Semialgebraic graphs are graphs whose vertices are points in $\mathbb{R}^d$, and adjacency between two vertices is determined by the truth value of a semialgebraic predicate of constant complexity. We show how to harness polynomial partitioning methods to construct compact adjacency labeling schemes for families of semialgebraic graphs. That is, we show that for any family of semialgebraic graphs, given a graph on $n$ vertices in this family, we can assign a label consisting of $O(n^{1-2/(d+1) + \varepsilon})$ bits to each vertex (where $\varepsilon > 0$ can be made arbitrarily small and the constant of proportionality depends on $\varepsilon$ and on the complexity of the adjacency-defining predicate), such that adjacency between two vertices can be determined solely from their two labels, without any additional information. We obtain for instance that unit disk graphs and segment intersection graphs have such labelings with labels of $O(n^{1/3 + \varepsilon})$ bits. This is in contrast to their natural implicit representation consisting of the coordinates of the disk centers or segment endpoints, which sometimes require exponentially many bits. It also improves on the best known bound of $O(n^{1-1/d}\log n)$ for $d$-dimensional semialgebraic families due to Alon (Discrete Comput. Geom., 2024), a bound that holds more generally for graphs with shattering functions bounded by a degree-$d$ polynomial. We also give new bounds on the size of adjacency labels for other families of graphs. In particular, we consider semilinear graphs, which are semialgebraic graphs in which the predicate only involves linear polynomials. We show that semilinear graphs have adjacency labels of size $O(\log n)$. We also prove that polygon visibility graphs, which are not semialgebraic in the above sense, have adjacency labels of size $O(\log^3 n)$.

翻译：半代数图是指其顶点为 $\mathbb{R}^d$ 中的点，且两个顶点之间的邻接关系由一个常数复杂度的半代数谓词的真值所决定的图。我们展示了如何利用多项式划分方法来为半代数图族构建紧凑的邻接标记方案。具体而言，我们证明对于任意半代数图族，给定该族中一个包含 $n$ 个顶点的图，我们可以为每个顶点分配一个由 $O(n^{1-2/(d+1) + \varepsilon})$ 比特组成的标记（其中 $\varepsilon > 0$ 可以任意小，比例常数依赖于 $\varepsilon$ 以及定义邻接关系的谓词的复杂度），使得两个顶点之间的邻接关系仅通过它们的两个标记即可确定，无需任何额外信息。例如，我们得到单位圆盘图和线段相交图具有此类标记，其标记大小为 $O(n^{1/3 + \varepsilon})$ 比特。这与它们自然的隐式表示（即圆盘中心或线段端点的坐标）形成对比，后者有时需要指数级的比特数。这也改进了 Alon (Discrete Comput. Geom., 2024) 针对 $d$ 维半代数图族提出的最佳已知界 $O(n^{1-1/d}\log n)$，该界更一般地适用于由 $d$ 次多项式界定的碎裂函数的图。我们还针对其他图族给出了邻接标记大小的新界。特别地，我们考虑了半线性图，即其中谓词仅涉及线性多项式的半代数图。我们证明半线性图具有大小为 $O(\log n)$ 的邻接标记。我们还证明了多边形可见性图（在上述意义上不属于半代数图）具有大小为 $O(\log^3 n)$ 的邻接标记。