Flexible realizations existence: NP-completeness on sparse graphs and algorithms

TL;DR

This paper investigates the computational complexity of determining flexible realizations of graphs in the plane, proving NP-completeness for sparse graphs, and introduces efficient algorithms for counting and finding NAC-colorings.

Contribution

It establishes NP-completeness of NAC-coloring existence on sparse graphs and develops faster algorithms for NAC-coloring detection and enumeration.

Findings

NP-completeness on graphs with maximum degree five

NP-completeness on graphs with average degree at most 4+ε

Linear-time counting of NAC-colorings for graphs with bounded treewidth

Abstract

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 NAC-coloring, which is a 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 $\varepsilon >0$. We also show that NAC-colorings can be counted in linear time for graphs with bounded 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.