Equality of graphs up to complementation

Abstract

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, 4le kle 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.