Abstract
We investigate the relative complexity of the graph isomorphism problem (GI) and problems related to the reconstruction of a graph from its vertex-deleted or edge-deleted subgraphs. We show that the problems are rather closely related for all amounts c of deletion:
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For all c ≥ 1, GI \(\equiv^{l}_{iso}\)VDC c , GI \(\equiv^{l}_{iso}\)EDC c , GI \(\leq^{l}_{m}\)LVD c , and GI \(\equiv^{p}_{iso}\)LED c .
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For all c ≥ 1 and k ≥ 2, \({\rm GI} \equiv^{p}_{iso} {k\textnormal{-}\rm VDC}_c\) and \({\rm GI} \equiv^{p}_{iso} {k\textnormal{-}\rm EDC}_c\).
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For all c ≥ 1 and k ≥ 2, \({\rm GI} \leq^{l}_{m} {k\textnormal{-}\rm LVD}_c\). In particular, for all c ≥ 1, \({\rm GI} \equiv^{p}_{iso} {2\textnormal{-}\rm LVD}_c\).
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For all c ≥ 1 and k ≥ 2, \({\rm GI} \equiv^{p}_{iso} {k\textnormal{-}\rm LED}_c\).
For many of these, even the c = 1 cases were not known.
Similar to the definition of reconstruction numbers vrn ∃ (G) [10] and ern ∃ (G) (see p. 120 of [17]), we introduce two new graph parameters, vrn ∀ (G) and ern ∀ (G), and give an example of a family {G n } n ≥ 4 of graphs on n vertices for which vrn ∃ (G n ) < vrn ∀ (G n ). For every k ≥ 2 and n ≥ 1, we show there exists a collection of k graphs on (2k − − 1+1)n+k vertices with 2n 1-vertex-preimages, i.e., one has families of graph collections whose number of 1-vertex-preimages is huge relative to the size of the graphs involved.
Supported in part by grants NSF-CCR-9322513, NSF-INT-9815095/DAAD-315-PPP-gü-ab, and NSF-CCR-0311021.
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References
Bondy, J., Hemminger, R.: Graph reconstruction—a survey. Journal of Graph Theory 1, 227–268 (1977)
Bondy, J.: On Ulam’s conjecture for separable graphs. Pacific Journal of Mathematics 31, 281–288 (1969)
Bondy, J.: A graph reconstructor’s manual. In: Surveys in Combinatorics. London Mathematical Society Lecture Notes Series 66, pp. 221–252. Cambridge University Press, Cambridge (1991)
Bryant, R.: On a conjecture concerning the reconstruction of graphs. Journal of Combinatorial Theory 11, 139–141 (1971)
Carrillo, H., Lipman, D.: The multiple sequence alignment problem in biology. SIAM Journal of Applied Mathematics 48, 1073–1082 (1988)
Fenner, S., Fortnow, L., Kurtz, S.: Gap-definable counting classes. Journal of Computer and System Sciences 48(1), 116–148 (1994)
Harary, F.: On the reconstruction of a graph from a collection of subgraphs. In: Fiedler, M. (ed.) Theory of Graphs and its Applications, pp. 47–52. Czechoslovak Academy of Sciences, Prague (1964)
Hemaspaandra, L., Ogihara, M.: The Complexity Theory Companion. Springer, Heidelberg (2002)
Harary, F., Palmer, E.: The reconstruction of a tree from its maximal subtrees. Canadian Journal of Mathematics 18, 803–810 (1966)
Harary, F., Plantholt, M.: The graph reconstruction number. J. Graph Theory 9, 451–454 (1985)
Kadin, J.: Restricted Turing Reducibilities and the Structure of the Polynomial Time Hierarchy. PhD thesis, Cornell University, USA (1988)
Kelly, P.: On Isometric Transformations. PhD thesis, University of Wisconsin, USA (1942)
Kelly, P.: A congruence theorem for trees. Pacific Journal of Mathematics 7, 961–968 (1957)
Kratsch, D., Hemaspaandra, L.: On the complexity of graph reconstruction. Mathematical Systems Theory 27(3), 257–273 (1994)
Köbler, J., Schöning, U., Torán, J.: The Graph Isomorphism Problem: Its Structural Complexity. Birkhäuser, Basel (1993)
Lauri, J.: Proof of Harary’s conjecture on the reconstruction of trees. Discrete Mathematics 43, 79–90 (1983)
Lauri, J., Scapellato, R.: Topics in Graph Automorphism and Reconstruction. London Mathematical Society, Cambridge University Press (2003)
Lozano, A., Torán, J.: On the non-uniform complexity of the graph isomorphism problem. In: Proceedings of the 7th Structure in Complexity Theory Conference, June 1992, pp. 118–129. IEEE Computer Society Press, Los Alamitos (1992)
Manvel, B.: Some basic observations on Kelly’s conjecture for graphs. Discrete Mathematics 8, 181–185 (1974)
Mansfield, A.: The relationship between the computational complexities of the legitimate deck and isomorphism problems. Quart. J. Math. Ser. 33(2), 345–347 (1982)
Manvel, B.: Reconstruction of graphs: Progress and prospects. Congressus Numerantium 63, 177–187 (1988)
Myrvold, W.: The ally-reconstruction number of a disconnected graph. Ars Combinat. 28, 123–127 (1989)
Myrvold, W.: The ally-reconstruction number of a tree with five or more vertices is three. Journal of Graph Theory 14, 149–166 (1990)
Nash-Williams, C.S.J.A.: The reconstruction problem. In: Beineke, L., Wilson, R. (eds.) Selected Topics in Graph Theory, pp. 205–236. Academic Press, London (1978)
Nýdl, V.: Graph reconstruction from subgraphs. Discrete Mathematics 235, 335–341 (2001)
Ranjan, D., Rohatgi, P.: On randomized reductions to sparse sets. In: Proceedings of the 7th Structure in Complexity Theory Conference, June 1992, pp. 239–242. IEEE Computer Society Press, Los Alamitos (1992)
Ulam, S.: A Collection of Mathematical Problems. Interscience Publishers, New York (1960)
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Hemaspaandra, E., Hemaspaandra, L.A., Radziszowski, S., Tripathi, R. (2004). Complexity Results in Graph Reconstruction. In: Fiala, J., Koubek, V., Kratochvíl, J. (eds) Mathematical Foundations of Computer Science 2004. MFCS 2004. Lecture Notes in Computer Science, vol 3153. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-28629-5_20
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