1. A note on the invertibility of oriented graphs
- Author
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Behague, Natalie, Johnston, Tom, Morrison, Natasha, and Ogden, Shannon
- Subjects
Mathematics - Combinatorics - Abstract
For an oriented graph $D$ and a set $X\subseteq V(D)$, the inversion of $X$ in $D$ is the graph obtained from $D$ by reversing the orientation of each edge that has both endpoints in $X$. Define the inversion number of $D$, denoted $inv(D)$, to be the minimum number of inversions required to obtain an acyclic oriented graph from $D$. We show that $inv({D_1 \rightarrow D_2}) > inv(D_1)$, for any oriented graphs $D_1$ and $D_2$ such that $inv(D_1) = inv(D_2) \ge 1$. This resolves a question of Aubian, Havet, H\"{o}rsch, Klingelhoefer, Nisse, Rambaud and Vermande. Our proof proceeds via a natural connection between the graph inversion number and the subgraph complementation number., Comment: 10 pages
- Published
- 2024