.gam3<-vector() for (i in 1:(n-2)) for (j in (i+1):(n-1)) for (k in (j+1):n) {if (Idom.num3CSstd.tri(Xp[i,],Xp[j,],Xp[k,],Xp,t)==1) ind.gam3<-rbind(ind.gam3,c(i,j,k))} ind.gam3 #Examples for Idom.num4CSstd.tri t<-1.5 n<-10 #try also 10, 20 (it may take longer for larger n) set.seed(1) Xp<-runif.std.tri(n)$gen.points Idom.num4CSstd.tri(Xp[1,],Xp[2,],Xp[3,],Xp[4,],Xp,t) ind.gam4<-vector() for (i in 1:(n-3)) for (j in (i+1):(n-2)) for (k in (j+1):(n-1)) for (l in (k+1):n) {if (Idom.num4CSstd.tri(Xp[i,],Xp[j,],Xp[k,],Xp[l,],Xp,t)==1) ind.gam4<-rbind(ind.gam4,c(i,j,k,l))} ind.gam4 Idom.num4CSstd.tri(c(.2,.2),Xp[2,],Xp[3,],Xp[4,],Xp,t,ch.data.pnts = FALSE) #gives an error message if ch.data.pnts = TRUE since not all points are data points in Xp #Examples for Idom.num5CSstd.tri t<-1.5 n<-10 #try also 10, 20 (it may take longer for larger n) set.seed(1) Xp<-runif.std.tri(n)$gen.points Idom.num5CSstd.tri(Xp[1,],Xp[2,],Xp[3,],Xp[4,],Xp[5,],Xp,t) ind.gam5<-vector() for (i1 in 1:(n-4)) for (i2 in (i1+1):(n-3)) for (i3 in (i2+1):(n-2)) for (i4 in (i3+1):(n-1)) for (i5 in (i4+1):n) {if (Idom.num5CSstd.tri(Xp[i1,],Xp[i2,],Xp[i3,],Xp[i4,],Xp[i5,],Xp,t)==1) ind.gam5<-rbind(ind.gam5,c(i1,i2,i3,i4,i5))} ind.gam5 Idom.num5CSstd.tri(c(.2,.2),Xp[2,],Xp[3,],Xp[4,],Xp[5,],Xp,t,ch.data.pnts = FALSE) #gives an error message if ch.data.pnts = TRUE since not all points are data points in Xp #Examples for Idom.num6CSstd.tri t<-1.5 n<-10 #try also 10, 20 (it may take longer for larger n) set.seed(1) Xp<-runif.std.tri(n)$gen.points Idom.num6CSstd.tri(Xp[1,],Xp[2,],Xp[3,],Xp[4,],Xp[5,],Xp[6,],Xp,t) ind.gam6<-vector() for (i1 in 1:(n-5)) for (i2 in (i1+1):(n-4)) for (i3 in (i2+1):(n-3)) for (i4 in (i3+1):(n-2)) for (i5 in (i4+1):(n-1)) for (i6 in (i5+1):n) {if (Idom.num6CSstd.tri(Xp[i1,],Xp[i2,],Xp[i3,],Xp[i4,],Xp[i5,],Xp[i6,],Xp,t)==1) ind.gam6<-rbind(ind.gam6,c(i1,i2,i3,i4,i5,i6))} ind.gam6 Idom.num6CSstd.tri(c(.2,.2),Xp[2,],Xp[3,],Xp[4,],Xp[5,],Xp[6,],Xp,t,ch.data.pnts = FALSE) #gives an error message if ch.data.pnts = TRUE since not all points are data points in Xp ``` ## See Also `Idom.num1CSstd.tri`, `Idom.num2PEtri` and `Idom.num2PEtetra` ## Author(s) Elvan Ceyhan

funsCSGamTe function