Reconstruction of Graphs

A graph is reconstructible if it is determined up to isomorphism from the collection of all its one-vertex deleted unlabeled subgraphs. One of the foremost unsolved problems in Graph Theory is the Reconstruction Conjecture, which asserts that every graph G on at least three vertices is reconstructible. In 1980’s, tremendous work was done and many significant results have been produced on the problem and its variations. During the last three decades, work on it has slowed down gradually. P. J. Kelly (1957) first noted that trees are reconstructible; but the proof is quite lengthy. A short proof, due to Greenwell and Hemminger (1973), was given which is based on a simple, but powerful, counting theorem. This chapter deals with the counting theorem and its subsequent applications; also it ends up with a reduction of the Reconstruction Conjecture using distance and connectedness, which may lead to the final solution of the conjecture.