Combinatorial reconstruction problems

Let X be a (finite or infinite) set and let G be a (finite or infinite) group of automorphisms of X. Thus G acts on X and for every g E G the sequence k&x is a permutation of X. For every subset Y of X and every g E G, let g Y be the set of all elements gy, for y E Y. Clearly 1 g Y 1 = 1 Y 1 for every finite Y, and this defines an action of the group G on the power set of X. The orbit of a subset Y of X is, as usual, the set YG = {g Y: g E G}. We say that two subsets Y and 2 of X are G-equivalent iff there is an element g of the group G mapping Y into 2, i.e., iff ZE p. Let R be a set of representatives of these orbits, i.e., a family consisting of a unique member of each orbit. For an integer k 2 1 and for a subset Y of X of cardinality m > k, the k-deck of Y is the function d = d,,: R -+ (0, 1,2, . ...} where for each Y E R, d(r) is the number of subsets of cardinality m k of Y which are G-equivalent to r. Notice that CrER d(r) = (7) and that if Y and 2 are G-equivalent then 4,/c = dZ,k, i.e., the k-deck of Y is determined by the orbit of Y under the action of G. Several problems in combinatorics and in graph theory, known as reconstruction problems, are special cases of the following general problem: