Expressing graphs as symmetric differences of cliques Calum Buchanan*, Christopher Purcell, & Puck Rombach

The -build number Upper bounds Minimum rank Forbidden subgraphs

Any finite simple graph G = (V,E) can be repre- A number of upper bounds for c2(G) are obtained Let M = M(C ) be a clique-incidence matrix. The c2(G) ≤ k is hereditary. We sented by a collection of cliques in the complete by its equivalence to the minimum dimension of a The off-diagonal entry (i, j) of MM T (mod 2) is have shown in [2] that it is defined by a finite graph on V whose symmetric difference is G. faithful orthogonal representation of G over F2. 0 if and only if the vertices corresponding to the set of minimal forbidden induced subgraphs. For For instance, consider {{u, v} | uv ∈ E}. But we Given a clique construction C of G, assign to ith and jth rows of M are nonadjacent. odd k, we have c2(G) = k whenever mr(G, F2) = can often do better. each v an incidence vector with a 1 in the The minimum rank of G over F is the minimum k, and c2(G) ≤ k whenever mr(G, F2) < k. Thus, ith slot if v appears in the ith clique in C , and rank over all matrices in Fn×n whose off-diagonal the sets of minimal forbidden induced subgraphs Example 1. a 0 otherwise. The equivalence follows, as two zeros match those of the adjacency matrix of G. for {G : mr(G, F2) ≤ k} and {G : c2(G) ≤ k} are T vectors are orthogonal over F2 if and only if they Since rank(MM ) ≤ rank(M) ≤ c2(G), we have the same. = 4 share an even number of 1’s. This is not the case when k is even. We exhibit We denote by M(C ) the clique-incidence matrix mr(G, F2) ≤ c2(G). (2) the sets of minimal forbidden induced subgraphs whose rows are the aforementioned vectors. For for c (G) ≤ 2 and mr(G, ) ≤ 2, labeled A and Question. What is the minimum cardinality of 2 F2 example, the matrix corresponding to the con- Theorem 6. For any forest F and field F, we have B respectively, below. such a collection of cliques? c (F ) = mr(F, ). struction C in Example 1 is 2 F

Definition 2. A clique construction of G is a col-  1 0  There is a close relationship between c2(G) and A B lection of subsets of V such that, for each pair 1 1 C M(C ) = 1 1 . (1) mr(G, F2): the numbers differ by at most 1, and 1 1 u, v ∈ V , uv ∈ E if and only if u and v appear do so only if c2(G) is odd. On the other hand, together an odd number of times in C . The min- Propositions 3 and 4 are corollaries of Theorems 1 these invariants have important differences. The imum cardinality of a clique construction of G is and 3 in [4], obtained by this equivalence. minimum rank of G with components G1,...,G` full house P` the clique-build number of G, denoted by c2(G). is mr(Gi, ), but this is not the case for c2(G). Let n denote the order of a graph G. 1 F K3,3 While c (W ) = 3 and c (K ) = 1, we have 2 5 2 2 3K2 P4 Equivalent problems Proposition 3. For any graph G, c2(G) ≤ n − 1. P ∨ P P + K 3 3 Proposition 4. For any graph G (n > 2) other 3 2 The problem of expressing a graph G as a sum 4 4 = than P , c (G) ≤ n − 2. of cliques modulo 2 was posed by Vatter [1]. n 2 W5 dart n Subgraph complementation [4] Theorem 5 ([2]). For any graph G with vertex Theorem 7. For any graph G, the following are Replace an induced subgraph of G by its graph cover number τ(G), c2(G) ≤ 2τ(G). complement. equivalent. Proof. For each v ∈ V (G), we can build edges to Faithful orthogonal representations i. c (G) = mr(G, ) + 1; any subset S ⊆ N(v) in two steps using cliques 2 F2 Given a field , assign to each vertex of G a vector references F S and S ∪ {v}. Given a minimum vertex cover U from d so that two vertices are adjacent if and ii. there is a unique matrix A of minimum rank F of G, or set of vertices spanning the edges of G, over which fits G, and every diagonal en- [1] V. Vatter. Terminology for expressing a graph as a only if they are represented by non-orthogonal F2 sum of cliques (mod 2), URL (version: 2018-12-15): build the edges incident to each u ∈ U which have vectors. Lovasz [3] introduced these representa- try of A is 0; https://mathoverflow.net/q/317716 not already been built in at most two steps. [2] C. Buchanan, C. Purcell, and P. Rombach. Express- tions over R. iii. there is an optimal clique construction of G ing graphs as symmetric differences of cliques of the , arXiv:2101.06180, 2021. Dot product representations Notice that c2(G) < 2τ(G) if any of the cliques in which every vertex appears an even num- we use are singletons, that is, if some vertex in [3] L. Lov´asz. On the Shannon capacity of a graph. IEEE Orthogonal representations in which the dot ber of times; Transactions on Information Theory, 1979. product of two vectors representing adjacent ver- U has only one neighbor outside of U. [4] V. Alekseev and V. Lozin. On orthogonal representa- iv. for every component G0 of G, c (G0) = tices is 1. 2 tions of graphs. Discrete Mathematics, 2001. mr(G0) + 1.