  We study a new kind of proximity graphs called proportional-edge proximity
catch digraphs (PCDs)in a randomized setting. PCDs are a special kind of random
catch digraphs that have been developed recently and have applications in
statistical pattern classification and spatial point pattern analysis. PCDs are
also a special type of intersection digraphs; and for one-dimensional data, the
proportional-edge PCD family is also a family of random interval catch
digraphs. We present the exact (and asymptotic) distribution of the domination
number of this PCD family for uniform (and non-uniform) data in one dimension.
We also provide several extensions of this random catch digraph by relaxing the
expansion and centrality parameters, thereby determine the parameters for which
the asymptotic distribution is non-degenerate. We observe sudden jumps (from
degeneracy to non-degeneracy or from a non-degenerate distribution to another)
in the asymptotic distribution of the domination number at certain parameter
values.
