Free edge lengths in plane graphs
Free edge lengths in plane graphs
复制标题
平面图中的自由边长
DOI：
10.1007/s00454-015-9704-z
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发表时间：
2015
影响因子：
0.8
通讯作者：
and Csaba Tóth
中科院分区：
文献类型：
--
作者：
Zachary Abel;Robert Connelly;Sarah Eisenstat;Radoslav Fulek;Filip Morić;Yoshio Okamoto;Tibor Szabó;and Csaba Tóth
研究了度量约束对平面图可实现性的影响。设G是平面图H的一个子图（其中H是G的“宿主”）.图G在H中是自由的，如果对于G的边的每一个正长度的选择，宿主H都有实现这些长度的平面直线嵌入;如果G的边长的所有约束仅依赖于G，而与宿主H的附加边无关，则G在H中是外自由的。我们刻画了所有在每个宿主H中自由的平面图G，G ∈ H，以及所有在每个宿主H中可外自由的平面图G，G ∈ H。循环G = Ck的情况提供了著名的木匠规则问题的新版本。即使圈Ck，k ≥ 4，在所有的三角剖分中都不是外切自由的，但我们发现“非退化”的边长总是可实现的，如果圈可以用两种不同的方式展平（成一条线），则边长被认为是退化的。我们证明了在一个4-连通三角网（没有分离三角形）中每个星星都是自由的。
We study the impact of metric constraints on the realizability of planar graphs. Let G be a subgraph of a planar graph H (where H is the "host" of G). The graph G is free in H if for every choice of positive lengths for the edges of G, the host H has a planar straight-line embedding that realizes these lengths; and G is extrinsically free in H if all constraints on the edge lengths of G depend on G only, irrespective of additional edges of the host H.We characterize all planar graphs G that are free in every host H, G ⊆ H, and all the planar graphs G that are extrinsically free in every host H, G ⊆ H. The case of cycles G = Ck provides a new version of the celebrated carpenter's rule problem. Even though cycles Ck, k ≥ 4, are not extrinsically free in all triangulations, it turns out that "nondegenerate" edge lengths are always realizable, where the edge lengths are considered degenerate if the cycle can be flattened (into a line) in two different ways.Separating triangles, and separating cycles in general, play an important role in our arguments. We show that every star is free in a 4-connected triangulation (which has no separating triangle).