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.

中文翻译：

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图子空间上施同构定理的一个简短证明

在他的博士论文中，Shih 描述了两个图之间的关系，其中第一个图的循环空间包含在第二个图的循环空间中，并且循环空间的维度相差一个 [7]。然而，这个结果从未出现在参考出版物中。结果，这个定理没有得到应有的重视。我们对这个结果给出了一个更简单、更简短的证明，以及一些独立的重述。此外，我们还提到了对偶周期拟阵、偶切拟阵和框架拟阵的应用。