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Graphs with flexible labelings allowing injective realizations
Discrete Mathematics ( IF 0.8 ) Pub Date : 2020-06-01 , DOI: 10.1016/j.disc.2019.111713 Georg Grasegger , Jan Legerský , Josef Schicho
Discrete Mathematics ( IF 0.8 ) Pub Date : 2020-06-01 , DOI: 10.1016/j.disc.2019.111713 Georg Grasegger , Jan Legerský , Josef Schicho
Abstract We consider realizations of a graph in the plane such that the distances between adjacent vertices satisfy the constraints given by an edge labeling. If there are infinitely many such realizations, counted modulo rigid motions, the labeling is called flexible. The existence of a flexible labeling, possibly non-generic, has been characterized combinatorially by the existence of a so called NAC-coloring. Nevertheless, the corresponding realizations are often non-injective. In this paper, we focus on flexible labelings with infinitely many injective realizations. We provide a necessary combinatorial condition on existence of such a labeling based also on NAC-colorings of the graph. By introducing new tools for the construction of such labelings, we show that the necessary condition is also sufficient up to 8 vertices, but this is not true in general for more vertices.
中文翻译:
带有灵活标签的图形允许单射实现
摘要 我们考虑平面中图的实现,使得相邻顶点之间的距离满足边标记给出的约束。如果有无限多个这样的实现,计算模刚性运动,则标记称为灵活。灵活标签的存在,可能是非通用的,其特征在于存在所谓的 NAC 着色。然而,相应的实现通常是非内射的。在本文中,我们专注于具有无限多内射实现的灵活标签。我们还基于图形的 NAC 着色为存在此类标记提供了必要的组合条件。通过引入用于构建此类标签的新工具,我们表明必要条件也足以达到 8 个顶点,
更新日期:2020-06-01
中文翻译:
带有灵活标签的图形允许单射实现
摘要 我们考虑平面中图的实现,使得相邻顶点之间的距离满足边标记给出的约束。如果有无限多个这样的实现,计算模刚性运动,则标记称为灵活。灵活标签的存在,可能是非通用的,其特征在于存在所谓的 NAC 着色。然而,相应的实现通常是非内射的。在本文中,我们专注于具有无限多内射实现的灵活标签。我们还基于图形的 NAC 着色为存在此类标记提供了必要的组合条件。通过引入用于构建此类标签的新工具,我们表明必要条件也足以达到 8 个顶点,