A Short Proof of Shih’s Isomorphism Theorem on Graphic Subspaces


In his PhD thesis Shih characterized the relationship between two graphs, where the cycle space of the first is included in the cycle space of the second and the dimension of the cycle spaces differ by one [7]. However, this result never appeared in a refereed publication. As a consequence this theorem has not received the attention it deserves. We give a simpler and shorter proof of this result as well as several restatements that are of independent interest. In addition, we mention applications to even-cycle matroids, even-cut matroids and frame matroids.

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We would like to thank Bert Gerards for bringing this problem to our attention, Neil Robertson for informing us of the existence of Shih’s Theorem, and Irene Pivotto and Paul Wollan for numerous discussions on this topic. We are also grateful to the referees for their suggestions and comments.

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Correspondence to Bertrand Guenin.

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Supported by NSERC grant 238811 and ONR grant N00014-12-1-0049.

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Ferchiou, Z., Guenin, B. A Short Proof of Shih’s Isomorphism Theorem on Graphic Subspaces. Combinatorica 40, 805–837 (2020). https://doi.org/10.1007/s00493-020-3972-9

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Mathematics Subject Classification (2010)

  • 05C22
  • 05C50
  • 05B35