Reconstruction conjecture

 Are graphs uniquely determined by their subgraphs?

Informally, the reconstruction conjecture in graph theory says that graphs are determined uniquely by their subgraphs. It is due to Kelly[1] and Ulam.[2]

Formal statements

Given a graph $G = (V,E)$, a vertex-deleted subgraph of $G$ is a subgraph formed by deleting exactly one vertex from $G$. Clearly, it is an induced subgraph of $G$.

For a graph $G$, the deck of G, denoted $D(G)$, is the multiset of all vertex-deleted subgraphs of $G$. Each graph in $D(G)$ 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[3] 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 $G = (V,E)$, an edge-deleted subgraph of $G$ is a subgraph formed by deleting exactly one edge from $G$.

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

• Edge Reconstruction Conjecture: (Harary, 1964)[3] Any two graphs with at least four edges and having the same edge-decks are isomorphic.

Verification

Both the reconstruction and set reconstruction conjectures have been verified for all graphs with at most 11 vertices (McKay[4]).

In a probabilistic sense, it has been shown (Bollobás[5]) that almost all graphs are reconstructible. This means that the probability that a randomly chosen graph on $n$ vertices is not reconstructible goes to 0 as $n$ 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.

• Regular graphs[6] - Regular Graphs are reconstructible by direct application of some of the facts that can be recognized from the deck of a graph. Given an $n$-regular graph $G$ and its deck $D(G)$, 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 $d(G)$, $G_i$. This graph contains some number of vertices with a degree of $n$ and $n$ vertices with a degree of $n-1$. We can add a vertex to this graph and then connect it to the $n$ vertices of degree $n-1$ to create an $n$-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.[7]
• Trees[6]
• Disconnected graphs[6]
• Unit interval graphs [8]
• Separable graphs without end vertices
• Maximal planar graphs
• Maximal outerplanar graphs
• Outerplanar graphs
• Critical blocks

Recognizable properties

In context of the reconstruction conjecture, a graph property is called recognizable if one can determine the property from the deck of a graph. The following properties of graphs are recognizable:

• Order of the graph – The order of a graph $G$, $|V(G)|$ is recognizable from $D(G)$ as the multiset $D(G)$ contains each subgraph of $G$ created by deleting one vertex of $G$. Hence $|V(G)| = |D(G)|$ [7]
• Number of edges of the graph – The number of edges in a graph $G$ with $n$ vertices, $|E(G)|$ is recognizable. First note that each edge of $G$ occurs in $n-2$ members of $D(G)$. This is true by the definition of $D(G)$ which ensures that each edge is included every time that each of the vertices it is incident with is included in a member of $D(G)$, so an edge will occur in every member of $D(G)$ except for the two in which its endpoints are deleted. Hence, $|E(G)| = \sum \frac{q_i}{n-2}$ where $q_i$ is the number of edges in the ith member of $D(G)$ [7]
• Degree sequence – The degree sequence of a graph $G$ is recognizable because the degree of every vertex is recognizable. To find the degree of a vertex $v_i$, we will examine the graph created by deleting it, $G_i$. This graph contains all of the edges not incident with $v_i$,so if $q_i$ is the number of edges in $G_i$ and $q$ is the number of edges in $G$, then $q - q_i = \deg(v_i)$. If we can tell the degree of every vertex in the graph, we can tell the degree sequence of the graph.[7]
• Tutte polynomial
• Planarity
• The types of spanning trees in a graph
• Chromatic polynomial
• Being a perfect graph or an interval graph, or some other subclasses of perfect graphs [8]

Reduction

The reconstruction conjecture is true if all 2-connected graphs are reconstructible [9]

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[10]) and non-tournaments (Stockmeyer[11]). A tournament is reconstructible if it is not strongly connected.[12] A weaker version of the reconstruction conjecture has been conjectured for digraphs, see New digraph reconstruction conjecture.
• Hypergraphs (Kocay[13]).
• Infinite graphs. Let T be a tree on an infinite number of vertices such that every vertex has infinite degree. The counterexample is T and 2T. The question of reconstructibility for locally finite infinite graphs is still open.

For further information on this topic, see the survey by Nash-Williams.[14]

References

1. ^ Kelly, P. J., A congruence theorem for trees, Pacific J. Math. 7 (1957), 961–968.
2. ^ Ulam, S. M., A collection of mathematical problems, Wiley, New York, 1960.
3. ^ a b Harary, F., On the reconstruction of a graph from a collection of subgraphs. In Theory of Graphs and its Applications (Proc. Sympos. Smolenice, 1963). Publ. House Czechoslovak Acad. Sci., Prague, 1964, pp. 47–52.
4. ^ McKay, B. D., Small graphs are reconstructible, Australas. J. Combin. 15 (1997), 123–126.
5. ^ Bollobás, B., Almost every graph has reconstruction number three, J. Graph Theory 14 (1990), 1–4.
6. ^ a b c Harary, F. (1974), "A survey of the reconstruction conjecture", A survey of the reconstruction conjecture, Graphs and Combinatorics. Lecture Notes in Mathematics 406, Springer, pp. 18–28, doi:10.1007/BFb0066431
7. ^ a b c d Wall, Nicole. "The Reconstruction Conjecture". Retrieved 2014-03-31.
8. ^ a b von Rimscha, M.: Reconstructibility and perfect graphs. Discrete Mathematics 47, 283–291 (1983)
9. ^ Yang Yongzhi:The reconstruction conjecture is true if all 2-connected graphs are reconstructible. Journal of graph theory 12, 237–243 (1988)
10. ^ Stockmeyer, P. K., The falsity of the reconstruction conjecture for tournaments, J. Graph Theory 1 (1977), 19–25.
11. ^ Stockmeyer, P. K., A census of non-reconstructable digraphs, I: six related families, J. Combin. Theory Ser. B 31 (1981), 232–239.
12. ^ Harary, F. and Palmer, E., On the problem of reconstructing a tournament from sub-tournaments, Monatsh. Math. 71 (1967), 14–23.
13. ^ Kocay, W. L., A family of nonreconstructible hypergraphs, J. Combin. Theory Ser. B 42 (1987), 46–63.
14. ^ Nash-Williams, C. St. J. A., The Reconstruction Problem, in Selected topics in graph theory, 205–236 (1978).