Stable Cuts in Graph Rigidity

• Stable cuts are vertex separators that are stable sets, meaning they contain no adjacent vertices and disconnect the graph upon removal.
• They form a combinatorial bridge by inducing NAC-colourings that certify non-generic flexible realizations in minimally rigid planar graphs.
• Stable cuts enable efficient detection of flexibility via polynomial-time algorithms, while their absence in denser graphs poses NP-completeness challenges.

Stable cuts are vertex separators that are simultaneously stable sets. For a simple graph $G=(V,E)$, a set $X\subseteq V$ is a stable cut if $G-X$ is disconnected and no two vertices of $X$ are adjacent. In recent rigidity theory, stable cuts have become a central combinatorial interface between sparse graph structure, NAC-colourings, and flexible realisations in the plane. In particular, stable cuts underpin the characterization that a minimally rigid graph has a flexible realisation with positive edge lengths if and only if it is not a $2$-tree, confirming a conjecture of Grasegger, Legerský and Schicho (Clinch et al., 2024).

1. Definition and terminological scope

A stable cut is defined by combining two standard notions. A set $X\subseteq V(G)$ is a cut if $G-X$ is disconnected, and it is a stable set if no two vertices of $X$ are adjacent. Equivalently, a separation $\{G_1,G_2\}$ of $G$ is a stable separation if $X\subseteq V$0 is a stable cut in $X\subseteq V$1. In the graph-theoretic literature on separators, the synonymous term stable cutset is also standard (Clinch et al., 2024).

This notion is structurally different from several other uses of “stable cut” in the literature. In particular, in cut-optimization and game-theoretic work, a stable cut may denote a cut whose weight cannot be increased by changing the side of a single vertex, i.e. a locally-optimal cut. That usage concerns local optimality in weighted cut problems rather than independent vertex separators, and the two concepts should not be conflated (Lampis, 2021).

Within rigidity theory, the stable-cut notion is valuable because it is purely combinatorial but interacts directly with geometric realisability. The key role of stable cuts is not that they encode rigidity counts by themselves, but that they induce NAC-colourings and therefore certify the existence of non-generic flexible behaviour in regimes where generic rigidity may still hold.

2. Extremal structure at the $X\subseteq V$2 edge threshold

The sparse extremal theory of stable cuts begins with the edge bound conjectured by Caro and proved by Chen and Yu: every graph on $X\subseteq V$3 vertices with at most $X\subseteq V$4 edges has a stable cutset. The critical case is therefore $X\subseteq V$5, the Laman count for minimal rigidity in the plane (Rauch et al., 2024).

For graphs with exactly $X\subseteq V$6 edges, Le and Pfender identified the obstruction class. Define a family $X\subseteq V$7 recursively: $X\subseteq V$8 lies in $X\subseteq V$9; and if $G-X$0 while $G-X$1 is either a triangle or the $G-X$2-prism, then any graph obtained by gluing $G-X$3 and $G-X$4 along an entire edge or along a $G-X$5-cycle also lies in $G-X$6. The Le–Pfender theorem states that if $G-X$7 has $G-X$8 vertices and $G-X$9, then exactly one of the following holds: $X$0 has a stable cut, or $X$1 (Clinch et al., 2024).

This characterization is the combinatorial backbone of the rigidity applications. Its proof uses induction on $X$2, together with the fact that the allowed gluing operations preserve the property of having no stable cut. Later work revisited the extremal classification of graphs without stable cutsets and filled a gap in the original Le–Pfender argument, thereby restoring a fully rigorous description of the extremal obstruction family (Rauch et al., 2024).

The significance of this threshold is twofold. First, it is the exact boundary after the Chen–Yu range $X$3, so it isolates the first nontrivial sparse graphs that can avoid stable cuts. Second, it coincides with the combinatorial count for minimally rigid graphs in the plane, which is why stable cuts become a rigidity-theoretic invariant rather than only a separator notion.

3. Stable cuts and flexible graphs in the plane

A $X$4-dimensional realisation of a graph $X$5 is a pair $X$6, where $X$7 maps the vertices of $X$8 to $X$9. A realisation is flexible if it can be continuously deformed while keeping the edge lengths fixed, and rigid otherwise. Similarly, a graph is flexible if its generic realisations are flexible, and rigid otherwise. In the planar setting considered here, a graph is flexible exactly when it fails Laman’s condition for minimal rigidity (Clinch et al., 2024).

A central theorem states that every flexible graph in the plane has at least one stable cut. More precisely, if $2$0 is flexible and $2$1 lie in distinct rigid components, then there exists a stable cut $2$2 that separates $2$3 and $2$4, and moreover meets each rigid component in at most one vertex. As a corollary, if $2$5 is $2$6-connected, then for every vertex $2$7 there is a stable cut disjoint from $2$8 (Clinch et al., 2024).

The proof combines contraction arguments with rigidity-matroid structure. One inductively contracts a triangle at a flexible vertex, maintaining flexibility or producing a neighbourhood that is already a stable set, and then uses a rank argument involving $2$9-circuits to force a separator. This moves stable cuts from an extremal graph-theoretic statement into a theorem about all flexible planar graphs.

This theorem strictly strengthens the earlier $X\subseteq V(G)$0 edge-count result. Any graph with fewer than $X\subseteq V(G)$1 edges is flexible and therefore has a stable cut, but the result is not restricted to sparse graphs of that kind: it asserts that flexibility itself forces a stable-cut structure.

4. NAC-colourings and non-generic flexibility

A NAC-colouring is a surjective $X\subseteq V(G)$2-colouring $X\subseteq V(G)$3 such that no cycle is almost monochromatic, meaning that no cycle has exactly one edge of the opposite colour. Grasegger, Legerský and Schicho proved that a connected graph admits a flexible quasi-injective realisation in $X\subseteq V(G)$4 if and only if it has a NAC-colouring (Clinch et al., 2024).

Stable cuts provide a direct mechanism for constructing such colourings. If $X\subseteq V(G)$5 is a stable separation with $X\subseteq V(G)$6, one colours all edges of $X\subseteq V(G)$7 red and all edges of $X\subseteq V(G)$8 blue. Since $X\subseteq V(G)$9 is independent, at each vertex in $G-X$0 all incident edges lie in just one side, so there is no almost-monochromatic cycle. Hence every graph with a stable cut admits a NAC-colouring (Clinch et al., 2024).

Combining this lemma with the Le–Pfender classification yields a rigidity consequence of particular importance: every minimally rigid graph on $G-X$1 vertices that is not a $G-X$2-tree has a NAC-colouring, and therefore a non-generic flex. Equivalently, a minimally rigid graph has a flexible realisation with positive edge lengths if and only if it is not a $G-X$3-tree. Here $G-X$4-trees are the graphs built by stacking triangles, or equivalently by repeatedly attaching triangles along edges.

The $G-X$5-tree case is the exceptional one. Such graphs are minimally rigid, have no stable cut, admit no NAC-colouring, and therefore have no non-generic flex. This gives a complete classification of minimally rigid graphs according to whether positive-length flexible realisations exist.

5. Algorithms, complexity, and counting questions

Stable cuts admit constructive algorithms in the regimes where the structural theory applies. By standard pebble-game algorithms, the rigid components of a graph can be computed in $G-X$6 time. Moreover, the recursive contraction-based proof of the stable-cut theorem can be implemented to find an explicit stable cut separating two chosen vertices in distinct rigid components in $G-X$7 time (Clinch et al., 2024).

The decision problem also exhibits a sharp complexity boundary across density regimes. Deciding the existence of a stable cut is easy when $G-X$8, by the Chen–Yu theorem, and when $G-X$9, by the Le–Pfender characterization. For denser graphs, however, the problem becomes NP-complete as soon as $X$0 (Clinch et al., 2024).

The same work also studies the number of NAC-colourings. It provides an upper bound on the number of NAC-colourings for arbitrary graphs, and it constructs families of graphs, including rigid and minimally rigid ones, for which this number is exponential in the number of vertices (Clinch et al., 2024). This shows that the combinatorial flexibility encoded by NAC-colourings can be abundant even in graph classes where generic rigidity is present.

Algorithmically, stable cuts therefore serve as polynomial-time certificates of non-generic flexibility in the critical sparse regime. Counting-wise, they sit inside a broader landscape in which the existence of one stable cut is only the first layer of a potentially exponentially large family of admissible NAC-colourings.

6. Examples, later developments, and probabilistic behaviour

Examples clarify both the strength and the limits of the concept. Trees, cycles, and more generally any graph with fewer than $X$1 edges are flexible and hence have a stable cut. The $X$2-prism furnishes a standard rigidity-theoretic example: it is rigid by Laman’s theorem but has a flexible non-generic realisation, and it also has a stable cut; specifically, the two opposite vertices of degree $X$3 form an independent separating set (Clinch et al., 2024).

Subsequent work has extended the theory beyond deterministic sparse structure. In the random graph process, the property of having no stable cut has an exact hitting time: with high probability, the random graph first has no stable cut precisely when every vertex lies in a triangle. Equivalently, in $X$4 there is a sharp threshold at

$X$5

for the disappearance of stable cuts, and the same hitting time governs the disappearance of NAC-colourings (Clinch et al., 7 Oct 2025).

This probabilistic result places stable cuts within the broader threshold picture for rigidity and flexibility in random graphs. It shows that “flexibility via stable cuts” disappears at the precise moment when the last local obstruction, a vertex outside every triangle, vanishes. A plausible implication is that stable cuts capture a genuinely local separator mechanism whose extinction can be tracked by a simple triangle condition, even though the rigidity phenomena they influence are global.

Taken together, these developments establish stable cuts as a unifying notion across separator theory, extremal sparse graph structure, and planar rigidity. They link an independent-vertex-separator condition to NAC-colourings, to flexible quasi-injective realisations, and to the precise obstruction class at the Laman edge count.