18 research outputs found
New Parameters for Beyond-Planar Graphs
Parameters for graphs appear frequently throughout the history of research in this field. They represent very important measures for the properties of graphs and graph drawings, and are often a main criterion for their classification and their aesthetic perception. In this direction, we provide new results for the following graph parameters:
– The segment complexity of trees;
– the membership of graphs of bounded vertex degree to certain graph classes;
– the maximal complete and complete bipartite graphs contained in certain graph classes beyond-planarity;
– the crossing number of graphs;
– edge densities for outer-gap-planar graphs and for bipartite gap-planar graphs with certain properties;
– edge densities and inclusion relationships for 2-layer graphs, as well as characterizations for complete bipartite graphs in the 2-layer setting
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
Min--planar Drawings of Graphs
The study of nonplanar drawings of graphs with restricted crossing
configurations is a well-established topic in graph drawing, often referred to
as beyond-planar graph drawing. One of the most studied types of drawings in
this area are the -planar drawings , where each edge cannot
cross more than times. We generalize -planar drawings, by introducing
the new family of min--planar drawings. In a min--planar drawing edges
can cross an arbitrary number of times, but for any two crossing edges, one of
the two must have no more than crossings. We prove a general upper bound on
the number of edges of min--planar drawings, a finer upper bound for ,
and tight upper bounds for . Also, we study the inclusion relations
between min--planar graphs (i.e., graphs admitting min--planar drawings)
and -planar graphs.Comment: Appears in the Proceedings of the 31st International Symposium on
Graph Drawing and Network Visualization (GD 2023
On RAC Drawings of Graphs with Two Bends per Edge
It is shown that every -vertex graph that admits a 2-bend RAC drawing in
the plane, where the edges are polylines with two bends per edge and any pair
of edges can only cross at a right angle, has at most edges for . This improves upon the previous upper bound of ; this is the first
improvement in more than 12 years. A crucial ingredient of the proof is an
upper bound on the size of plane multigraphs with polyline edges in which the
first and last segments are either parallel or orthogonal.Comment: Presented at the 31st International Symposium on Graph Drawing and
Network Visualization (GD 2023
Product structure of graph classes with strongly sublinear separators
We investigate the product structure of hereditary graph classes admitting
strongly sublinear separators. We characterise such classes as subgraphs of the
strong product of a star and a complete graph of strongly sublinear size. In a
more precise result, we show that if any hereditary graph class
admits separators, then for any fixed
every -vertex graph in is a subgraph
of the strong product of a graph with bounded tree-depth and a complete
graph of size . This result holds with if
we allow to have tree-depth . Moreover, using extensions of
classical isoperimetric inequalties for grids graphs, we show the dependence on
in our results and the above bound are
both best possible. We prove that -vertex graphs of bounded treewidth are
subgraphs of the product of a graph with tree-depth and a complete graph of
size , which is best possible. Finally, we investigate the
conjecture that for any hereditary graph class that admits
separators, every -vertex graph in is a
subgraph of the strong product of a graph with bounded tree-width and a
complete graph of size . We prove this for various classes
of interest.Comment: v2: added bad news subsection; v3: removed section "Polynomial
Expansion Classes" which had an error, added section "Lower Bounds", and
added a new author; v4: minor revisions and corrections