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

TL;DR

This paper analyzes the distribution of the domination number in a new class of random interval catch digraphs called proportional-edge PCDs, providing exact and asymptotic results for uniform and non-uniform data.

Contribution

It introduces the distribution analysis of the domination number for proportional-edge PCDs, including extensions and parameter conditions affecting asymptotic behavior.

Findings

Exact and asymptotic distribution formulas derived.

Identification of parameter values causing distribution jumps.

Extensions relaxing parameters and analyzing their effects.

Abstract

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.