On the Complexity of Reconstructing H-free Graphs from Their Star Systems

Abstract

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 4 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 5 vertices. This yields a complete dichotomy for paths and cycles.