The graph reconstruction conjecture asserts that every finite simple graph on
at least three vertices can be reconstructed up to isomorphism from its deck -
the collection of its vertex-deleted subgraphs. Kocay's Lemma is an important
tool in graph reconstruction. Roughly speaking, given the deck of a graph G
and any finite sequence of graphs, it gives a linear constraint that every
reconstruction of G must satisfy.
Let ψ(n) be the number of distinct (mutually non-isomorphic) graphs on
n vertices, and let d(n) be the number of distinct decks that can be
constructed from these graphs. Then the difference ψ(n)−d(n) measures
how many graphs cannot be reconstructed from their decks. In particular, the
graph reconstruction conjecture is true for n-vertex graphs if and only if
ψ(n)=d(n).
We give a framework based on Kocay's lemma to study this discrepancy. We
prove that if M is a matrix of covering numbers of graphs by sequences of
graphs, then d(n)≥rankR(M). In particular, all
n-vertex graphs are reconstructible if one such matrix has rank ψ(n). To
complement this result, we prove that it is possible to choose a family of
sequences of graphs such that the corresponding matrix M of covering numbers
satisfies d(n)=rankR(M).Comment: 12 pages, 2 figure