103 research outputs found
Optimal Acyclic Hamiltonian Path Completion for Outerplanar Triangulated st-Digraphs (with Application to Upward Topological Book Embeddings)
Given an embedded planar acyclic digraph G, we define the problem of "acyclic
hamiltonian path completion with crossing minimization (Acyclic-HPCCM)" to be
the problem of determining an hamiltonian path completion set of edges such
that, when these edges are embedded on G, they create the smallest possible
number of edge crossings and turn G to a hamiltonian digraph. Our results
include:
--We provide a characterization under which a triangulated st-digraph G is
hamiltonian.
--For an outerplanar triangulated st-digraph G, we define the st-polygon
decomposition of G and, based on its properties, we develop a linear-time
algorithm that solves the Acyclic-HPCCM problem with at most one crossing per
edge of G.
--For the class of st-planar digraphs, we establish an equivalence between
the Acyclic-HPCCM problem and the problem of determining an upward 2-page
topological book embedding with minimum number of spine crossings. We infer
(based on this equivalence) for the class of outerplanar triangulated
st-digraphs an upward topological 2-page book embedding with minimum number of
spine crossings and at most one spine crossing per edge.
To the best of our knowledge, it is the first time that edge-crossing
minimization is studied in conjunction with the acyclic hamiltonian completion
problem and the first time that an optimal algorithm with respect to spine
crossing minimization is presented for upward topological book embeddings
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
Convexity in partial cubes: the hull number
We prove that the combinatorial optimization problem of determining the hull
number of a partial cube is NP-complete. This makes partial cubes the minimal
graph class for which NP-completeness of this problem is known and improves
some earlier results in the literature.
On the other hand we provide a polynomial-time algorithm to determine the
hull number of planar partial cube quadrangulations.
Instances of the hull number problem for partial cubes described include
poset dimension and hitting sets for interiors of curves in the plane.
To obtain the above results, we investigate convexity in partial cubes and
characterize these graphs in terms of their lattice of convex subgraphs,
improving a theorem of Handa. Furthermore we provide a topological
representation theorem for planar partial cubes, generalizing a result of
Fukuda and Handa about rank three oriented matroids.Comment: 19 pages, 4 figure
Strip Planarity Testing of Embedded Planar Graphs
In this paper we introduce and study the strip planarity testing problem,
which takes as an input a planar graph and a function and asks whether a planar drawing of exists
such that each edge is monotone in the -direction and, for any
with , it holds . The problem has strong
relationships with some of the most deeply studied variants of the planarity
testing problem, such as clustered planarity, upward planarity, and level
planarity. We show that the problem is polynomial-time solvable if has a
fixed planar embedding.Comment: 24 pages, 12 figures, extended version of 'Strip Planarity Testing'
(21st International Symposium on Graph Drawing, 2013
Monotone Grid Drawings of Planar Graphs
A monotone drawing of a planar graph is a planar straight-line drawing of
where a monotone path exists between every pair of vertices of in some
direction. Recently monotone drawings of planar graphs have been proposed as a
new standard for visualizing graphs. A monotone drawing of a planar graph is a
monotone grid drawing if every vertex in the drawing is drawn on a grid point.
In this paper we study monotone grid drawings of planar graphs in a variable
embedding setting. We show that every connected planar graph of vertices
has a monotone grid drawing on a grid of size , and such a
drawing can be found in O(n) time
05191 Abstracts Collection -- Graph Drawing
From 08.05.05 to 13.05.05, the Dagstuhl Seminar 05191 ``Graph Drawing\u27\u27 was held
in the International Conference and Research Center (IBFI), Schloss Dagstuhl.
During the seminar, several participants presented their current
research, and ongoing work and open problems were discussed. Abstracts of
the presentations given during the seminar as well as abstracts of
seminar results and ideas are put together in this paper. The first section
describes the seminar topics and goals in general.
Links to extended abstracts or full papers are provided, if available
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