Excluding a group-labelled graph

This paper contains a first step towards extending the Graph Minors Project of Robertson and Seymour to group-labelled graphs. For a finite abelian group Γ and Γ-labelled graph G, we describe the class of Γ-labelled graphs that do not contain a minor isomorphic to G

Tangles, tree-decompositions, and grids in matroids

A tangle in a matroid is an obstruction to small branch-width. In particular, the maximum order of a tangle is equal to the branch-width. We prove that: (i) there is a tree-decomposition of a matroid that “displays” all of the maximal tangles, and (ii) when M is representable over a finite field, each tangle of sufficiently large order “dominates” a large grid-minor. This extends results of Robertson and Seymour concerning Graph Minors

Quasi-graphic matroids

Frame matroids and lifted-graphic matroids are two interesting generalizations of graphic matroids. Here we introduce a new generalization, quasi-graphic matroids, that unifies these two existing classes. Unlike frame matroids and lifted-graphic matroids, it is easy to certify that a matroid is quasi-graphic. The main result of the paper is that every 3-connected representable quasi-graphic matroid is either a lifted-graphic matroid or a rame matroid

Branch-width and well-quasi-ordering in matroids and graphs

AbstractWe prove that a class of matroids representable over a fixed finite field and with bounded branch-width is well-quasi-ordered under taking minors. With some extra work, the result implies Robertson and Seymour's result that graphs with bounded tree-width (or equivalently, bounded branch-width) are well-quasi-ordered under taking minors. We will not only derive their result from our result on matroids, but we will also use the main tools for a direct proof that graphs with bounded branch-width are well-quasi-ordered under taking minors. This proof also provides a model for the proof of the result on matroids, with all specific matroid technicalities stripped off

On inequivalent representations of matroids over non-prime fields

For each finite field $F$ of prime order there is a constant $c$ such that every 4-connected matroid has at most $c$ inequivalent representations over $F$. We had hoped that this would extend to all finite fields, however, it was not to be. The $(m,n)$-mace is the matroid obtained by adding a point freely to $M(K_{m,n})$. For all $n \geq 3$, the $(3,n)$-mace is 4-connected and has at least $2n$ representations over any field $F$ of non-prime order $q \geq 9$. More generally, for $n \geq m$, the $(m,n)$-mace is vertically $(m+1)$-connected and has at least $2n$ inequivalent representations over any finite field of non-prime order $q\geq m^m$

Node-balancing by edge-increments

Suppose you are given a graph $G=(V,E)$ with a weight assignment $w:V\rightarrow\mathbb{Z}$ and that your objective is to modify $w$ using legal steps such that all vertices will have the same weight, where in each legal step you are allowed to choose an edge and increment the weights of its end points by $1$. In this paper we study several variants of this problem for graphs and hypergraphs. On the combinatorial side we show connections with fundamental results from matching theory such as Hall's Theorem and Tutte's Theorem. On the algorithmic side we study the computational complexity of associated decision problems. Our main results are a characterization of the graphs for which any initial assignment can be balanced by edge-increments and a strongly polynomial-time algorithm that computes a balancing sequence of increments if one exists.Comment: 10 page

Claw-free t-perfect graphs can be recognised in polynomial time

A graph is called t-perfect if its stable set polytope is defined by non-negativity, edge and odd-cycle inequalities. We show that it can be decided in polynomial time whether a given claw-free graph is t-perfect

On the odd-minor variant of Hadwiger's conjecture

A {\it $K_l$ -expansion} consists of $l$ vertex-disjoint trees, every two of which are joined by an edge. We call such an expansion {\it odd} if its vertices can be two-coloured so that the edges of the trees are bichromatic but the edges between trees are monochromatic. We show that, for every $l$, if a graph contains no odd $K_l$ -expansion then its chromatic number is $O(l \sqrt{\log l})$. In doing so, we obtain a characterization of graphs which contain no odd $K_l$ -expansion which is of independent interest. We also prove that given a graph and a subset $S$ of its vertex set, either there are $k$ vertex-disjoint odd paths with endpoints in $S$, or there is a set X of at most $2k − 2$ vertices such that every odd path with both ends in $S$ contains a vertex in $X$. Finally, we discuss the algorithmic implications of these results