7,579 research outputs found
Complete Graph Minors and the Graph Minor Structure Theorem
The graph minor structure theorem by Robertson and Seymour shows that every
graph that excludes a fixed minor can be constructed by a combination of four
ingredients: graphs embedded in a surface of bounded genus, a bounded number of
vortices of bounded width, a bounded number of apex vertices, and the
clique-sum operation. This paper studies the converse question: What is the
maximum order of a complete graph minor in a graph constructed using these four
ingredients? Our main result answers this question up to a constant factor
Shallow Minors, Graph Products and Beyond Planar Graphs
The planar graph product structure theorem of Dujmovi\'{c}, Joret, Micek,
Morin, Ueckerdt, and Wood [J. ACM 2020] states that every planar graph is a
subgraph of the strong product of a graph with bounded treewidth and a path.
This result has been the key tool to resolve important open problems regarding
queue layouts, nonrepetitive colourings, centered colourings, and adjacency
labelling schemes. In this paper, we extend this line of research by utilizing
shallow minors to prove analogous product structure theorems for several beyond
planar graph classes. The key observation that drives our work is that many
beyond planar graphs can be described as a shallow minor of the strong product
of a planar graph with a small complete graph. In particular, we show that
powers of planar graphs, -planar, -cluster planar, fan-planar and
-fan-bundle planar graphs have such a shallow-minor structure. Using a
combination of old and new results, we deduce that these classes have bounded
queue-number, bounded nonrepetitive chromatic number, polynomial -centred
chromatic numbers, linear strong colouring numbers, and cubic weak colouring
numbers. In addition, we show that -gap planar graphs have at least
exponential local treewidth and, as a consequence, cannot be described as a
subgraph of the strong product of a graph with bounded treewidth and a path
On Brambles, Grid-Like Minors, and Parameterized Intractability of Monadic Second-Order Logic
Brambles were introduced as the dual notion to treewidth, one of the most
central concepts of the graph minor theory of Robertson and Seymour. Recently,
Grohe and Marx showed that there are graphs G, in which every bramble of order
larger than the square root of the treewidth is of exponential size in |G|. On
the positive side, they show the existence of polynomial-sized brambles of the
order of the square root of the treewidth, up to log factors. We provide the
first polynomial time algorithm to construct a bramble in general graphs and
achieve this bound, up to log-factors. We use this algorithm to construct
grid-like minors, a replacement structure for grid-minors recently introduced
by Reed and Wood, in polynomial time. Using the grid-like minors, we introduce
the notion of a perfect bramble and an algorithm to find one in polynomial
time. Perfect brambles are brambles with a particularly simple structure and
they also provide us with a subgraph that has bounded degree and still large
treewidth; we use them to obtain a meta-theorem on deciding certain
parameterized subgraph-closed problems on general graphs in time singly
exponential in the parameter.
The second part of our work deals with providing a lower bound to Courcelle's
famous theorem, stating that every graph property that can be expressed by a
sentence in monadic second-order logic (MSO), can be decided by a linear time
algorithm on classes of graphs of bounded treewidth. Using our results from the
first part of our work we establish a strong lower bound for tractability of
MSO on classes of colored graphs
Are there any good digraph width measures?
Several different measures for digraph width have appeared in the last few
years. However, none of them shares all the "nice" properties of treewidth:
First, being \emph{algorithmically useful} i.e. admitting polynomial-time
algorithms for all \MS1-definable problems on digraphs of bounded width. And,
second, having nice \emph{structural properties} i.e. being monotone under
taking subdigraphs and some form of arc contractions. As for the former,
(undirected) \MS1 seems to be the least common denominator of all reasonably
expressive logical languages on digraphs that can speak about the edge/arc
relation on the vertex set.The latter property is a necessary condition for a
width measure to be characterizable by some version of the cops-and-robber game
characterizing the ordinary treewidth. Our main result is that \emph{any
reasonable} algorithmically useful and structurally nice digraph measure cannot
be substantially different from the treewidth of the underlying undirected
graph. Moreover, we introduce \emph{directed topological minors} and argue that
they are the weakest useful notion of minors for digraphs
- …