We prove the following: Let G and \(G'\) be two graphs on the same set V of v vertices, and let k be an integer, \(4\le k\le v-4\). If for all k-element subsets K of V, the induced subgraphs \(G_{\restriction K}\) and \(G'_{\restriction K}\) have the same numbers of 3-homogeneous subsets, the same numbers of \(P_4\)’s, and the same numbers of claws or co-claws, then \(G'\) is equal to G or to the complement \(\overline{G}\) of G. We give also a similar result whenever the same numbers are modulo a prime.

中文翻译：

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图的相等性直至补全

我们证明如下结果：设ģ和\（G'\）是在同一组两个曲线图V的v顶点，并让ķ是整数，\（4 \文件ķ\文件V-4 \） 。如果对于所有ķ -元素子集ķ的V，感应子图\（G _ {\限制ķ} \）和\（G'_ {\限制ķ} \）有3个均匀子集的相同的数字，相同的数的\（P_4 \）的，和爪或共爪的相同的数字，然后\（G'\）等于ģ或补体\（{G} \ \划线）的ģ。每当相同的数字以质数为模时，我们也会给出相似的结果。