Paul Wollan

An improved linear edge bound for graph linkages

A graph is said to be k-linked if it has at least 2k vertices and for every sequence S1,...,Sk, t1,...,tk of distinct vertices there exist disjoint paths P1,...,Pk such that the ends of Pi are si and ti. Bollobas and Thomason showed that if a simple graph G on n vertices is 2k-connected and G has at least 11kn edges, then G is k-linked. We give a relatively simple inductive proof of the stronger statement that 8kn edges and 2k-connectivity suffice, and then with more effort improve the edge bound to 5kn.

Finding topological subgraphs is fixed-parameter tractable

We prove that for every fixed undirected graph <i>H</i>, there is an O(|V(G)|³) time algorithm that, given a graph <i>G</i>, tests if <i>G</i> contains <i>H</i> as a topological subgraph (that is, a subdivision of <i>H</i> is subgraph of <i>G</i>). 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 <i>H</i> we obtain an O(|V(G)|³) time algorithm that tests if there is an immersion of <i>H</i> into a given graph <i>G</i>. This answers another open question raised by Downey and Fellows in 1992.

Proper minor-closed families are small

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!cn graphs with vertex-set {1, 2,...,n}. This answers a question of Welsh.

Generation of simple quadrangulations of the sphere

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 Nakamotos proof. In the case of 3-connectivity, a corollary of our theorem matches a theorem of Batagelj. However, Batageljs 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.

A simpler algorithm and shorter proof for the graph minor decomposition

At the core of the Robertson-Seymour theory of graph minors lies a powerful decomposition theorem which captures, for any fixed graph H, the common structural features of all the graphs which do not contain H as a minor. Robertson and Seymour used this result to prove Wagners Conjecture that finite graphs are well-quasi-ordered under the graph minor relation, as well as give a polynomial time algorithm for the disjoint paths problem when the number of the terminals is fixed. The theorem has since found numerous applications, both in graph theory and theoretical computer science. The original proof runs more than 400 pages and the techniques used are highly non-trivial. In this paper, we give a simplified algorithm for finding the decomposition based on a new constructive proof of the decomposition theorem for graphs excluding a fixed minor H. The new proof is both dramatically simpler and shorter, making these results and techniques more accessible. The algorithm runs in time O(n3), as does the original algorithm of Robertson and Seymour. Moreover, our proof gives an explicit bound on the constants in the O notation, whereas the original proof of Robertson and Seymour does not.

The structure of graphs not admitting a fixed immersion

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.

K6 minors in large 6-connected graphs

Jorgensen conjectured that every 6-connected graph with no K_6 minor has a vertex whose deletion makes the graph planar. We prove the conjecture for all sufficiently large graphs.

Finite connectivity in infinite matroids

We introduce a connectivity function for infinite matroids with properties similar to the connectivity function of a finite matroid, such as submodularity and invariance under duality. As an application we use it to extend Tuttes Linking Theorem to finitary and cofinitary matroids.

Packing cycles with modularity constraints

We prove that for all positive integers k, there exists an integer N =N(k) such that the following holds. Let G be a graph and let Γ an abelian group with no element of order two. Let γ: E(G)→Γ be a function from the edges of G to the elements of Γ. A non-zero cycle is a cycle C such that Σe∈E(C)γ(e) ≠ 0 where 0 is the identity element of Γ. Then G either contains k vertex disjoint non-zero cycles or there exists a set X ⊆ V (G) with |X| ≤N(k) such that G−X contains no non-zero cycle.An immediate consequence is that for all positive odd integers m, a graph G either contains k vertex disjoint cycles of length not congruent to 0 mod m, or there exists a set X of vertices with |X| ≤ N(k) such that every cycle of G-X has length congruent to 0 mod m. No such value N(k) exists when m is allowed to be even, as examples due to Reed and Thomassen show.

The Graph Minor Algorithm with Parity Conditions

We generalize the seminal Graph Minor algorithm of Robertson and Seymour to the parity version. We give polynomial time algorithms for the following problems:\begin{enumerate}\itemthe parity