3 research outputs found
Generating spherical multiquadrangulations by restricted vertex splittings and the reducibility of equilibrium classes
A quadrangulation is a graph embedded on the sphere such that each face is
bounded by a walk of length 4, parallel edges allowed. All quadrangulations can
be generated by a sequence of graph operations called vertex splitting,
starting from the path P_2 of length 2. We define the degree D of a splitting S
and consider restricted splittings S_{i,j} with i <= D <= j. It is known that
S_{2,3} generate all simple quadrangulations.
Here we investigate the cases S_{1,2}, S_{1,3}, S_{1,1}, S_{2,2}, S_{3,3}.
First we show that the splittings S_{1,2} are exactly the monotone ones in the
sense that the resulting graph contains the original as a subgraph. Then we
show that they define a set of nontrivial ancestors beyond P_2 and each
quadrangulation has a unique ancestor.
Our results have a direct geometric interpretation in the context of
mechanical equilibria of convex bodies. The topology of the equilibria
corresponds to a 2-coloured quadrangulation with independent set sizes s, u.
The numbers s, u identify the primary equilibrium class associated with the
body by V\'arkonyi and Domokos. We show that both S_{1,1} and S_{2,2} generate
all primary classes from a finite set of ancestors which is closely related to
their geometric results.
If, beyond s and u, the full topology of the quadrangulation is considered,
we arrive at the more refined secondary equilibrium classes. As Domokos,
L\'angi and Szab\'o showed recently, one can create the geometric counterparts
of unrestricted splittings to generate all secondary classes. Our results show
that S_{1,2} can only generate a limited range of secondary classes from the
same ancestor. The geometric interpretation of the additional ancestors defined
by monotone splittings shows that minimal polyhedra play a key role in this
process. We also present computational results on the number of secondary
classes and multiquadrangulations.Comment: 21 pages, 11 figures and 3 table