471 research outputs found
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 of prime order there is a constant such that every 4-connected matroid has at most inequivalent representations over . We had hoped that this would extend to all finite fields, however, it was not to be. The -mace is the matroid obtained by adding a point freely to . For all , the -mace is 4-connected and has at least representations over any field of non-prime order . More generally, for , the -mace is vertically -connected and has at least inequivalent representations over any finite field of non-prime order
Node-balancing by edge-increments
Suppose you are given a graph with a weight assignment
and that your objective is to modify 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 .
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 -expansion} consists of 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 , if a graph contains no odd -expansion then its chromatic number is . In doing so, we obtain a characterization of graphs which contain no odd -expansion which is of independent interest. We also prove that given a graph and a subset of its vertex set, either there are vertex-disjoint odd paths with endpoints in , or there is a set X of at most vertices such that every odd path with both ends in contains a vertex in . Finally, we discuss the algorithmic implications of these results
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