A note on inverting the dijoin of oriented graphs
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| Format: | Preprint |
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2024
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| _version_ | 1866914139494940672 |
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| author | Behague, Natalie Johnston, Tom Morrison, Natasha Ogden, Shannon |
| author_facet | Behague, Natalie Johnston, Tom Morrison, Natasha Ogden, Shannon |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2404_10663 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A note on inverting the dijoin of oriented graphs Behague, Natalie Johnston, Tom Morrison, Natasha Ogden, Shannon Combinatorics 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. |
| title | A note on inverting the dijoin of oriented graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2404.10663 |