Techniques in matroid reconstruction (Q1363660)

The question of whether the isomorphism class of a finite simple graph is determined by its set (or ``deck'') of induced subgraphs is a fundamental open question in graph theory. The matroid analogues and generalizations of this question, reconstruction matroids or their invariants, are both natural and important. Roughly speaking, one may show a graph or matroid is reconstructible if either, on the one hand, there is a recognizable local structure that can be used as a benchmark, e.g. an isthmus, or there is a global regularity condition that aides in the reconstruction, e.g. \(k\)-valent graphs. The main result of this paper, that the matroids of Dowling lattices are reconstructible from their decks of hyperplanes, falls in the latter class, while the proof techniques fall in the former. The paper also contains a readable introduction to the problem, a discussion of general techniques, and many references.