Generating spherical multiquadrangulations by restricted vertex splittings and the reducibility of equilibrium classes

A spherical quadrangulation is a loopless graph embedded on the sphere such that each face is bounded by a walk of length 4, parallel edges allowed. The family of the isomorphism classes of quadrangulations can be generated by a sequence of graph operations called vertex splitting, starting from the path graph P2 with three vertices and two edges. P2 is also referred to as the common ancestor of all quadrangulations. We define the degree 1 ≤ D ≤ bd/2c of a splitting S (where d is the degree of the split vertex) and consider restricted splittings Si,j with 1 ≤ i ≤ D ≤ j ≤ bd/2c. As Brinkmann and coworkers have recently pointed out, restricted splittings S2,3 generate all simple quadrangulations. Here we investigate the cases S1,2, S1,3, S1,1, S2,2, S3,3. First we show that the restricted splittings S1,2 are exactly the monotone ones in the sense that the resulting graph contains the original as a subgraph. Then we proceed to show that they define a set of nontrivial ancestors beyond P2 and each quadrangulation has a unique ancestor. Our results have a direct geometric interpretation in the context of mechanical equilibria of convex bodies. The latter can be defined as the scalar distance R(θ, φ) measured from the center of gravity and the MorseSmale complex associated with the gradient of R corresponds to a 2coloured quadrangulation with independent set sizes s, u. The numbers s, u of coloured vertices identify the primary equilibrium class associated with the body by Várkonyi and Domokos. We show that the S1,1 and S2,2 splittings generate all primary equilibrium classes (in case of S1,1 from a single ancestor, in case of S2,2 from a finite nontrivial set of ancestors). This is closely related to the geometric results of Várkonyi and Domokos ∗[email protected], Dept. of Control Engineering and Information Technology, Budapest University of Technology and Economics, H-1117 Magyar tudósok körútja 2., Budapest, Hungary †[email protected], Dept. of Mechanics, Materials & Structures, Budapest University of Technology and Economics, H-1521, Műegyetem rakpart 1-3. K.II.42., Budapest, Hungary ‡[email protected], Dept. of Mechanics, Materials & Structures, Budapest University of Technology and Economics, H-1521, Műegyetem rakpart 1-3. K.II.42., Budapest, Hungary 1 ar X iv :1 20 6. 16 98 v1 [ cs .D M ] 8 J un 2 01 2 where they show that specific geometric transformations can generate all equilibrium classes. If, beyond the numbers s, u, the full topology of the quadrangulation is considered, we arrive at the more refined secondary equilibrium classes. As Domokos, Lángi and Szabó showed recently, one can create the geometric counterparts of unrestricted splittings to generate all secondary classes. Our results show that restricted, monotone splittings S1,2, while adequate to generate all primary classes from one single ancestor, can only generate a limited range of secondary equilibrium 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 some computational results on the cardinality of secondary equilibrium classes and multiquadrangulations.