The orthogonality dimension of a graph over $\mathbb{R}$ is the smallest integer $d$ for which one can assign to every vertex a nonzero vector in $\mathbb{R}^d$ such that every two adjacent vertices receive orthogonal vectors. For an integer $d$, the $d$-Ortho-Dim$_\mathbb{R}$ problem asks to decide whether the orthogonality dimension of a given graph over $\mathbb{R}$ is at most $d$. We prove that for every integer $d \geq 3$, the $d$-Ortho-Dim$_\mathbb{R}$ problem parameterized by the vertex cover number $k$ admits a kernel with $O(k^{d-1})$ vertices and bit-size $O(k^{d-1} \cdot \log k)$. We complement this result by a nearly matching lower bound, showing that for any $\varepsilon > 0$, the problem admits no kernel of bit-size $O(k^{d-1-\varepsilon})$ unless $\mathsf{NP} \subseteq \mathsf{coNP/poly}$. We further study the kernelizability of orthogonality dimension problems in additional settings, including over general fields and under various structural parameterizations.

翻译：图在实数域 $\mathbb{R}$ 上的正交维数是指最小的整数 $d$，使得可以为每个顶点分配一个 $\mathbb{R}^d$ 中的非零向量，且任意两个相邻顶点所对应的向量正交。对于整数 $d$，$d$-Ortho-Dim$_\mathbb{R}$ 问题旨在判定给定图在 $\mathbb{R}$ 上的正交维数是否不超过 $d$。我们证明，对于任意整数 $d \geq 3$，以顶点覆盖数 $k$ 为参数的 $d$-Ortho-Dim$_\mathbb{R}$ 问题存在一个具有 $O(k^{d-1})$ 个顶点且比特规模为 $O(k^{d-1} \cdot \log k)$ 的核。我们通过一个近乎匹配的下界补充了这一结果，表明对于任意 $\varepsilon > 0$，除非 $\mathsf{NP} \subseteq \mathsf{coNP/poly}$，否则该问题不存在比特规模为 $O(k^{d-1-\varepsilon})$ 的核。我们进一步研究了正交维数问题在其他设定下的核化性质，包括在一般域上的情形以及在不同结构参数化下的表现。