4. Vertex deletion and self-stress
Section 2.2 of Zheng (2026) measures what deleting one
vertex does to the self-stress space, over a coefficient field that shrinks
with the graph. The measurement is the step of the induction that proves
the stress–codimension inequality
s + \operatorname{trdeg}_k K \le 3|V|: each step deletes a vertex v of
degree at most three, and equation (2.6) below expresses the resulting change
in s + \operatorname{trdeg}_k K - 3|V| through two local numbers, a kernel
dimension u and a transcendence degree \delta_v, both defined in this
chapter.
We follow the paper's order. An exact sequence relates the self-stress space
of F to that of H = F - v; the tower formula for transcendence degree
does the same for the coordinate fields; and a local classification,
Lemma 2.3, shows that u + \delta_v \le 3 whenever \deg_F(v) \le 3,
except in one case: a degree-three vertex whose three neighbours are
collinear. The last two lemmas of the chapter show that in this exceptional
case the three rigidity rows on pairs of neighbours already lie in the row
space of the deleted graph; in the flags
chapter the case is retained as a collinearity flag and carried through the
remaining steps of the induction.