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.
Generation of simple quadrangulations of the sphere / Gunnar, Brinkmann; Sam, Greenberg; Catherine, Greenhill; Brendan D., Mckay; R., Thomas; Wollan, PAUL JOSEPH. - In: DISCRETE MATHEMATICS. - ISSN 0012-365X. - 305:1-3(2005), pp. 33-54. [10.1016/j.disc.2005.10.005]
Generation of simple quadrangulations of the sphere
WOLLAN, PAUL JOSEPH
2005
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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.