50 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)
Stack and Queue Layouts via Layered Separators
It is known that every proper minor-closed class of graphs has bounded
stack-number (a.k.a. book thickness and page number). While this includes
notable graph families such as planar graphs and graphs of bounded genus, many
other graph families are not closed under taking minors. For fixed and ,
we show that every -vertex graph that can be embedded on a surface of genus
with at most crossings per edge has stack-number ;
this includes -planar graphs. The previously best known bound for the
stack-number of these families was , except in the case
of -planar graphs. Analogous results are proved for map graphs that can be
embedded on a surface of fixed genus. None of these families is closed under
taking minors. The main ingredient in the proof of these results is a
construction proving that -vertex graphs that admit constant layered
separators have stack-number.Comment: Appears in the Proceedings of the 24th International Symposium on
Graph Drawing and Network Visualization (GD 2016
Mixed Linear Layouts of Planar Graphs
A -stack (respectively, -queue) layout of a graph consists of a total
order of the vertices, and a partition of the edges into sets of
non-crossing (non-nested) edges with respect to the vertex ordering. In 1992,
Heath and Rosenberg conjectured that every planar graph admits a mixed
-stack -queue layout in which every edge is assigned to a stack or to a
queue that use a common vertex ordering.
We disprove this conjecture by providing a planar graph that does not have
such a mixed layout. In addition, we study mixed layouts of graph subdivisions,
and show that every planar graph has a mixed subdivision with one division
vertex per edge.Comment: Appears in the Proceedings of the 25th International Symposium on
Graph Drawing and Network Visualization (GD 2017
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
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
The Queue-Number of Posets of Bounded Width or Height
Heath and Pemmaraju conjectured that the queue-number of a poset is bounded
by its width and if the poset is planar then also by its height. We show that
there are planar posets whose queue-number is larger than their height,
refuting the second conjecture. On the other hand, we show that any poset of
width has queue-number at most , thus confirming the first conjecture in
the first non-trivial case. Moreover, we improve the previously best known
bounds and show that planar posets of width have queue-number at most
while any planar poset with and has queue-number at most its
width.Comment: 14 pages, 10 figures, Appears in the Proceedings of the 26th
International Symposium on Graph Drawing and Network Visualization (GD 2018