A biclique decomposition of a graph is a partition of its edges into complete bipartite subgraphs. We consider graphs whose vertices can be ordered such that the neighborhood of every vertex is the union of a sublinear number of intervals. We observe that these graphs admit compact representations in the form of biclique decompositions of small size. Here, the size of a decomposition is measured as the sum of the number of vertices of its bicliques. Combining this result with the existence of suitable vertex orderings for graphs of low neighborhood complexity, as proven by Welzl in 1988, we recover and extend several known results up to logarithmic factors. These results include upper bounds on the Zarankiewicz problem, matrix multiplication, quantum circuit complexity, and shortest path algorithms in ``well-structured'' instances.

翻译：图的二部团分解是指将其边划分为完全二部子图。我们考虑这样一类图，其顶点可被排序，使得每个顶点的邻域是亚线性个区间的并集。我们观察到，这些图允许以小型二部团分解的形式进行紧凑表示。这里，分解的大小通过其二部团顶点数之和来度量。结合这一结果与Welzl于1988年证明的低邻域复杂度图中存在合适顶点排序的结论，我们恢复并扩展了若干已知结果，误差仅在对数因子内。这些结果包括Zarankiewicz问题的上界、矩阵乘法、量子电路复杂度以及“结构良好”实例中最短路径算法的上界。