One of the questions in Rigidity Theory is whether a realization of the vertices of a graph in the plane is flexible, namely, if it allows a continuous deformation preserving the edge lengths. A flexible realization of a connected graph in the plane exists if and only if the graph has a so called NAC-coloring, which is surjective edge coloring by two colors such that for each cycle either all the edges have the same color or there are at least two edges of each color. The question whether a graph has a NAC-coloring, and hence also the existence of a flexible realization, has been proven to be NP-complete. We show that this question is also NP-complete on graphs with maximum degree five and on graphs with the average degree at most $4+\varepsilon$ for every fixed $\varepsilon >0$. The existence of a NAC-coloring is fixed parameter tractable when parametrized by treewidth. Since the only existing implementation of checking the existence of a NAC-coloring is rather naive, we propose new algorithms along with their implementation, which is significantly faster. We also focus on searching all NAC-colorings of a graph, since they provide useful information about its possible flexible realizations.

翻译：刚性理论中的一个核心问题是：图中顶点在平面上的某种实现是否具有柔性，即是否存在保持边长度不变的连续形变。连通图在平面上存在柔性实现的充要条件是该图具有所谓的NAC着色——一种将边用两种颜色满射着色的方法，要求对任意环而言，要么所有边颜色相同，要么每种颜色至少出现两次。判断一个图是否存在NAC着色（进而判断柔性实现是否存在）已被证明是NP完全问题。本文证明该问题在最大度为五的图以及平均度不超过$4+\varepsilon$（对任意固定$\varepsilon >0$）的图上同样是NP完全的。当以树宽为参数时，NAC着色存在性属于固定参数可解问题。鉴于现有检验NAC着色存在的实现方法较为朴素，我们提出了新算法及其实现方案，其运行效率显著提升。由于NAC着色能为图的可能柔性实现提供重要信息，我们还重点研究了枚举图中所有NAC着色的方法。