Generation of simple quadrangulations of the sphere

Abstract

A simple quadrangulation of the sphere is a finite simple graph embedded on the sphere such that every face is bounded by a walk of 4 edges. We consider the following classes of simple quadrangulations: arbitrary, minimum degree 3, 3-connected, and 3-connected without non-facial 4-cycles. In each case, we show how the class can be generated by starting with some basic graphs in the class and applying a sequence of local modifications. The duals of our algorithms generate classes of quartic (4-regular) planar graphs. In the case of minimum degree 3, our result is a strengthening of a theorem of Nakamoto and almost implicit in Nakamoto's proof. In the case of 3-connectivity, a corollary of our theorem matches a theorem of Batagelj. However, Batagelj's proof contained a serious error which cannot easily be corrected. We also give a theoretical enumeration of rooted planar quadrangulations of minimum degree 3, and some counts obtained by a program of Brinkmann and McKay that implements our algorithm. © 2005 Elsevier B.V. All rights reserved

A Genealogy of Convex Solids Via Local and Global Bifurcations of Gradient Vector Fields

“…Face contractions on a graph in class Q 2 . As in [4], triangles incident to some vertices indicate that one or more edges may occur at that position around the vertex. Here p 1 and p 3 are sinks; analogous face contractions, each removing a source from the graph, can be performed by switching the colors.…”

“…As in [8], we denote by a quadrangulation a finite planar undirected multigraph on the 2-sphere in which each face is bounded by a closed walk of length 4 (cf. [1,4]). A multigraph contains no loops but may have multiple (parallel) edges, and it is usually permitted that the boundary of a face may contain a vertex or an edge of the graph more than once (e.g.…”

“…[1,4] or [22]). Applied to the face F defined in the previous paragraph, this operation results in the contraction of the vertices p 1 and p 3 into the same vertex, and the disappearance of F ; the modified graphs, depending on the 'shape' of the original face F are shown on the right of each panel in Figure 6.…”

“…We set (4) V µ1,µ2,α (x, y) = x 3 3 + y 2 2 + y 3 3 − αx 2 y − µ 1 x − µ 2 xy, with α ≥ (1/4) 1/3 , so that the vector field (3) becomes…”

Cube-contractions in 3-connected quadrangulations

“…Theorem 1.1 (Brinkmann et al [2]). Any K 3 -irreducible quadrangulation of the sphere is isomorphic to a pseudo double wheel.…”