Equality of graphs up to complementation

Jamel Dammak, Gérard Lopez, Hamza Si Kaddour
2020 Arabian Journal of Mathematics  
We prove the following: Let G and $$G'$$ G ′ be two graphs on the same set V of v vertices, and let k be an integer, $$4\le k\le v-4$$ 4 ≤ k ≤ v - 4 . If for all k-element subsets K of V, the induced subgraphs $$G_{\restriction K}$$ G ↾ K and $$G'_{\restriction K}$$ G ↾ K ′ have the same numbers of 3-homogeneous subsets, the same numbers of $$P_4$$ P 4 's, and the same numbers of claws or co-claws, then $$G'$$ G ′ is equal to G or to the complement $$\overline{G}$$ G ¯ of G. We give also a
more » ... ar result whenever the same numbers are modulo a prime.
doi:10.1007/s40065-020-00297-8 fatcat:nhhravphwrapxfyengeujiagu4