3. Sparse graphs and addable edges
Section 2.1 of Zheng (2026) defines the class of graphs on
which the paper's central estimate is proved: a simple graph is
(2,2)-sparse when every nonempty vertex set U spans at most 2|U| - 2
edges. The class enters the proof of the main theorem in its sufficiency
direction, that the partition condition implies generic rigidity. That
argument first selects, from the pins of a body–pin graph satisfying the
partition condition, a (2,2)-sparse subgraph of representative pins —
the selection lemma of Section 6.2 — and
then bounds the self-stresses of sparse graphs:
the stress–codimension inequality, the estimate at the
centre of the paper, is stated for exactly this class.
The inequality is proved by deleting one vertex at a time. A (2,2)-sparse
graph on n vertices has at most 2n - 2 edges, so some vertex has degree
at most three, and deleting a vertex leaves a sparse graph; the deletion
argument therefore never leaves the class and always has a vertex of degree at
most three to remove. We first state sparsity and tightness, then the two
consequences of supermodularity that the deletion argument uses, and then
Lemma 2.1, which gives an addable edge among the three neighbours of a
deleted degree-three vertex. In the exceptional case of
the local classification those three
neighbours are collinear, and the induction continues on the smaller graph
with one such edge added back, which restores the self-stress dimension; when
all three neighbour edges are already present, a collinearity flag is created
instead (the flags chapter). The last two
sections describe combinatorics that the formalization contains and the paper
never mentions: a construction theorem for (2,2)-tight graphs, and a
transport lemma that moves sparsity along an embedding.