367 research outputs found
Unavoidable parallel minors of regular matroids
This is the post-print version of the Article - Copyright @ 2011 ElsevierWe prove that, for each positive integer k, every sufficiently large 3-connected regular matroid has a parallel minor isomorphic to M (K_{3,k}), M(W_k), M(K_k), the cycle matroid of the graph obtained from K_{2,k} by adding paths through the vertices of each vertex class, or the cycle matroid of the graph obtained from K_{3,k} by adding a complete graph on the vertex class with three vertices.This study is partially supported by a grant from the National Security Agency
Clustered Colouring in Minor-Closed Classes
The "clustered chromatic number" of a class of graphs is the minimum integer
such that for some integer every graph in the class is -colourable
with monochromatic components of size at most . We prove that for every
graph , the clustered chromatic number of the class of -minor-free graphs
is tied to the tree-depth of . In particular, if is connected with
tree-depth then every -minor-free graph is -colourable with
monochromatic components of size at most . This provides the first
evidence for a conjecture of Ossona de Mendez, Oum and Wood (2016) about
defective colouring of -minor-free graphs. If then we prove that 4
colours suffice, which is best possible. We also determine those minor-closed
graph classes with clustered chromatic number 2. Finally, we develop a
conjecture for the clustered chromatic number of an arbitrary minor-closed
class
Distributed Dominating Set Approximations beyond Planar Graphs
The Minimum Dominating Set (MDS) problem is one of the most fundamental and
challenging problems in distributed computing. While it is well-known that
minimum dominating sets cannot be approximated locally on general graphs, over
the last years, there has been much progress on computing local approximations
on sparse graphs, and in particular planar graphs.
In this paper we study distributed and deterministic MDS approximation
algorithms for graph classes beyond planar graphs. In particular, we show that
existing approximation bounds for planar graphs can be lifted to bounded genus
graphs, and present (1) a local constant-time, constant-factor MDS
approximation algorithm and (2) a local -time
approximation scheme. Our main technical contribution is a new analysis of a
slightly modified variant of an existing algorithm by Lenzen et al.
Interestingly, unlike existing proofs for planar graphs, our analysis does not
rely on direct topological arguments.Comment: arXiv admin note: substantial text overlap with arXiv:1602.0299
Ramsey Theory Using Matroid Minors
This thesis considers a Ramsey Theory question for graphs and regular matroids. Specifically, how many elements N are required in a 3-connected graphic or regular matroid to force the existence of certain specified minors in that matroid? This question cannot be answered for an arbitrary collection of specified minors. However, there are results from the literature for which the number N exists for certain collections of minors. We first encode totally unimodular matrix representations of certain matroids. We use the computer program MACEK to investigate this question for certain classes of specified minors
Universal graphs with forbidden wheel minors
Let be any wheel graph and the class of all countable
graphs not containing as a minor. We show that there exists a graph in
which contains every graph in as an induced
subgraph.Comment: 14 pages, 2 figure
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