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 S 1 , S 2 , …, S n of an n ‐element set V does there exist a graph G = ( V, E ) with { N [ v ]: v ∈ V } = { S 1 , S 2 , …, S n }? 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