Modular Decomposition and the Reconstruction Conjecture

We prove that a large family of graphs which are decomposable with respect to the modular decomposition can be reconstructed from their collection of vertex-deleted subgraphs.Comment: 9 pages, 2 figure

Switching Reconstruction of Digraphs

Switching about a vertex in a digraph means to reverse the direction of every edge incident with that vertex. Bondy and Mercier introduced the problem of whether a digraph can be reconstructed up to isomorphism from the multiset of isomorphism types of digraphs obtained by switching about each vertex. Since the largest known non-reconstructible oriented graphs have 8 vertices, it is natural to ask whether there are any larger non-reconstructible graphs. In this paper we continue the investigation of this question. We find that there are exactly 44 non-reconstructible oriented graphs whose underlying undirected graphs have maximum degree at most 2. We also determine the full set of switching-stable oriented graphs, which are those graphs for which all switchings return a digraph isomorphic to the original

Reconstruction of Finite Truncated Semi-Modular Lattices

AbstractIn this paper, we prove reconstruction results for truncated lattices. The main results are that truncated lattices that contain a 4-crown and truncated semi-modular lattices are reconstructible. Reconstruction of the truncated lattices not covered by this work appears challenging. Indeed, the remaining truncated lattices possess very little lattice-typical structure. This seems to indicate that further progress on the reconstruction of truncated lattices is closely correlated with progress on reconstructing ordered sets in general

Reconstruction of the Ranks of the Nonextremal Cards and of Ordered Sets with a Minmax Pair of Pseudo-Similar Points

For every ordered set, we reconstruct the deck obtained by removal of the elements of rank r that are neither minimal nor maximal. Consequently, we also reconstruct the deck obtained by removal of the extremal, that is, minimal or maximal, elements. Finally, we reconstruct the ordered sets with a minmax pair of pseudo-similar points

Graph reconstruction numbers

The Reconstruction Conjecture is one of the most important open problems in graph theory today. Proposed in 1942, the conjecture posits that every simple, finite, undirected graph with more than three vertices can be uniquely reconstructed up to isomorphism given the multiset of subgraphs produced by deleting each vertex of the original graph. Related to the Reconstruction Conjecture, reconstruction numbers concern the minimum number of vertex deleted subgraphs required to uniquely identify a graph up to isomorphism. During the summer of 2004, Jennifer Baldwin completed an MS project regarding reconstruction numbers. In it, she calculated reconstruction numbers for all graphs G where 2 \u3c |V(G)| \u3c 9. This project expands the computation of reconstruction numbers up to all graphs with ten vertices and a specific class of graphs with eleven vertices. Whereas Jennifer\u27s project focused on a statistical analysis of reconstruction number results, we instead focus on theorizing the causes of high reconstruction numbers. Accordingly, this project establishes the reasons behind all high existential reconstruction numbers identified within the set of all graphs G where 2 \u3c |V(G)| \u3c 11 and identifies new classes of graphs that have large reconstruction numbers. Finally, we consider 2-reconstructibility - the ability to reconstruct a graph G from the multiset of subgraphs produced by deleting each combination of 2 vertices from G. The 2-reconstructibility of all graphs with nine or less vertices was tested, identifying two graphs in this range with five vertices as the highest order graphs that are 2-nonreconstructible

Polynomial reconstruction for certain subclasses of disconnected graphs

The Reconstruction Conjecture (RC) and the Polynomial Reconstruction Problem (PRP) are two of the open problems in algebraic graph theory. They have been resolved successfully for a number of different classes and subclasses of graphs. This paper gives proofs for a positive conclusion for the polynomial reconstruction of the following three subclasses of the class of disconnected graphs. These subclasses are disconnected graphs with two unicyclic components, the bidegreed disconnected graphs with regular components and the disconnected graphs with a wheel as one component.peer-reviewe