A note on inverting the dijoin of oriented graphs
Natalie Behague, Tom Johnston, Natasha Morrison, Shannon Ogden
TL;DR
This work studies how the inversion number $ ext{inv}(D)$, defined via inverting vertex subsets, behaves under the dijoin operation $D_1 ightarrow D_2$. They prove the sharp inequality $ ext{inv}({D_1 ightarrow D_2}) > ext{inv}(D_1)$ whenever $ ext{inv}(D_1)= ext{inv}(D_2) ext{≥ }1$, resolving a question posed by Aubian et al. by leveraging a deep connection between inversion and subgraph complementation through the concept of tournament minimum rank $ ext{tmr}(D)$. Specifically, they show for any tournament $D$ that $ ext{inv}(D) ext{in}igligl\{ ext{tmr}(D), ext{tmr}(D)+1igriglig angle$, with the '+1' case characterized by a zero diagonal in all minimum-rank matrices, a key ingredient in the proof. The results unify inversion with the subgraph complementation framework and offer pathways for further investigations on dijoin operations and their extremal behavior in terms of $ ext{tmr}$ and related matrix-rank parameters.
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 $\mathrm{inv}(D)$, to be the minimum number of inversions required to obtain an acyclic oriented graph from $D$. The dijoin, denoted $D_1\rightarrow D_2$, of two oriented graphs $D_1$ and $D_2$ is constructed by taking vertex-disjoint copies of $D_1$ and $D_2$ and adding all edges from $D_1$ to $D_2$. We show that $\mathrm{inv}({D_1 \rightarrow D_2}) > \mathrm{inv}(D_1)$, for any oriented graphs $D_1$ and $D_2$ such that $\mathrm{inv}(D_1) = \mathrm{inv}(D_2) \ge 1$. This resolves a question of Aubian, Havet, Hörsch, Klingelhoefer, Nisse, Rambaud and Vermande. Our proof proceeds via a natural connection between the graph inversion number and the subgraph complementation number.