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. 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.

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) G = v1v1 v2v2 v3v3 v4v4 D( G) = v1v1 v1v1 v1v1 v2v2 v2v2 v2v2 v3v3 v3v3 v3v3 v4v4 v4v4 v4v4

Graph reconstruction D( G) = Can we obtain G from D(G)?

Graph reconstruction G = D( G) = Can we obtain G from D(G)?

Graph Reconstruction Conjecture Every graph with at least three vertices is reconstructible Unproven Unproven Verified for regular graphs (graphs in which all vertices have same number of edges). Verified for regular graphs (graphs in which all vertices have same number of edges). Verified for all graphs with at most 11 vertices. 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. 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. For almost all graphs, there exist 3 cards that uniquely determine the graph.

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

Reconstruction Number Logic 1. Get all cards of a given deck 2. Extend the cards 3. 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 Bollobás used proof via probability to prove  rn(G) is at least 3

Finding Reconstruction Number G = v1v1 v2v2 v3v3 v4v4 D( G) = v1v1 v1v1 v1v1 v2v2 v2v2 v2v2 v3v3 v3v3 v3v3 v4v4 v4v4 v4v4 1. Get all cards of a given deck

Finding Reconstruction Number 1. Get all cards of a given deck 2. Extend the cards D( G) = v1v1 v1v1 v1v1 v2v2 v2v2 v2v2 v3v3 v3v3 v3v3 v4v4 v4v4 v4v4 =>

Finding Reconstruction Number D( G) = v1v1 v1v1 v1v1 v2v2 v2v2 v2v2 v3v3 v3v3 v3v3 v4v4 v4v4 v4v4 => 1. Get all cards of a given deck 2. Extend the cards 3. The intersection of the extended cards is the solution

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