An integer distance graph is a graph G(Z,D) with the set of integers as vertex set and an edge joining two vertices u and?v if and only if ∣u - v∣D where D is a subset of the positive integers. It is known that x(G(Z,D) )=4 where P is a set of Prime numbers. So we can allocate the subsets D of P to four classes, accordingly as is 1 or 2 or 3 or 4. In this paper we have considered the open problem of characterizing class three and class four sets when the distance set D is not only a subset of primes P but also a special class of primes like Additive primes, Deletable primes, Wedderburn-Etherington Number primes, Euclid-Mullin sequence primes, Motzkin primes, Catalan primes, Schroder primes, Non-generous primes, Pell primes, Primeval primes, Primes of Binary Quadratic Form, Smarandache-Wellin primes, and Highly Cototient number primes. We also have indicated the membership of a number of special classes of prime numbers in class 2 category.
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