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On the complexity of graph reconstruction

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Abstract

In the wake of the resolution of the four-color conjecture, the graph reconstruction conjecture has emerged as one focal point of graph theory. This paper considers thecomputational complexity of decision problems (Deck Checking andLegitimate Deck), construction problems (Preimage Construction), and counting problems (Preimage Counting) related to the graph reconstruction conjecture. We show that:

$${\text{GRAPH}} {\text{ISOMORPHISM}} \leqslant _m^l LEGITIMATE DECK, and$$
(1.)
$${\text{GRAPH}} {\text{ISOMORPHISM}} \equiv _{iso}^l DECK CHECKING.$$
(2)

Relatedly, we display the first natural GI-hard NP set lacking obvious padding functions. Finally, we show thatLegitimate Deck, Preimage Construction, andPreimage Counting are solvable in polynomial time for graphs of bounded degree, partialk-trees for any fixedk, and graphs of bounded genus, in particular for planar graphs.

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Authors and Affiliations

  1. Fakultät für Mathematik und Informatik, Friedrich-Schiller-Universität, 07740, Jena, Germany

    Dieter Kratsch

  2. Department of Computer Science, University of Rochester, 14627, Rochester, NY, USA

    Lane A. Hemaspaandra

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  1. Dieter Kratsch
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  2. Lane A. Hemaspaandra
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Additional information

The work of L. A. Hemaspaandra was supported in part by the National Science Foundation under Grants CCR-8809174/CCR-8996198, CCR-8967604, and NSF-INT-9116781/JSPS-ENGR-207. Work done in part while visiting Friedrich-Schiller-Universität and Universität Ulm.

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Kratsch, D., Hemaspaandra, L.A. On the complexity of graph reconstruction. Math. Systems Theory 27, 257–273 (1994). https://doi.org/10.1007/BF01578845

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