Reconstruction Conjecture - Formal Statements

Formal Statements

Given a graph, a vertex-deleted subgraph of is a subgraph formed by deleting exactly one vertex from . Clearly, it is an induced subgraph of .

For a graph, the deck of G, denoted, is the multiset of all vertex-deleted subgraphs of . Each graph in is called a card. Two graphs that have the same deck are said to be hypomorphic.

With these definitions, the conjecture can be stated as:

Reconstruction Conjecture: Any two hypomorphic graphs on at least three vertices are isomorphic.

(The requirement that the graphs have at least three vertices is necessary because both graphs on two vertices have the same decks.)

Harary suggested a stronger version of the conjecture:

Set Reconstruction Conjecture: Any two graphs on at least four vertices with the same sets of vertex-deleted subgraphs are isomorphic.

Given a graph, an edge-deleted subgraph of is a subgraph formed by deleting exactly one edge from .

For a graph, the edge-deck of G, denoted, is the multiset of all edge-deleted subgraphs of . Each graph in is called an edge-card.

Edge Reconstruction Conjecture: (Harary, 1964) Any two graphs with at least four edges and having the same edge-decks are isomorphic.