61 research outputs found
Surfaces have (asymptotic) dimension 2
The asymptotic dimension is an invariant of metric spaces introduced by
Gromov in the context of geometric group theory. When restricted to graphs and
their shortest paths metric, the asymptotic dimension can be seen as a large
scale version of weak diameter colorings (also known as weak diameter network
decompositions), i.e. colorings in which each monochromatic component has small
weak diameter.
In this paper, we prove that for any , the class of graphs excluding
as a minor has asymptotic dimension at most 2. This implies that the
class of all graphs embeddable on any fixed surface (and in particular the
class of planar graphs) has asymptotic dimension 2, which gives a positive
answer to a recent question of Fujiwara and Papasoglu. Our result extends from
graphs to Riemannian surfaces. We also prove that graphs of bounded pathwidth
have asymptotic dimension at most 1 and graphs of bounded layered pathwidth
have asymptotic dimension at most 2. We give some applications of our
techniques to graph classes defined in a topological or geometrical way, and to
graph classes of polynomial growth. Finally we prove that the class of bounded
degree graphs from any fixed proper minor-closed class has asymptotic dimension
at most 2. This can be seen as a large scale generalization of the result that
bounded degree graphs from any fixed proper minor-closed class are 3-colorable
with monochromatic components of bounded size. This also implies that
(infinite) Cayley graphs avoiding some minor have asymptotic dimension at most
2, which solves a problem raised by Ostrovskii and Rosenthal.Comment: 35 pages, 4 figures - v3: correction of the statements of Theorem
5.2, Corollary 5.3 and Theorem 5.9. Most of the results in this paper have
been merged to arXiv:2012.0243
Parameterized Complexity of Vertex Splitting to Pathwidth at most 1
Motivated by the planarization of 2-layered straight-line drawings, we
consider the problem of modifying a graph such that the resulting graph has
pathwidth at most 1. The problem Pathwidth-One Vertex Explosion (POVE) asks
whether such a graph can be obtained using at most vertex explosions, where
a vertex explosion replaces a vertex by deg degree-1 vertices, each
incident to exactly one edge that was originally incident to . For POVE, we
give an FPT algorithm with running time and an
kernel, thereby improving over the -kernel by Ahmed et al. [GD 22] in a
more general setting. Similarly, a vertex split replaces a vertex by two
distinct vertices and and distributes the edges originally incident
to arbitrarily to and . Analogously to POVE, we define the
problem variant Pathwidth-One Vertex Splitting (POVS) that uses the split
operation instead of vertex explosions. Here we obtain a linear kernel and an
algorithm with running time . This answers an open
question by Ahmed et al. [GD22].
Finally, we consider the problem Vertex Splitting (-VS), which
generalizes the problem POVS and asks whether a given graph can be turned into
a graph of a specific graph class using at most vertex splits. For
graph classes that can be tested in monadic second-order graph logic
(MSO), we show that the problem -VS can be expressed as an MSO
formula, resulting in an FPT algorithm for -VS parameterized by if
additionally has bounded treewidth. We obtain the same result for the
problem variant using vertex explosions
The product structure of squaregraphs
A squaregraph is a plane graph in which each internal face is a 4-cycle and each internal vertex has degree at least 4. This paper proves that every squaregraph is isomorphic to a subgraph of the semistrong product of an outerplanar graph and a path. We generalise this result for infinite squaregraphs, and show that this is best possible in the sense that “outerplanar graph” cannot be replaced by “forest”
Notes on Graph Product Structure Theory
It was recently proved that every planar graph is a subgraph of the strong
product of a path and a graph with bounded treewidth. This paper surveys
generalisations of this result for graphs on surfaces, minor-closed classes,
various non-minor-closed classes, and graph classes with polynomial growth. We
then explore how graph product structure might be applicable to more broadly
defined graph classes. In particular, we characterise when a graph class
defined by a cartesian or strong product has bounded or polynomial expansion.
We then explore graph product structure theorems for various geometrically
defined graph classes, and present several open problems.Comment: 19 pages, 0 figure
IST Austria Thesis
Algorithms in computational 3-manifold topology typically take a triangulation as an input and return topological information about the underlying 3-manifold. However, extracting the desired information from a triangulation (e.g., evaluating an invariant) is often computationally very expensive. In recent years this complexity barrier has been successfully tackled in some cases by importing ideas from the theory of parameterized algorithms into the realm of 3-manifolds. Various computationally hard problems were shown to be efficiently solvable for input triangulations that are sufficiently “tree-like.”
In this thesis we focus on the key combinatorial parameter in the above context: we consider the treewidth of a compact, orientable 3-manifold, i.e., the smallest treewidth of the dual graph of any triangulation thereof. By building on the work of Scharlemann–Thompson and Scharlemann–Schultens–Saito on generalized Heegaard splittings, and on the work of Jaco–Rubinstein on layered triangulations, we establish quantitative relations between the treewidth and classical topological invariants of a 3-manifold. In particular, among other results, we show that the treewidth of a closed, orientable, irreducible, non-Haken 3-manifold is always within a constant factor of its Heegaard genus
Clustered 3-Colouring Graphs of Bounded Degree
A (not necessarily proper) vertex colouring of a graph has "clustering"
if every monochromatic component has at most vertices. We prove that planar
graphs with maximum degree are 3-colourable with clustering
. The previous best bound was . This result for
planar graphs generalises to graphs that can be drawn on a surface of bounded
Euler genus with a bounded number of crossings per edge. We then prove that
graphs with maximum degree that exclude a fixed minor are 3-colourable
with clustering . The best previous bound for this result was
exponential in .Comment: arXiv admin note: text overlap with arXiv:1904.0479
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