A Computational Verification of the Reconstruction Conjecture on Small Graphs
Alternative Title
Abstract
The reconstruction conjecture is perhaps the foremost unsolved problem in graph theory. It states that a simple undirected graph on at least three vertices is determined up to isomorphism by its collection of vertex-deleted subgraphs. The collection of vertex-deleted subgraphs is called the Deck of the Graph, and a graph which is uniquely determined by its deck is said to be reconstructible. Various types of graphs are known to be reconstructible such as regular graphs, trees, and disconnected graphs, but the reconstructability of
graphs in general is an unresolved problem. The goal of our research is to provide computer verification of the reconstruction
conjecture for graphs on up to nine vertices. Nine may seem like a small number, but there are over 68 billion graphs on nine vertices. Many of these graphs are isomorphic, and graph isomorphism is a central idea related to graph reconstruction. Two graphs are isomorphic if there is a bijection between the vertex sets of the graphs that preserves adjacency. A brute force approach to checking whether two graphs of order 9 are isomorphic requires 9! = 362880 equivalence checks. With numbers this large, verification of the reconstruction conjecture is extremely time consuming, even with fast computers and efficient algorithms. Based on specific isomorphism theorems we were able to prove and efficient coding, we have written a program that can verify the reconstruction conjecture for graphs on eight vertices in four seconds, on nine vertices in 20 minutes, and we project that we could resolve the conjecture for graphs on 10 vertices in roughly a month. A secondary goal of this research is to investigate the maximum deck intersection size for two non-isomorphic graphs. While it may be true that all graphs are determined by their decks of vertex deleted subgraphs, an interesting question is how many vertex-deleted subgraphs two non-isomorphic graphs can have in common. We report our findings related to this question for graphs on nine vertices.