Some work towards the proof of the reconstruction conjecture

In this paper we define five families of simple graphs (m̸F1,m̸F2,m̸F3,m̸F4 and m̸F5) such that the reconstruction conjecture is true if reconstruction is proved for either the families m̸F1,m̸F2 and m̸F3 or the families m̸F3,m̸F4 and m̸F5. These families are quite restrictive in that each has diameter two or three. Then we prove that families m̸F1,m̸F2,m̸F3,m̸F4 and m̸F5 are recognizable by proving that graphs of diameter two and edge-minimal graphs of diameter two are recognizable. The reconstruction conjecture is thus reduced to showing weak reconstructibility for either of the two sets of families.