Reconstruction conjecture

In graph theory, informally, the reconstruction conjecture says that graphs are determined uniquely by their subgraphs. It is due to Kelly and Ulam.

Formal statements

thumbnail|right|300px|A graph and the associated deck of single-vertex-deleted subgraphs. Note some of the cards show isomorphic graphs.

Given a graph <math>G = (V,E)</math>, a vertex-deleted subgraph of <math>G</math> is a subgraph formed by deleting exactly one vertex from <math>G</math>. By definition, it is an induced subgraph of <math>G</math>.

For a graph <math>G</math>, the deck of G, denoted <math>D(G)</math>, is the multiset of isomorphism classes of all vertex-deleted subgraphs of <math>G</math>. Each graph in <math>D(G)</math> 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 <math>G = (V,E)</math>, an edge-deleted subgraph of <math>G</math> is a subgraph formed by deleting exactly one edge from <math>G</math>.

For a graph <math>G</math>, the edge-deck of G, denoted <math>ED(G)</math>, is the multiset of all isomorphism classes of edge-deleted subgraphs of <math>G</math>. Each graph in <math>ED(G)</math> is called an edge-card.

Edge Reconstruction Conjecture: (Harary, 1964)

Verification

Both the reconstruction and set reconstruction conjectures have been verified for all graphs with at most 13 vertices by Brendan McKay.

In a probabilistic sense, it has been shown by Béla Bollobás that almost all graphs are reconstructible. This means that the probability that a randomly chosen graph on <math>n</math> vertices is not reconstructible goes to 0 as <math>n</math> goes to infinity. In fact, it was shown that not only are almost all graphs reconstructible, but in fact that the entire deck is not necessary to reconstruct them — almost all graphs have the property that there exist three cards in their deck that uniquely determine the graph.

Reconstructible graph families

The conjecture has been verified for a number of infinite classes of graphs (and, trivially, their complements).

Regular graphs - Regular Graphs are reconstructible by direct application of some of the facts that can be recognized from the deck of a graph. Given an <math>n</math>-regular graph <math>G</math> and its deck <math>D(G)</math>, we can recognize that the deck is of a regular graph by recognizing its degree sequence. Let us now examine one member of the deck <math>D(G)</math>, <math>G_i</math>. This graph contains some number of vertices with a degree of <math>n</math> and <math>n</math> vertices with a degree of <math>n-1</math>. We can add a vertex to this graph and then connect it to the <math>n</math> vertices of degree <math>n-1</math> to create an <math>n</math>-regular graph which is isomorphic to the graph which we started with. Therefore, all regular graphs are reconstructible from their decks. A particular type of regular graph which is interesting is the complete graph.
Trees
Maximal planar graphs
Maximal outerplanar graphs
Outerplanar graphs
Critical blocks

Reduction

The reconstruction conjecture is true if all 2-connected graphs are reconstructible.

Duality

The vertex reconstruction conjecture obeys the duality that if <math>G</math> can be reconstructed from its vertex deck <math>D(G)</math>, then its complement <math>G'</math> can be reconstructed from <math>D(G')</math> as follows: Start with <math>D(G')</math>, take the complement of every card in it to get <math>D(G)</math>, use this to reconstruct <math>G</math>, then take the complement again to get <math>G'</math>.

Edge reconstruction does not obey any such duality: Indeed, for some classes of edge-reconstructible graphs it is not known if their complements are edge reconstructible.

Other structures

It has been shown that the following are not in general reconstructible:

Digraphs: Infinite families of non-reconstructible digraphs are known, including tournaments (Stockmeyer) and non-tournaments (Stockmeyer). A tournament is reconstructible if it is not strongly connected. A weaker version of the reconstruction conjecture has been conjectured for digraphs, see new digraph reconstruction conjecture.
Hypergraphs, including all k-uniform hypergraphs for k>2 (Kocay).
Infinite graphs. If T is the tree where every vertex has countably infinite degree, then the union of two disjoint copies of T is hypomorphic, but not isomorphic, to T.(Fisher)
Locally finite graphs, which are graphs where every vertex has finite degree. The question of reconstructibility for locally finite infinite trees (the Harary-Schwenk-Scott conjecture from 1972) was a longstanding open problem until 2017, when a non-reconstructible tree of maximum degree 3 was found by Bowler et al.