We investigate the structure of two-dimensional partial cubes, i.e., of isometric subgraphs of hypercubes whose vertex set defines a set family of VC-dimension at most 2. Equivalently, those are the partial cubes which are not contractible to the 3-cube $Q_3$ (here contraction means contracting the edges corresponding to the same coordinate of the hypercube). We show that our graphs can be obtained from two types of combinatorial cells (gated cycles and gated full subdivisions of complete graphs) via amalgams. The cell structure of two-dimensional partial cubes enables us to establish a variety of results. In particular, we prove that all partial cubes of VC-dimension 2 can be extended to ample aka lopsided partial cubes of VC-dimension 2, yielding that the set families defined by such graphs satisfy the sample compression conjecture by Littlestone and Warmuth (1986). Furthermore we point out relations to tope graphs of COMs of low rank and region graphs of pseudoline arrangements.

中文翻译：

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二维部分立方体

我们研究了二维局部立方体的结构，即超立方体的等距子图的结构，其顶点集定义了最多 2 个 VC 维的集合族。等效地，那些是不能收缩为 3 立方体的局部立方体Q_3$（这里收缩是指收缩超立方体相同坐标对应的边）。我们表明，我们的图可以通过汞齐从两种类型的组合单元（门控循环和完整图的门控完整细分）中获得。二维局部立方体的单元结构使我们能够建立各种结果。特别是，我们证明了 VC 维 2 的所有部分立方体都可以扩展为充足的 VC 维 2 的不平衡部分立方体，得出由这些图定义的集合族满足 Littlestone 和 Warmuth (1986) 的样本压缩猜想。此外，我们指出了与低秩 COM 的顶图和伪线排列的区域图的关系。