  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.
