Constructing reflection-symmetric flexible realisations of graphs

We study reflection-symmetric realisations of symmetric graphs in the plane that allow a continuous symmetry and edge-length preserving deformation. To do so, we identify a necessary combinatorial condition on graphs with reflection-symmetric flexible realisations. This condition is based on a specific type of edge colouring, where edges are assigned one of three colours in a symmetric way. From some of these colourings we also construct concrete reflection-symmetric realisations with their corresponding symmetry preserving motion. We study also a specific class of reflection-symmetric realisations consisting of triangles and parallelograms.

💡 Research Summary

The paper investigates planar graph realizations that preserve a reflection symmetry while allowing a continuous, edge‑length‑preserving deformation. Building on the well‑established theory of NAC‑colourings (two‑colour edge assignments that forbid “almost cycles”), the authors introduce a three‑colour scheme—red, blue, and gold—tailored to reflection‑symmetric graphs, called RS‑colourings.

A pseudo‑RS‑colouring must satisfy four conditions: (i) both red and blue appear, (ii) converting all gold edges to blue yields a NAC‑colouring, (iii) converting all gold edges to red also yields a NAC‑colouring, and (iv) under the graph’s reflection σ, the colour of an edge e and its mirror σ(e) are opposite (red ↔ blue) while gold edges are fixed. An RS‑colouring is a pseudo‑RS‑colouring that either contains no “almost red‑blue cycle” (a cycle with exactly one gold edge) or, if such a cycle exists, admits a certificate pseudo‑RS‑colouring in which two edges of the cycle receive different colours.

The authors prove that deciding the existence of a pseudo‑RS‑colouring is NP‑complete, by reduction from the NAC‑colouring problem, and that deciding an RS‑colouring is NP‑hard. This establishes the combinatorial difficulty of the problem.

The central theoretical contribution is the proof that any reflection‑symmetric flexible framework necessarily induces an RS‑colouring. To this end, they model the configuration space of a framework as an algebraic variety V_s(G,p) defined by the edge‑length equations together with the reflection constraints. An irreducible algebraic motion M ⊂ V_s(G,p) captures a non‑trivial flex. By introducing valuations ν on the function field C(M), they assign a numerical “weight” to each directed edge via ν(W_uv), where W_uv = (x_u−x_v)+i(y_u−y_v). Choosing a threshold α>0, edges with ν(W_uv)>α are coloured red, those with ν(W_uv)<−α blue, and the intermediate ones gold. Lemma 4.5 shows that this construction always yields a pseudo‑RS‑colouring, and Lemma 4.6 demonstrates that any such colouring is in fact an RS‑colouring (the valuation guarantees the required certificate for any almost red‑blue cycle). Consequently, the existence of a reflection‑symmetric flex implies the existence of an RS‑colouring—a necessary combinatorial condition.

The paper also provides explicit constructions for several small graphs, illustrating how to realize the prescribed colours geometrically and to exhibit a concrete reflection‑symmetric flex. In particular, they focus on frameworks built solely from triangles and parallelograms, a class previously studied for rigidity. By applying the RS‑colouring framework, they construct reflection‑symmetric flexible realizations of such structures, showing that the added symmetry imposes new constraints compared with the rotation‑symmetric case.

In the concluding section, the authors note that while they have identified a necessary condition (RS‑colouring) and supplied constructive examples, a full characterization (i.e., a sufficient condition) remains open. They suggest future work on algorithmic methods for finding RS‑colourings, extensions to three‑dimensional reflection symmetry, and integration with other symmetry groups (rotations, glide reflections). Overall, the paper bridges combinatorial graph colouring theory with geometric rigidity, delivering a novel tool for detecting and constructing reflection‑symmetric flexible frameworks.