Move-r: Move-Reduced Torus Graphs

• Move-r is a condition for torus-embedded bipartite graphs ensuring that no reduction (via spider and contraction-uncontraction moves) is available in any move-equivalent configuration.
• It employs strands, homology, and decorated Newton polygons—enhanced by cyclic compositions—to encode complex combinatorial and topological invariants.
• The move-reduced criterion minimizes the number of contractible faces, providing a toric generalization of minimality concepts from Goncharov–Kenyon and Postnikov’s theories.

Searching arXiv for "Move-r" and closely related terms to ground the article in current papers. Move-r most directly denotes the move-reduced condition for bipartite graphs embedded in a torus: a graph $\Gamma\subset \mathbb{T}^2$ is move-reduced if no graph move-equivalent to $\Gamma$ under the spider move $(\mathrm{M1})$ and the contraction-uncontraction move $(\mathrm{M2})$ admits any of the reductions $(\mathrm{R1})$ parallel edge reduction, $(\mathrm{R2})$ leaf reduction, or $(\mathrm{R3})$ dipole reduction. In this formulation, the object of study is not mere local irreducibility, but irreducibility after all allowed local equivalence moves. The torus theory developed in “Move-reduced graphs on a torus” characterizes such graphs through strands and decorated Newton polygons, and classifies their move-equivalence classes by a modular invariant; it extends the minimal torus graphs of Goncharov–Kenyon and gives a toric analog of Postnikov’s results on a disk (Galashin et al., 2022).

1. Definition and local move system

Let $\Gamma$ be a bipartite graph embedded in the torus $\mathbb{T}^2$. The basic equivalence moves are the spider move $(\mathrm{M1})$ and the contraction-uncontraction move $\Gamma$0, with black and white colors interchangeable. A graph is move-equivalent to another if they are related by a sequence of $\Gamma$1 and $\Gamma$2 moves.

The reduction moves are distinct from the equivalence moves. They are $\Gamma$3 parallel edge reduction, $\Gamma$4 leaf reduction, and $\Gamma$5 dipole reduction. The decisive point is that move-reduced means not only that none of these reductions is immediately visible on $\Gamma$6, but that none becomes available after first applying any sequence of equivalence moves. A common misreading is to treat move-reduced as a purely local condition on a single drawing; the definition excludes that interpretation.

The main torus characterization theorem assumes that $\Gamma$7 has a perfect matching. That assumption is essential. The paper explicitly exhibits a counterexample showing that the theory fails without it, so the perfect-matching hypothesis is structural rather than technical.

2. Strands, homology, and decorated Newton polygons

The analysis is organized around strands, also called zig-zag paths. A strand is a walk in $\Gamma$8 that turns maximally right at black vertices and maximally left at white vertices. Each strand $\Gamma$9 determines a homology class

$(\mathrm{M1})$0

and the sum of all strand homology classes is zero: $(\mathrm{M1})$1

These homology classes determine the Newton polygon $(\mathrm{M1})$2: the unique convex integral polygon whose counterclockwise boundary vectors are the $(\mathrm{M1})$3’s in some order. From $(\mathrm{M1})$4 one then obtains a weakly decorated Newton polygon

$(\mathrm{M1})$5

where for each edge $(\mathrm{M1})$6, the decoration $(\mathrm{M1})$7 is a partition of

$(\mathrm{M1})$8

when $(\mathrm{M1})$9.

For a partition $(\mathrm{M2})$0, the excess is

$(\mathrm{M2})$1

and for a collection $(\mathrm{M2})$2,

$(\mathrm{M2})$3

When $(\mathrm{M2})$4 is move-reduced, the decoration strengthens. Parallel strands corresponding to a fixed side of $(\mathrm{M2})$5 never meet, so their cyclic order can be recorded. This yields, on each edge, a cyclic composition

$(\mathrm{M2})$6

considered up to cyclic rotation, and hence a strongly decorated Newton polygon

$(\mathrm{M2})$7

This passage from weak to strong decoration is the key combinatorial refinement behind the classification theory (Galashin et al., 2022).

3. Characterization by minimal contractible-face count

The first main theorem gives an exact geometric criterion for move-reducedness. Let $(\mathrm{M2})$8 be a bipartite graph embedded in $(\mathrm{M2})$9 with weakly decorated Newton polygon $(\mathrm{R1})$0, and assume that $(\mathrm{R1})$1 has a perfect matching. Then the following are equivalent:

1. $(\mathrm{R1})$2 is move-reduced.
2. $(\mathrm{R1})$3 has

$(\mathrm{R1})$4

contractible faces, no contractible connected components, and no leaf vertices.

Moreover, if $(\mathrm{R1})$5 is move-reduced and $(\mathrm{R1})$6 are distinct parallel strands, then $(\mathrm{R1})$7 and $(\mathrm{R1})$8 do not share any vertices or edges (Galashin et al., 2022).

This theorem identifies move-reducedness with exact minimality of the number of contractible faces. The quantity

$(\mathrm{R1})$9

is the minimal possible number of contractible faces among all graphs with the same weakly decorated Newton polygon. In that sense, move-reducedness is a sharp toric minimality condition rather than a heuristic simplification criterion.

The paper also proves a triple-crossing-diagram counterpart: $(\mathrm{R2})$0 This reformulation shows that the same minimality principle survives translation into the triple-crossing language.

4. Square moves, strong decoration, and the modular invariant

The central equivalence move is the spider move $(\mathrm{R2})$1, which in plabic or triple-crossing language is the local square move. The contraction-uncontraction move $(\mathrm{R2})$2 is used to pass between bipartite graphs and plabic graphs. Move-equivalence preserves the Newton polygon and, for move-reduced graphs, also preserves the strong decoration together with an additional modular invariant.

For a strongly decorated polygon $(\mathrm{R2})$3, the set of move-reduced graphs $(\mathrm{R2})$4 with $(\mathrm{R2})$5 is not always a single move-equivalence class. Instead it is a union of

$(\mathrm{R2})$6

move-equivalence classes. Here $(\mathrm{R2})$7 is the rotation number of the cyclic composition $(\mathrm{R2})$8, defined as the smallest $(\mathrm{R2})$9 such that rotating the induced partition of $(\mathrm{R3})$0 by $(\mathrm{R3})$1 leaves it unchanged. Concretely, if

$(\mathrm{R3})$2

then $(\mathrm{R3})$3 is the least $(\mathrm{R3})$4 with

$(\mathrm{R3})$5

where $(\mathrm{R3})$6 is the cyclic shift.

The classes are separated by a modular invariant

$(\mathrm{R3})$7

The classification theorem states that

$(\mathrm{R3})$8

A frequent misconception is that the decorated Newton polygon alone should determine the move-equivalence class. The torus case is subtler: the strong decoration is necessary, and even that does not suffice without the extra invariant $(\mathrm{R3})$9. The modular splitting is therefore an intrinsic feature of toric move theory, not an artifact of proof technique (Galashin et al., 2022).

5. Position within torus and planar graph theories

The paper is explicitly framed as an extension of two established theories. In the framework of Goncharov–Kenyon, a torus graph is minimal if no strand self-intersects. Such graphs are automatically move-reduced. The move-reduced notion is strictly broader, because strands in a move-reduced graph may self-intersect, and these self-intersections are encoded first by the weak decoration through partitions and later by the strong decoration through cyclic compositions. In the minimal case, all parts of all decorations are $\Gamma$0, so

$\Gamma$1

and the characterization reduces to the Goncharov–Kenyon setting.

The comparison with Postnikov’s disk theory is equally direct. For bipartite graphs or plabic graphs in a disk, Postnikov characterized reduced graphs and showed that equivalence classes are classified by positroids. In that setting, move-reduced coincides with reduced; the moves are the planar analogs of the torus moves; every boundary matching is realizable; and any two reduced diagrams with the same boundary matching are equivalent. The torus case departs from this pattern because classification is no longer determined solely by a boundary matching or a Newton polygon. The strong decoration and the modular invariant are genuinely toric phenomena.

A plausible implication is that the torus theory should be read not as a routine periodic extension of disk results, but as a setting in which homological data and cyclic ordering data interact in a substantially richer way.

6. Examples, edge cases, and terminological scope

The paper gives a graph $\Gamma$2 that is move-reduced but not minimal in the Goncharov–Kenyon sense. This establishes that move-reducedness is not a restatement of strand-minimality.

It also gives an example showing that two move-reduced graphs can have the same strongly decorated Newton polygon and still fail to be move-equivalent. In Figure 1, the graphs $\Gamma$3 have the same $\Gamma$4, but different modular invariants: $\Gamma$5 This is the canonical demonstration that the modular invariant is indispensable.

The basic non-example is a move-reduced graph with a single simple zero-homology strand when the Newton polygon is a point. Such a graph has no perfect matching and unequal numbers of black and white vertices. This is why the perfect-matching assumption is required in the main theorem (Galashin et al., 2022).

The label Move-r is also used in unrelated arXiv literatures. In the Move programming-language literature, it denotes the Move borrow checker, a modular, intraprocedural static reference-safety analysis run as a load-time bytecode verification pass (Blackshear et al., 2022), and a defense-in-depth runtime safety layer on Aptos that dynamically enforces type safety, ability safety, and reference safety (Gao et al., 16 Jun 2026). In compressed indexing, related work studies move structures for RLBWT-derived permutations and improves balancing and length capping for $\Gamma$6, $\Gamma$7, $\Gamma$8, and $\Gamma$9 (Brown et al., 11 Feb 2026, Brown et al., 23 Mar 2026). These usages are distinct from the move-reduced condition on torus-embedded bipartite graphs.