graph reconstruction conjecture. proposed by s.m. ulan & p.j. kelly in 1941: the conjecture...

Graph Reconstruction Graph Reconstruction ConjectureConjecture

Graph Reconstruction Conjecture Graph Reconstruction Conjecture

Proposed by S.M. Ulan & P.J. Kelly in 1941:Proposed by S.M. Ulan & P.J. Kelly in 1941:

The conjecture states that every graph with at The conjecture states that every graph with at least 3 vertices is reconstructible; a graph G least 3 vertices is reconstructible; a graph G is reconstructible if it is defined by its vertex-is reconstructible if it is defined by its vertex-

deleted subgraphs. deleted subgraphs.

DefinitionsDefinitions

DECK = the multi-subset of vertex-deleted DECK = the multi-subset of vertex-deleted subgraphs of Gsubgraphs of G

CARD = a vertex-deleted subgraph of GCARD = a vertex-deleted subgraph of G

(we do not worry about graphs that are different (we do not worry about graphs that are different through labelling nodes differently, but are isomorphic)through labelling nodes differently, but are isomorphic)

G =

v1 v2

v3v4

D(D(G) = G) =

v1v1

v1v2v2 v2

v3 v3v3v4 v4

v4

Graph reconstructionGraph reconstruction

D(D(G) = G) =

Can we obtain G from D(G)?Can we obtain G from D(G)?

Graph reconstructionGraph reconstruction

G =

D(D(G) = G) =

Can we obtain G from D(G)?Can we obtain G from D(G)?

Graph Reconstruction ConjectureGraph Reconstruction ConjectureEvery graph with at least three vertices is reconstructibleEvery graph with at least three vertices is reconstructible UnprovenUnproven Verified for Verified for regular graphsregular graphs (graphs in which all vertices (graphs in which all vertices

have same number of edges).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 Bollobás: Used probability to show that almost all

graphs are reconstructible; probability that a randomly graphs are reconstructible; probability that a randomly chosen graph with chosen graph with n n vertices is not reconstructible vertices is not reconstructible approaches zero as approaches zero as n n approaches infinity.approaches infinity. For almost all graphs, there exist 3 cards that uniquely For almost all graphs, there exist 3 cards that uniquely

determine the graph.determine the graph.

Reconstruction NumberReconstruction Number

rn(G)rn(G)

The number of cards required to reconstruct The number of cards required to reconstruct the original graph. the original graph.

> 2 because at least 2 different graphs can be > 2 because at least 2 different graphs can be generated from a deck of 2 (difference generated from a deck of 2 (difference between the 2 graphs is 1 edge)between the 2 graphs is 1 edge)

Reconstruction Number LogicReconstruction Number Logic

1.1. Get all cards of a given deckGet all cards of a given deck

2.2. Extend Extend the cardsthe cards

3.3. The intersection of the extended cards is the The intersection of the extended cards is the solutionsolution

Conjecture: Conjecture: rn(G)rn(G) is at most ( is at most (n/2 +1) n/2 +1) • Bollobás used proof via probability to prove Bollobás used proof via probability to prove

rn(G)rn(G) is at least 3 is at least 3

Finding Reconstruction NumberFinding Reconstruction Number

G =v1 v2

v3v4

D(D(G) = G) = v1

v1

v1v2

v2 v2v3 v3v3v4

v4v4



1.1. Get all cards of a given deckGet all cards of a given deck2.2. Extend Extend the cardsthe cards

D(D(G) = G) = v1

v1

v1v2

v2 v2v3 v3v3v4

v4v4

=>

=>

=>


D(D(G) = G) = v1

v1

v1v2

v2 v2v3 v3v3v4

v4v4

=>

=>

=>


2.2. Extend Extend the cardsthe cards

3.3. The intersection of the extended cards is the solutionThe intersection of the extended cards is the solution

Proven generally not reconstructible:Proven generally not reconstructible: Digraphs (AKA Directed graph)Digraphs (AKA Directed graph)

Hypergraphs: edges can connect to any Hypergraphs: edges can connect to any number of vertices. number of vertices.

Infinite Graphs: Infinitely many vertices Infinite Graphs: Infinitely many vertices and/or edgesand/or edges

graph reconstruction conjecture. proposed by s.m. ulan & p.j. kelly in 1941: the conjecture...

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