We are interested in embedding trees T with maximum degree at most four in a rectangular grid, such that the vertices of T correspond to grid points, while edges of T correspond to non-intersecting straight segments of the grid lines. Such embeddings are called straight models. While each edge is represented by a straight segment, a path of T is represented in the model by the union of the segments corresponding to its edges, which may consist of a path in the model having several bends. The aim is to determine a straight model of a given tree T minimizing the maximum number of bends over all paths of T. We provide a quadratic-time algorithm for this problem. We also show how to construct straight models that have k as its minimum number of bends and with the least number of vertices possible. As an application of our algorithm, we provide an upper bound on the number of bends of EPG models of graphs that are both VPT and EPT.

中文翻译：

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网格嵌入中树的最小路径弯曲数

我们感兴趣的是在矩形网格中嵌入最大度数最多为 4 的树 T，这样 T 的顶点对应于网格点，而 T 的边缘对应于网格线的非相交直线段。这种嵌入称为直接模型。虽然每条边由一条直线段表示，但模型中的路径 T 由与其边对应的段的并集表示，这可能由模型中具有多个弯曲的路径组成。目的是确定给定树 T 的直线模型，最小化 T 的所有路径上的最大弯曲次数。我们为此问题提供了二次时间算法。我们还展示了如何构建以 k 作为其最小弯曲数和尽可能少的顶点数的直线模型。作为我们算法的应用，