10 research outputs found
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
Parameterized Algorithms for Queue Layouts
An -queue layout of a graph consists of a linear order of its vertices
and a partition of its edges into queues, such that no two independent
edges of the same queue nest. The minimum such that admits an -queue
layout is the queue number of . We present two fixed-parameter tractable
algorithms that exploit structural properties of graphs to compute optimal
queue layouts. As our first result, we show that deciding whether a graph
has queue number and computing a corresponding layout is fixed-parameter
tractable when parameterized by the treedepth of . Our second result then
uses a more restrictive parameter, the vertex cover number, to solve the
problem for arbitrary .Comment: Appears in the Proceedings of the 28th International Symposium on
Graph Drawing and Network Visualization (GD 2020
On Families of Planar DAGs with Constant Stack Number
A -stack layout (or -page book embedding) of a graph consists of a
total order of the vertices, and a partition of the edges into sets of
non-crossing edges with respect to the vertex order. The stack number of a
graph is the minimum such that it admits a -stack layout.
In this paper we study a long-standing problem regarding the stack number of
planar directed acyclic graphs (DAGs), for which the vertex order has to
respect the orientation of the edges. We investigate upper and lower bounds on
the stack number of several families of planar graphs: We prove constant upper
bounds on the stack number of single-source and monotone outerplanar DAGs and
of outerpath DAGs, and improve the constant upper bound for upward planar
3-trees. Further, we provide computer-aided lower bounds for upward (outer-)
planar DAGs
Four Pages Are Indeed Necessary for Planar Graphs
An embedding of a graph in a book consists of a linear order of its vertices
along the spine of the book and of an assignment of its edges to the pages of
the book, so that no two edges on the same page cross. The book thickness of a
graph is the minimum number of pages over all its book embeddings. Accordingly,
the book thickness of a class of graphs is the maximum book thickness over all
its members. In this paper, we address a long-standing open problem regarding
the exact book thickness of the class of planar graphs, which previously was
known to be either three or four. We settle this problem by demonstrating
planar graphs that require four pages in any of their book embeddings, thus
establishing that the book thickness of the class of planar graphs is four
Directed Acyclic Outerplanar Graphs Have Constant Stack Number
The stack number of a directed acyclic graph is the minimum for which
there is a topological ordering of and a -coloring of the edges such
that no two edges of the same color cross, i.e., have alternating endpoints
along the topological ordering. We prove that the stack number of directed
acyclic outerplanar graphs is bounded by a constant, which gives a positive
answer to a conjecture by Heath, Pemmaraju and Trenk [SIAM J. Computing, 1999].
As an immediate consequence, this shows that all upward outerplanar graphs have
constant stack number, answering a question by Bhore et al. [GD 2021] and
thereby making significant progress towards the problem for general upward
planar graphs originating from Nowakowski and Parker [Order, 1989]. As our main
tool we develop the novel technique of directed -partitions, which might be
of independent interest. We complement the bounded stack number for directed
acyclic outerplanar graphs by constructing a family of directed acyclic 2-trees
that have unbounded stack number, thereby refuting a conjecture by N\"ollenburg
and Pupyrev [arXiv:2107.13658, 2021]