On the Complexity of Reconstructing H -free Graphs from Their Star Systems
نویسندگان
چکیده
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 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 remains 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. Department of Informatics, University of Bergen, N-5020 Bergen, Norway. Email: {fomin, daniello, federico, telle}@ii.uib.no. Dept. of Applied Mathematics and Institute for Theoretical Computer Science, Charles University, Praha, Czech Republic. Supported by Czech research grant 1M0545.
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