REALIZABILITY AND UNIQUENESS IN GRAPHS

Consider a finite graph G(V,E). Let us associate to G a finite list P(G) of invariants. To any P the following two natural problems arise: (R) Realizability. Given P, when is P = P(G) for some graph G?, (U) Uniqueness. Suppose P(G) = P(H) for graphs G and H. When does this imply G congruent to H? The best studied questions in this context are the degree realization problem for (R) and the reconstruction conjecture for (U). We discuss the problems (R) and (U) for the degree sequence and the size sequence of induced subgraphs for undirected and directed graphs, concentrating on the complexity of the corresponding decision problems and their connection to a natural search problem on graphs.