Equality of graphs up to complementation

We prove the following: Let G and G′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G'$$\end{document} be two graphs on the same set V of v vertices, and let k be an integer, 4≤k≤v-4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$4\le k\le v-4$$\end{document}. If for all k-element subsets K of V, the induced subgraphs G↾K\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_{\restriction K}$$\end{document} and G↾K′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G'_{\restriction K}$$\end{document} have the same numbers of 3-homogeneous subsets, the same numbers of P4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_4$$\end{document}’s, and the same numbers of claws or co-claws, then G′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G'$$\end{document} is equal to G or to the complement G¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{G}$$\end{document} of G. We give also a similar result whenever the same numbers are modulo a prime.

Theorem 1.1 [4] Let v, k be two integers with 4 ≤ k ≤ ϑ(v). Then, for every pair of graphs G and G on the same set V of v vertices, if G K and G K have the same number of edges, up to complementation, and the same number of 3-homogeneous subsets, for all k-element subsets K of V , then G = G or G = G. Theorem 1.2 [4] Let k be an integer, 4 ≤ k ≤ v − 2, k ≡ 0 (mod 4). Let G and G be two graphs on the same set V of v vertices.
We assume that e(G K ) has the same parity as e(G K ) for all k-element subsets K of V . Then G = G or G = G.
It is also shown in [4] that two graphs G and G on the same set of v vertices are equal up to complementation whenever they are k-hypomorphic up to complementation and 4 ≤ k ≤ v − 4 (for the case k = v − 3, the conclusion obtained is G G or G G [5]). Later, an extension of this result was obtained for uniform hypergraphs [9]. Theorem 1.3 [9] Let h be a non-negative integer. There are two non-negative integers k and t, k ≤ t such that two h-uniform hypergraphs H and H on the same set V of vertices, |V | ≥ t, are equal up to complementation whenever H and H are k-hypomorphic up to complementation.
In this paper, we look for similar results on graphs if the conditions on the restrictions G K and G K are that they have the same numbers of 3-homogeneous subsets, the same numbers of P 4 's, and the same numbers of claws or co-claws, we obtain Theorem 1.4. Whenever the same numbers (cited above) are modulo a prime, we obtain Theorem 1.7. This last theorem is a consequence of Theorem 1.4 and Proposition 4.3 which is inspired by a result of Pouzet on monomorphic relations (Proposition 4.2).
Let G := (V, E) be a graph. We set: Our first result is: Theorem 1.4 Let G, G be two graphs on the same finite set V of v ≥ 4 vertices. (4) (G ) and S (4) (4) (G K ) and s (4) The proof of Theorem 1.4 will be given in Sect. 3. The following propositions show that Theorem 1.4 is optimal.
Proposition 1.5 For every integer v ≥ 4 and every integer k, 4 ≤ k ≤ v, there are two graphs G and G on the same set of v vertices, nonisomorphic up to complementation, satisfying p (4) (G K ) = p (4) (G K ) and s (4) Proof Let V := {1, 2, . . . , v} with v ≥ 4. Let G and G be two graphs on the same vertex set V defined by G {2,3,...,v} and G {2,3,...,v} are complete graphs, Clearly  Fig. 3). Clearly, G and G are not isomorphic up to complementation. We have h (3) Let k, p be positive integers, the decomposition of k = As an application of Theorem 1.4, our second result is: Theorem 1.7 Let G, G be two graphs on the same set V of v vertices. Let p be a prime number, p ≥ 5 and The proof of Theorem 1.7 will be given in Sect. 4.

Sketch of the proofs of Theorems 1.4 and 1.7
Let G, G be two graphs on the same vertex set V . The Boolean sum G+G of G and G is the graph U on V whose edges are pairs e of vertices such that e ∈ E(G) if and only if e / ∈ E(G ). Indeed, G = G or G = G amounts to the fact that U is either the empty graph or the complete graph. The intersection graph of G and G is the graph

(G) and e(G ) have the same parity if and only if e(U ) is even.
To prove (1) of Theorem 1.4, we consider U := G+G . From Theorem 1.2, it is sufficient to prove that e(G X ) has the same parity as e(G X ) for all 4-element subsets X of V that is, from Observation 2.1, e(U X ) is even for all 4-element subsets X of V . For this we proceed by contradiction, assuming that e(U X ) is odd. Theorem 3.1 says that U X is a path of length 1 or 3. If the length is one, we study two cases. If the length is 3, we conclude using Theorem 3.2.
To prove item (2), and also Theorem 1.7, we will prove that item (1) of Theorem 1.4 holds. For this, we will use linear algebra. The incidence matrix W t k (v) used by Wilson [13], or more simply W t k , is defined as follows : Let V be a finite set, with v elements. Given non-negative integers t ≤ k ≤ v, let W t k be the v by v k matrix of 0's and 1's, the rows of which are indexed by the t-element subsets T of V , the columns are indexed by the k-element subsets K of V , and where the entry W t k (T, K ) is 1 if T ⊆ K and is 0 otherwise. The matrix transpose of W t k is denoted t W t k . We denote by rank Q W t k , the rank of W t k over the field Q. Whenever p is a prime, we denote by rank p W t k , the rank of W t k over the field F p , and by K er p ( t W t k ) the kernel of t W t k in F p .
First, rank Q W t k is given by Theorem 2.2 due to Gottlieb [7].
Theorem 2.2 [7] For t ≤ min(k, v − k), the rank of W t k over the field Q of rational numbers is v t and thus K er( t W t k ) = {0}.
Let G := (V, E) be a graph with v vertices, v ≥ 6. Let t ∈ {3, 4}, and k be an integer, k ≤ v and t ≤ min(k, v − k).
be an enumeration of the k-element subsets of V . We set: We have: Observation 2.3 Let G and G be two graphs on the same finite set V of v vertices.
Now for the end of the proof of item (2), from Observation 2.
For the other parameters, we do the same. Thus, we get the hypotheses of item (1).
For Theorem 1.7, there are two cases according to the value of k 0 . If k 0 ≥ 4, we proceed as above. If k 0 = 0, we use Theorem 4.1 which gives the dimension and a basis of K er p ( t W t k ).

Proof of Theorem 1.4
A description of the Boolean sum G+G , of graphs G and G having the same 3-element homogeneous subsets, is given by Theorem 3.1 below.
We denote by P 9 the Paley graph on 9 vertices (cf. Fig. 2). Note that P 9 is isomorphic to its complement P 9 .
Theorem 3.1 [10] Let U be a graph. The following properties are equivalent: (1) There are two graphs G and G having the same 3-element homogeneous subsets such that U := G+G ; (2) Either (i) U is an induced subgraph of P 9 , or (ii) the connected components of U , or of its complement U , are cycles of even length or paths.  For graphs G and G having the same 3-element homogeneous subsets, Theorem 3.2 below gives the form of their restrictions on a connected component of G+G .

Theorem 3.2 [5] Let G and G be two graphs on the same vertex set V and U := G+G . We assume H (3) (G) = H (3) (G ) and U not connected. If C is a connected component of U of cardinality n, then the pair
The graphs mentioned in the conclusion of Theorem 3.2 are defined as follows (see Fig. 3). Let n ≥ 2. Let X n be an n-element set, v 0 , · · · , v n−1 be an enumeration of X n , X 0 n := {v i ∈ X n : i ≡ 0 (mod 2)} and X 1 n := X n \ X 0 n . Set R n := [X 0 n ] 2 ∪ [X 1 n ] 2 , S n := {{v 2i , v 2i+1 } : 2i + 1 < n}, S n := {{v 2i+1 , v 2i+2 } : 2i + 2 < n}. Let M n and M n be the graphs with vertex set X n and edge sets E(M n ) := R n ∪ S n and E(M n ) := R n ∪ S n , respectively. Let M n := (X n , R n ∪ S n ∪ {{v 0 , v n−1 }}) for n even, n ≥ 4. Finally, let Let G = (V, E) be a graph. For x = y ∈ V , x ∼ G y means {x, y} ∈ E, x G y means {x, y} / ∈ E. For X, Y ⊆ V , X ∼ G Y signifies that for every x ∈ X and y ∈ Y , x ∼ G y. Similarly, X G Y signifies that for every x ∈ X and y ∈ Y , x G y. Whenever X = {x}, X ∼ G Y and X G Y are, respectively, denoted x ∼ G Y and x G Y . Now we prove Theorem 1.4.
(1) Let U := G+G . Using Theorem 1.2, it is sufficient to prove that e(G X ) has the same parity as e(G X ) for all 4-element subsets X of V that is, from Observation 2.1, e(U X ) is even for all 4-element subsets X of V . Let X := {v 0 , v 1 , v 2 , v 3 } be a subset of V . By contradiction, we assume that e(U X ) is odd, then e(U X ) ∈ {1, 3, 5}. As H (3) We get a contradiction with P (4) (G) = P (4) (4) (G ) and conclude using (1). The following result is one of the keys to our proof.
(2) t = t t ( p) p t ( p) and k = k( p) i=t ( p)+1 k i p i if and only if dim K er p ( t W t k ) = 1 and {(1, 1, . . . , 1)} is a basis of K er p ( t W t k ).
Let k ≥ 1 be an integer and G be a graph. We say that G is k-monomorphic (resp. k-monomorphic up to complementation) if G X G Y (resp. G X G Y or G X G Y ) for all k-element subsets X and Y of V . The notion of monomorphy was introduced by Fraïssé [6].
The following result on monomorphy, due to Pouzet, is very useful since it is the origin of Proposition 4.3, which is a key in the proof of Theorem 1.7.