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We study a new kind of proximity graphs called proportional-edge proximitycatch digraphs (PCDs)in a randomized setting. PCDs are a special kind of randomcatch digraphs that have been developed recently and have applications instatistical pattern classification and spatial point pattern analysis. PCDs arealso a special type of intersection digraphs; and for one-dimensional data, theproportional-edge PCD family is also a family of random interval catchdigraphs. We present the exact (and asymptotic) distribution of the dominationnumber 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 theexpansion and centrality parameters, thereby determine the parameters for whichthe asymptotic distribution is non-degenerate. We observe sudden jumps (fromdegeneracy to non-degeneracy or from a non-degenerate distribution to another)in the asymptotic distribution of the domination number at certain parametervalues.

The Distribution of the Domination Number of a Family of Random Interval Catch Digraphs