Isomorphic factorizations VIII: Bisectable trees

Abstract

A tree is called even if its line set can be partitioned into two isomorphic subforests; it is bisectable if these forests are trees. The problem of deciding whether a given tree is even is known (Graham and Robinson) to be NP-hard. That for bisectability is now shown to have a polynomial time algorithm. This result is contained in the proof of a theorem which shows that if a treeS is bisectable then there is a unique treeT that accomplishes the bipartition. With the help of the uniqueness ofT and the observation that the bisection ofS into two copies ofT is unique up to isomorphism, we enumerate bisectable trees.

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Visiting Professor, University of Newcastle, 1976 and 1977 when this work was begun.

Visiting Scholar, University of Michigan, 1981–82 on leave from Newcastle University (Australia) when this work was completed. The research was supported by grants from the Australian Research Grants Commission. The computing reported herein was performed by A. Nymeyer.

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Harary, F., Robinson, R.W. Isomorphic factorizations VIII: Bisectable trees. Combinatorica 4, 169–179 (1984). https://doi.org/10.1007/BF02579217

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AMS subject classification (1980)

  • 05 C 30
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  • 05 B 30