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

We prove that for every fixed undirected graph H, there is an O(|V(G)| 3) time algorithm that, given a graph G, tests if G contains H as a topological subgraph (that is, a subdivision of H is subgraph of G). This shows that topological subgraph testing is fixed-parameter tractable, resolving a longstanding open question of Downey and Fellows from 1992. As a corollary, for every H we obtain an O(|V(G)| 3) time algorithm that tests if there is an immersion of H into a given graph G. This answers another open question raised by Downey and Fellows in 1992. © 2011 ACM

We present an easy structure theorem for graphs which do not admit an immersion of the complete graph K t . The theorem motivates the definition of a variation of tree decompositions based on edge cuts instead of vertex cuts which we call tree-cut decompositions. We give a definition for the width of tree-cut decompositions, and using this definition along with the structure theorem for excluded clique immersions, we prove that every graph either has bounded tree-cut width or admits an immersion of a large wall.

We give axiomatic foundations for infinite matroids with duality, in terms of independent sets, bases, circuits, closure and rank. Continuing work of Higgs and Oxley, this completes the solution to a problem of Rado of 1966. (C) 2013 Henning Bruhn, Reinhard Diestel, Matthias Kriesell, Rudi Pendavingh and Paul Wollan. Published by Elsevier Inc. All rights reserved

We prove that for every proper minor-closed class I of graphs there exists a constant c such that for every integer n the class I includes at most n ! c(n) graphs with vertex-set (1,2,...,n). This answers a question of Welsh. (c) 2006 Robin Thomas. Published by Elsevier Inc. All rights reserved