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：半代数图是一类顶点为$\mathbb{R}^d$中点的图，顶点间的邻接关系由常复杂度的半代数谓词的真值决定。我们展示了如何利用多项式划分方法为半代数图族构造紧凑的邻接标记方案。具体而言，对任意半代数图族，给定该族中一个包含$n$个顶点的图，可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$\varepsilon > 0$可任意小，比例常数取决于$\varepsilon$及邻接定义谓词的复杂度），使得仅凭两个顶点的标记即可判定其邻接关系，无需额外信息。例如，单位圆盘图和线段相交图存在此类标记，其标记长度为$O(n^{1/3 + \varepsilon})$比特。这与其自然隐式表示（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比特数。该结果也改进了Alon（Discrete Comput. Geom., 2024）对$d$维半代数图族的最佳已知界$O(n^{1-1/d}\log n)$——该界更广泛地适用于破碎函数被$d$次多项式界定的图。此外，我们还给出了其他图族邻接标记大小的新界。特别地，考虑半线性图（即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。同时，我们证明多边形可见性图（虽不符合上述半代数定义）的邻接标记大小为$O(\log^3 n)$。