Graph Reconstruction Conjecture

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Graph Reconstruction Conjecture
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Graph Reconstruction Conjecture

• Proposed by S.M. Ulan P.J. Kelly in 1941
• The conjecture states that every graph with at
  least 3 vertices is reconstructible a graph G is
  reconstructible if it is defined by its
  vertex-deleted subgraphs.

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Definitions
DECK the multi-subset of vertex-deleted
subgraphs of G
CARD a vertex-deleted subgraph of G
(we do not worry about graphs that are different
through labelling nodes differently, but are
isomorphic)
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Graph reconstruction
Can we obtain G from D(G)?
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Graph reconstruction
Can we obtain G from D(G)?
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Graph Reconstruction Conjecture

• Every graph with at least three vertices is
  reconstructible
• Unproven
• Verified for regular graphs (graphs in which all
  vertices have same number of edges).
• Verified for all graphs with at most 11 vertices.
• Bollobás Used probability to show that almost
  all graphs are reconstructible probability that
  a randomly chosen graph with n vertices is not
  reconstructible approaches zero as n approaches
  infinity.
• For almost all graphs, there exist 3 cards that
  uniquely determine the graph.

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Reconstruction Number

• ?rn(G)
• The number of cards required to reconstruct the
  original graph.
• gt 2 because at least 2 different graphs can be
  generated from a deck of 2 (difference between
  the 2 graphs is 1 edge)

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Reconstruction Number Logic

• Get all cards of a given deck
• Extend the cards
• The intersection of the extended cards is the
  solution
• Conjecture ?rn(G) is at most (n/2 1)
• Bollobás used proof via probability to prove
  ?rn(G) is at least 3

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Finding Reconstruction Number

1. Get all cards of a given deck

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Finding Reconstruction Number

1. Get all cards of a given deck
2. Extend the cards

gt
gt
gt
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Finding Reconstruction Number

1. Get all cards of a given deck
2. Extend the cards
3. The intersection of the extended cards is the
   solution

gt
gt
gt
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• Proven generally not reconstructible
• Digraphs (AKA Directed graph)
• Hypergraphs edges can connect to any number of
  vertices.
• Infinite Graphs Infinitely many vertices and/or
  edges