12,424 research outputs found
Visualizing Co-Phylogenetic Reconciliations
We introduce a hybrid metaphor for the visualization of the reconciliations
of co-phylogenetic trees, that are mappings among the nodes of two trees. The
typical application is the visualization of the co-evolution of hosts and
parasites in biology. Our strategy combines a space-filling and a node-link
approach. Differently from traditional methods, it guarantees an unambiguous
and `downward' representation whenever the reconciliation is time-consistent
(i.e., meaningful). We address the problem of the minimization of the number of
crossings in the representation, by giving a characterization of planar
instances and by establishing the complexity of the problem. Finally, we
propose heuristics for computing representations with few crossings.Comment: This paper appears in the Proceedings of the 25th International
Symposium on Graph Drawing and Network Visualization (GD 2017
Low Ply Drawings of Trees
We consider the recently introduced model of \emph{low ply graph drawing}, in
which the ply-disks of the vertices do not have many common overlaps, which
results in a good distribution of the vertices in the plane. The
\emph{ply-disk} of a vertex in a straight-line drawing is the disk centered at
it whose radius is half the length of its longest incident edge. The largest
number of ply-disks having a common overlap is called the \emph{ply-number} of
the drawing.
We focus on trees. We first consider drawings of trees with constant
ply-number, proving that they may require exponential area, even for stars, and
that they may not even exist for bounded-degree trees. Then, we turn our
attention to drawings with logarithmic ply-number and show that trees with
maximum degree always admit such drawings in polynomial area.Comment: This is a complete access version of a paper that will appear in the
proceedings of GD201
Proximity Drawings of High-Degree Trees
A drawing of a given (abstract) tree that is a minimum spanning tree of the
vertex set is considered aesthetically pleasing. However, such a drawing can
only exist if the tree has maximum degree at most 6. What can be said for trees
of higher degree? We approach this question by supposing that a partition or
covering of the tree by subtrees of bounded degree is given. Then we show that
if the partition or covering satisfies some natural properties, then there is a
drawing of the entire tree such that each of the given subtrees is drawn as a
minimum spanning tree of its vertex set
On Vertex- and Empty-Ply Proximity Drawings
We initiate the study of the vertex-ply of straight-line drawings, as a
relaxation of the recently introduced ply number. Consider the disks centered
at each vertex with radius equal to half the length of the longest edge
incident to the vertex. The vertex-ply of a drawing is determined by the vertex
covered by the maximum number of disks. The main motivation for considering
this relaxation is to relate the concept of ply to proximity drawings. In fact,
if we interpret the set of disks as proximity regions, a drawing with
vertex-ply number 1 can be seen as a weak proximity drawing, which we call
empty-ply drawing. We show non-trivial relationships between the ply number and
the vertex-ply number. Then, we focus on empty-ply drawings, proving some
properties and studying what classes of graphs admit such drawings. Finally, we
prove a lower bound on the ply and the vertex-ply of planar drawings.Comment: Appears in the Proceedings of the 25th International Symposium on
Graph Drawing and Network Visualization (GD 2017
Strongly Monotone Drawings of Planar Graphs
A straight-line drawing of a graph is a monotone drawing if for each pair of
vertices there is a path which is monotonically increasing in some direction,
and it is called a strongly monotone drawing if the direction of monotonicity
is given by the direction of the line segment connecting the two vertices.
We present algorithms to compute crossing-free strongly monotone drawings for
some classes of planar graphs; namely, 3-connected planar graphs, outerplanar
graphs, and 2-trees. The drawings of 3-connected planar graphs are based on
primal-dual circle packings. Our drawings of outerplanar graphs are based on a
new algorithm that constructs strongly monotone drawings of trees which are
also convex. For irreducible trees, these drawings are strictly convex
Rectangular Layouts and Contact Graphs
Contact graphs of isothetic rectangles unify many concepts from applications
including VLSI and architectural design, computational geometry, and GIS.
Minimizing the area of their corresponding {\em rectangular layouts} is a key
problem. We study the area-optimization problem and show that it is NP-hard to
find a minimum-area rectangular layout of a given contact graph. We present
O(n)-time algorithms that construct -area rectangular layouts for
general contact graphs and -area rectangular layouts for trees.
(For trees, this is an -approximation algorithm.) We also present an
infinite family of graphs (rsp., trees) that require (rsp.,
) area.
We derive these results by presenting a new characterization of graphs that
admit rectangular layouts using the related concept of {\em rectangular duals}.
A corollary to our results relates the class of graphs that admit rectangular
layouts to {\em rectangle of influence drawings}.Comment: 28 pages, 13 figures, 55 references, 1 appendi
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