Isomorphic pastings and the two possible structures for a pair of graphs having the same deck

Abstract: When G denotes a graph, the unlabeled subgraph obtained by deleting a vertex from G is called a card of G and the collection of all cards of G is the deck of G. A graph having the same deck as G is called a hypomorph of G. A graph is called reconstructible if it is isomorphic to all its hypomorphs. Reconstruction Conjecture claims that all graphs are reconstructible and it is open. A representation of a hypomorph of G in terms of two of its cards, called pasting, is introduced. Isomorphic pastings of two cards is defined. In the case of a digraph, a card with which the degree triple of the deleted vertex is also given is called a degree associated card or dacard. Dadeck, dareconstructible digraphs, dapastings and isomorphic dapastings based on dacards are defined analogously. DARC claims that all digraphs are dareconstructible and it is also open. Results: Two hypomorphs G and H of a graph are isomorphic if and only if a pair of cards in their common deck is pasted isomorphically in both G and H. Either every pair of cards in their common deck is pasted isomorphically in both G and H, or no pair of cards is pasted isomorphically in both G and H. Results analogous to the above hold for dapastings in dahypomorphs of a digraph. Some results on pastings are proved and two graph parameters are reconstructed. The neighborhood degree quintuple of a vertex and a new family of digraphs are dareconstructible. New approaches for proving the reconstruction conjecture and DARC by the method of contradiction arise.