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arXiv · 2509.10232

A case of the dijoin conjecture on inverting oriented graphs

Natalie Behague·Patrick Gaudart-Wifling
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Abstract

For an oriented graph $D$, the inversion of $X\subseteq V(D)$ in $D$ is the graph obtained by reversing the orientation of all arcs with both ends in $X$. The inversion number $\mathrm{inv}(D)$ is the minimum number of inversions needed to obtain an acyclic oriented graph. We show that the dijoin conjecture of Bang-Jensen, da Silva and Havet, that $\mathrm{inv}(D_1\rightarrow D_2)=\mathrm{inv}(D_1)+\mathrm{inv}(D_2)$, is true in the case where $\mathrm{inv}(D_1)=2$ and $\mathrm{inv}(D_2)$ is even. We also characterise the cases $\mathrm{inv}(D_1)=2$ and $\mathrm{inv}(D_2)$ odd, for which the conjecture does and does not hold. We then go on to show a similar result for n-joins, in doing so we prove a conjecture of Alon, Powierski, Savery, Scott and Wilmer. Our proofs build on the idea of tournament minimum rank, introduced by Behague, Johnston, Morrison and Ogden.

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Natalie Behague, Patrick Gaudart-Wifling. 2025-09-12. A case of the dijoin conjecture on inverting oriented graphs. https://arxiv.org/abs/2509.10232

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