926 research outputs found
Stack-number is not bounded by queue-number
We describe a family of graphs with queue-number at most 4 but unbounded
stack-number. This resolves open problems of Heath, Leighton and Rosenberg
(1992) and Blankenship and Oporowski (1999)
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
Graph product structure for non-minor-closed classes
Dujmovi\'c et al. (FOCS 2019) recently proved that every planar graph is a
subgraph of the strong product of a graph of bounded treewidth and a path.
Analogous results were obtained for graphs of bounded Euler genus or
apex-minor-free graphs. These tools have been used to solve longstanding
problems on queue layouts, non-repetitive colouring, -centered colouring,
and adjacency labelling. This paper proves analogous product structure theorems
for various non-minor-closed classes. One noteable example is -planar graphs
(those with a drawing in the plane in which each edge is involved in at most
crossings). We prove that every -planar graph is a subgraph of the
strong product of a graph of treewidth and a path. This is the first
result of this type for a non-minor-closed class of graphs. It implies, amongst
other results, that -planar graphs have non-repetitive chromatic number
upper-bounded by a function of . All these results generalise for drawings
of graphs on arbitrary surfaces. In fact, we work in a much more general
setting based on so-called shortcut systems that are of independent interest.
This leads to analogous results for map graphs, string graphs, graph powers,
and nearest neighbour graphs.Comment: v2 Cosmetic improvements and a corrected bound for
(layered-)(tree)width in Theorems 2, 9, 11, and Corollaries 1, 3, 4, 6, 12.
v3 Complete restructur
Characterisations and Examples of Graph Classes with Bounded Expansion
Classes with bounded expansion, which generalise classes that exclude a
topological minor, have recently been introduced by Ne\v{s}et\v{r}il and Ossona
de Mendez. These classes are defined by the fact that the maximum average
degree of a shallow minor of a graph in the class is bounded by a function of
the depth of the shallow minor. Several linear-time algorithms are known for
bounded expansion classes (such as subgraph isomorphism testing), and they
allow restricted homomorphism dualities, amongst other desirable properties. In
this paper we establish two new characterisations of bounded expansion classes,
one in terms of so-called topological parameters, the other in terms of
controlling dense parts. The latter characterisation is then used to show that
the notion of bounded expansion is compatible with Erd\"os-R\'enyi model of
random graphs with constant average degree. In particular, we prove that for
every fixed , there exists a class with bounded expansion, such that a
random graph of order and edge probability asymptotically almost
surely belongs to the class. We then present several new examples of classes
with bounded expansion that do not exclude some topological minor, and appear
naturally in the context of graph drawing or graph colouring. In particular, we
prove that the following classes have bounded expansion: graphs that can be
drawn in the plane with a bounded number of crossings per edge, graphs with
bounded stack number, graphs with bounded queue number, and graphs with bounded
non-repetitive chromatic number. We also prove that graphs with `linear'
crossing number are contained in a topologically-closed class, while graphs
with bounded crossing number are contained in a minor-closed class
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