Graph Reconstruction

In 1941 Kelly and Ulam proposed the Graph Reconstruction Conjecture, and it has remained an open problem to this day[3]. The Graph Reconstruction Conjecture states simply that any simple finite undirected graph on 3 or more vertices can be uniquely identified (up to isomorphism) by the multiset of its unlabeled 1-vertex-deleted subgraphs. There are no known counter-examples to the Conjecture, and it is widely believed to be true, although yet unproven[2]. However, for some classes of graphs the conjecture has been proven to hold; specifically disconnected graphs, regular graphs, trees, and maximal planar graphs [2]. Through exhaustive computer search it has also been shown that all graphs of between 3 and 10 vertices are reconstructible [6]. From the original Graph Reconstruction Conjecture, there has sprung many new related problems. In 1964 the Edge Reconstruction Conjecture was formulated, which state that all simple undirected graphs with 4 or more edges can be uniquely identified (up to isomorphism) by the multiset of its unlabeled 1-edge-deleted subgraphs [5]. In 1957 Kelly introduced the concept of k-vertex-reconstruction, where k vertices are deleted to form each subgraph, rather than 1. Kelly further proved that for k > 0 there is a graph on 2k vertices which is not k-reconstructible [6], although behavior for larger graphs is unknown. More recently the question of if a graph is reconstructible, how many of its subgraphs are required to reconstruct it has been asked. This takes two forms, the Existential ( or Ally) Reconstruction Number (∃rn ), and the Universal ( or Adversarial) Reconstruction Number (∀rn ). The Existential Reconstruction Number is the minimum number of subgraphs required if they are carefully selected, while the Universal Reconstruction Number is the minimum number such that any multiset of subgraphs of that size can reconstruct the original graph. Similarly, the same question can be asked, but with the addition of the knowledge that a graph is in a certain class, leading to the concept of Class Reconstruction Numbers (Crn ). The concept of Edge Reconstruction Numbers exists as a 1-edge-deleted analogue to what is more specifically call Vertex Reconstruction Numbers (vrn ). Taken all together these concepts result in a veritable zoo of reconstruction numbers, many of which little is known about.