Reconstruction algorithm for permutation graphs

Authors:
Masashi Kiyomi;Toshiki Saitoh;Ryuhei Uehara
Affiliations:
School of Information Science, JAIST, Ishikawa, Japan;School of Information Science, JAIST, Ishikawa, Japan;School of Information Science, JAIST, Ishikawa, Japan
Venue:
WALCOM'10 Proceedings of the 4th international conference on Algorithms and Computation
Year:
2010

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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 n−1 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.