Abstract
PREIMAGE CONSTRUCTION problem by Kratsch and Hemaspaandra naturally arose from the famous graph reconstruction conjecture. It deals with the algorithmic aspects of the conjecture. We present an \(\mbox{\cal O}(n^8)\) time algorithm for PREIMAGE CONSTRUCTION on permutation graphs, where n is the number of graphs in the input. Since each graph of the input has nāāā1 vertices and \(\mbox{\cal O}(n^2)\) edges, the input size is \(\mbox{\cal O}(n^3)\). There are polynomial time isomorphism algorithms for permutation graphs. However the number of permutation graphs obtained by adding a vertex to a permutation graph is generally exponentially large. Thus exhaustive checking of these graphs does not achieve any polynomial time algorithm. Therefore reducing the number of preimage candidates is the key point.
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Kiyomi, M., Saitoh, T., Uehara, R. (2010). Reconstruction Algorithm for Permutation Graphs. In: Rahman, M.S., Fujita, S. (eds) WALCOM: Algorithms and Computation. WALCOM 2010. Lecture Notes in Computer Science, vol 5942. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-11440-3_12
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DOI: https://doi.org/10.1007/978-3-642-11440-3_12
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