Combinatorial reconstruction problems

Abstract

A general technique for tackling various reconstruction problems is presented and applied to some old and some new instances of such problems.

Finite Subsets of the Plane are 18-Reconstructible

“…In Section 3 we showed that finite subsets of S 1 are 6-reconstructible under the action of SO (1). In [32] we show that a similar result for S n−1 would prove Conjecture 1.…”

“…We define a pointed configuration C, x to be a coloured configuration in style (1), that is, a finite multiset C together with one distinguished element x ∈ supp(C), which has colour 1. An oriented configuration is, similarly, a coloured configuration in style (1, 1).…”

“…(For instance, the 2-deck is essentially the multiset of distances between pairs of points in S.) We want to know how large k must be so that S is determined up to rigid motion by its k-deck. Alon, Caro, Krasikov, and Roditty [1] proved that subsets of n points in the plane can be reconstructed from their (log 2 n + 1)-decks. Our first aim in this paper is to prove that every finite subset of the plane can be reconstructed from its 18-deck.…”

“…This in turn is proved by considering the circle as a limit (in an appropriate sense) of the groups Z n for n large. Alon, Caro, Krasikov and Roditty [1] proved that if Z n acts on itself then arbitrary subsets S are reconstructible from their (log 2 n + 1)-decks (see also Mnukhin [22,24]). Radcliffe and Scott [29] improved their bound substantially in the case of Z n acting on itself.…”

“…Some progress has been made in recent years in the general case where we have a group action G X providing the notion of isomorphism, and we wish to reconstruct a subset S of X from the multiset of isomorphism classes of its k-element subsets, known as the k-deck (see Alon, Caro, Krasikov, and Roditty [1], Babai [2], Cameron [7,9,8], Maynard and Siemons [20], Mnukhin [21,22,23], and [31]). Several authors [1,7,22] have noted that we can reconstruct S provided k > log 2 |G| + 1; the n log 2 n bound for edge reconstruction (Müller [25]; Lovász [18]) also follows from this. In general, however, much smaller decks may suffice (see [28,31]).…”