# The domination number of Proportional Edge Proximity Catch Digraph (PE-PCD) with non-degeneracy centers - multiple interval case Returns the domination number, a minimum dominating set of PE-PCD whose vertices are the 1D data set `Xp`, and the domination numbers for partition intervals based on `Yp` when PE-PCD is constructed with vertex regions based on non-degeneracy centers. `Yp` determines the end points of the intervals (i.e., partition the real line via intervalization). If there are duplicates of `Yp` points, only one point is retained for each duplicate value, and a warning message is printed. PE proximity regions are defined with respect to the intervals based on `Yp` points with expansion parameter $r \ge 1$ and vertex regions in each interval are based on the centrality parameter `c` which is one of the 2 values of `c` (i.e., $c \in \{(r-1)/r,1/r\}$) that renders the asymptotic distribution of domination number to be non-degenerate for a given value of `r` in $(1,2)$ and `c` is center of mass for $r=2$. These values are called non-degeneracy centrality parameters and the corresponding centers are called nondegeneracy centers. ```r PEdom.num1Dnondeg(Xp, Yp, r) ``` ## Arguments - `Xp`: A set of 1D points which constitute the vertices of the PE-PCD. - `Yp`: A set of 1D points which constitute the end points of the intervals which partition the real line. - `r`: A positive real number which serves as the expansion parameter in PE proximity region; must be in $(1,2]$ here. ## Returns A `list` with three elements - **dom.num**: Domination number of PE-PCD with vertex set `Xp` and expansion parameter $r in (1,2]$ and centrality parameter $c \in \{(r-1)/r,1/r\}$. - **mds**: A minimum dominating set of the PE-PCD. - **ind.mds**: The data indices of the minimum dominating set of the PE-PCD whose vertices are `Xp` points. - **int.dom.nums**: Domination numbers of the PE-PCD components for the partition intervals. ## Examples ```r a<-0; b<-10 r<-1.5 #nx is number of X points (target) and ny is number of Y points (nontarget) nx<-15; ny<-4; #try also nx<-40; ny<-10 or nx<-1000; ny<-10; set.seed(1) Xp<-runif(nx,a,b) Yp<-runif(ny,a,b) PEdom.num1Dnondeg(Xp,Yp,r) PEdom.num1Dnondeg(Xp,Yp,r=1.25) ``` ## See Also `PEdom.num.nondeg` ## Author(s) Elvan Ceyhan

PEdom.num1Dnondeg function