30 research outputs found
Odd-Minors I: Excluding small parity breaks
Given a graph class~, the -blind-treewidth of a
graph~ is the smallest integer~ such that~ has a tree-decomposition
where every bag whose torso does not belong to~ has size at
most~. In this paper we focus on the class~ of bipartite graphs
and the class~ of planar graphs together with the odd-minor
relation. For each of the two parameters, -blind-treewidth and
-blind-treewidth, we prove an analogue of the
celebrated Grid Theorem under the odd-minor relation. As a consequence we
obtain FPT-approximation algorithms for both parameters. We then provide
FPT-algorithms for \textsc{Maximum Independent Set} on graphs of bounded
-blind-treewidth and \textsc{Maximum Cut} on graphs of bounded
-blind-treewidth
Obstructions for bounded branch-depth in matroids
DeVos, Kwon, and Oum introduced the concept of branch-depth of matroids as a
natural analogue of tree-depth of graphs. They conjectured that a matroid of
sufficiently large branch-depth contains the uniform matroid or the
cycle matroid of a large fan graph as a minor. We prove that matroids with
sufficiently large branch-depth either contain the cycle matroid of a large fan
graph as a minor or have large branch-width. As a corollary, we prove their
conjecture for matroids representable over a fixed finite field and
quasi-graphic matroids, where the uniform matroid is not an option.Comment: 25 pages, 1 figur
A unified Erd\H{o}s-P\'{o}sa theorem for cycles in graphs labelled by multiple abelian groups
In 1965, Erd\H{o}s and P\'{o}sa proved that there is a duality between the
maximum size of a packing of cycles and the minimum size of a vertex set
hitting all cycles. Such a duality does not hold for odd cycles, and Dejter and
Neumann-Lara asked in 1988 to find all pairs of integers where
such a duality holds for the family of cycles of length modulo . We
characterise all such pairs, and we further generalise this characterisation to
cycles in graphs labelled with a bounded number of abelian groups, whose values
avoid a bounded number of elements of each group. This unifies almost all known
types of cycles that admit such a duality, and it also provides new results.
Moreover, we characterise the obstructions to such a duality in this setting,
and thereby obtain an analogous characterisation for cycles in graphs
embeddable on a fixed compact orientable surface.Comment: 37 pages, 2 figure
Product structure of graph classes with bounded treewidth
We show that many graphs with bounded treewidth can be described as subgraphs
of the strong product of a graph with smaller treewidth and a bounded-size
complete graph. To this end, define the "underlying treewidth" of a graph class
to be the minimum non-negative integer such that, for some
function , for every graph there is a graph with
such that is isomorphic to a subgraph of . We introduce disjointed coverings of graphs
and show they determine the underlying treewidth of any graph class. Using this
result, we prove that the class of planar graphs has underlying treewidth 3;
the class of -minor-free graphs has underlying treewidth (for ); and the class of -minor-free graphs has underlying
treewidth . In general, we prove that a monotone class has bounded
underlying treewidth if and only if it excludes some fixed topological minor.
We also study the underlying treewidth of graph classes defined by an excluded
subgraph or excluded induced subgraph. We show that the class of graphs with no
subgraph has bounded underlying treewidth if and only if every component of
is a subdivided star, and that the class of graphs with no induced
subgraph has bounded underlying treewidth if and only if every component of
is a star
Representations of Infinite Tree Sets
Tree sets are abstract structures that can be used to model various tree-shaped objects in combinatorics. Finite tree sets can be represented by finite graph-theoretical trees. We extend this representation theory to infinite tree sets. First we characterise those tree sets that can be represented by tree sets arising from infinite trees; these are precisely those tree sets without a chain of order type omega + 1. Then we introduce and study a topological generalisation of infinite trees which can have limit edges, and show that every infinite tree set can be represented by the tree set admitted by a suitable such tree-like space11Nsciescopu