On the complexity of reconstructing H-free graphs from their Star Systems

In the Star System problem we are given a set system and asked whether it is realizable by the multi-set of closed neighborhoods of some graph, i.e. given subsets S1, S2, …, Sn of an n-element set V does there exist a graph G = (V, E) with {N[v]: v∈V} = {S1, S2, …, Sn}? For a fixed graph H the H-free Star System problem is a variant of the Star System problem where it is asked whether a given set system is realizable by closed neighborhoods of a graph containing no H as an induced subgraph. We study the computational complexity of the H-free Star System problem. We prove that when H is a path or a cycle on at most four vertices the problem is polynomial time solvable. In complement to this result, we show that if H belongs to a certain large class of graphs the H-free Star System problem is NP-complete. In particular, the problem is NP-complete when H is either a cycle or a path on at least five vertices. This yields a complete dichotomy for paths and cycles. Copyright © 2010 John Wiley & Sons, Ltd. 68:113-124, 2011