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Discrete Seminar: Subgraph complementation and minimum rank

Author: Lona

Speaker: Calum Buchanan, University of Vermont <https://www.uvm.edu/~cjbuchan>Title: Subgraph complementation and minimum rankAbstract:It is possible to obtain any simple graph $G$ from any other graph $H$ on the same set of vertices by performing a sequence of subgraph complementations. That is, we can iteratively replace induced subgraphs by their graph complements until we obtain $G$ from $H$. We ask for the minimum number of subgraph complementations in such a sequence. When $H$ is the graph with no edges, we denote this parameter by $c_2(G)$. Finding $c_2(G)$ relates closely to the minimum rank problem. We show that $c_2 (G) = \operatorname{mr}(G,\mathbb{F}_2)$ when $\operatorname{mr}(G,\mathbb{F}_2)$ is odd or when $G$ is a forest; otherwise, $\operatorname{mr}(G,\mathbb{F}_2) \leq c_2 (G) \leq \operatorname{mr}(G,\mathbb{F}_2) + 1$. We then provide two conditions which are equivalent to having $c_2(G) = \operatorname{mr}(G,\mathbb{F}_2) + 1$. In this case, we can still interpret $\operatorname{mr}(G,\mathbb{F}_2)$ combinatorially using a variation of subgraph complementation. Finally, the class of gr! aphs $G$ with $c_2(G) \leq k$ is hereditary and finitely defined for any natural number $k$. We exhibit the sets of minimal forbidden induced subgraphs for small values of $k$. This is joint work with Christopher Purcell and Puck Rombach.

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