906 research outputs found
On Planar Greedy Drawings of 3-Connected Planar Graphs
A graph drawing is greedy if, for every ordered pair of vertices (x,y), there is a path from x to y such that the Euclidean distance to y decreases monotonically at every vertex of the path. Greedy drawings support a simple geometric routing scheme, in which any node that has to send a packet to a destination "greedily" forwards the packet to any neighbor that is closer to the destination than itself, according to the Euclidean distance in the drawing. In a greedy drawing such a neighbor always exists and hence this routing scheme is guaranteed to succeed.
In 2004 Papadimitriou and Ratajczak stated two conjectures related to greedy drawings. The greedy embedding conjecture states that every 3-connected planar graph admits a greedy drawing. The convex greedy embedding conjecture asserts that every 3-connected planar graph admits a planar greedy drawing in which the faces are delimited by convex polygons. In 2008 the greedy embedding conjecture was settled in the positive by Leighton and Moitra.
In this paper we prove that every 3-connected planar graph admits a planar greedy drawing. Apart from being a strengthening of Leighton and Moitra\u27s result, this theorem constitutes a natural intermediate step towards a proof of the convex greedy embedding conjecture
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
Euclidean Greedy Drawings of Trees
Greedy embedding (or drawing) is a simple and efficient strategy to route
messages in wireless sensor networks. For each source-destination pair of nodes
s, t in a greedy embedding there is always a neighbor u of s that is closer to
t according to some distance metric. The existence of greedy embeddings in the
Euclidean plane R^2 is known for certain graph classes such as 3-connected
planar graphs. We completely characterize the trees that admit a greedy
embedding in R^2. This answers a question by Angelini et al. (Graph Drawing
2009) and is a further step in characterizing the graphs that admit Euclidean
greedy embeddings.Comment: Expanded version of a paper to appear in the 21st European Symposium
on Algorithms (ESA 2013). 24 pages, 20 figure
Drawing Arrangement Graphs In Small Grids, Or How To Play Planarity
We describe a linear-time algorithm that finds a planar drawing of every
graph of a simple line or pseudoline arrangement within a grid of area
O(n^{7/6}). No known input causes our algorithm to use area
\Omega(n^{1+\epsilon}) for any \epsilon>0; finding such an input would
represent significant progress on the famous k-set problem from discrete
geometry. Drawing line arrangement graphs is the main task in the Planarity
puzzle.Comment: 12 pages, 8 figures. To appear at 21st Int. Symp. Graph Drawing,
Bordeaux, 201
Experimental Evaluation of Book Drawing Algorithms
A -page book drawing of a graph consists of a linear ordering of
its vertices along a spine and an assignment of each edge to one of the
pages, which are half-planes bounded by the spine. In a book drawing, two edges
cross if and only if they are assigned to the same page and their vertices
alternate along the spine. Crossing minimization in a -page book drawing is
NP-hard, yet book drawings have multiple applications in visualization and
beyond. Therefore several heuristic book drawing algorithms exist, but there is
no broader comparative study on their relative performance. In this paper, we
propose a comprehensive benchmark set of challenging graph classes for book
drawing algorithms and provide an extensive experimental study of the
performance of existing book drawing algorithms.Comment: Appears in the Proceedings of the 25th International Symposium on
Graph Drawing and Network Visualization (GD 2017
Some Results On Convex Greedy Embedding Conjecture for 3-Connected Planar Graphs
A greedy embedding of a graph into a metric space is a
function such that in the embedding for every pair of
non-adjacent vertices there exists another vertex adjacent
to which is closer to than . This notion of greedy
embedding was defined by Papadimitriou and Ratajczak (Theor. Comput. Sci.
2005), where authors conjectured that every 3-connected planar graph has a
greedy embedding (possibly planar and convex) in the Euclidean plane. Recently,
greedy embedding conjecture has been proved by Leighton and Moitra (FOCS 2008).
However, their algorithm do not result in a drawing that is planar and convex
for all 3-connected planar graph in the Euclidean plane. In this work we
consider the planar convex greedy embedding conjecture and make some progress.
We derive a new characterization of planar convex greedy embedding that given a
3-connected planar graph , an embedding x: V \to \bbbr^2 of is
a planar convex greedy embedding if and only if, in the embedding , weight
of the maximum weight spanning tree () and weight of the minimum weight
spanning tree (\func{MST}) satisfies \WT(T)/\WT(\func{MST}) \leq
(\card{V}-1)^{1 - \delta}, for some .Comment: 19 pages, A short version of this paper has been accepted for
presentation in FCT 2009 - 17th International Symposium on Fundamentals of
Computation Theor
Improved Compact Visibility Representation of Planar Graph via Schnyder's Realizer
Let be an -node planar graph. In a visibility representation of ,
each node of is represented by a horizontal line segment such that the line
segments representing any two adjacent nodes of are vertically visible to
each other. In the present paper we give the best known compact visibility
representation of . Given a canonical ordering of the triangulated , our
algorithm draws the graph incrementally in a greedy manner. We show that one of
three canonical orderings obtained from Schnyder's realizer for the
triangulated yields a visibility representation of no wider than
. Our easy-to-implement O(n)-time algorithm bypasses the
complicated subroutines for four-connected components and four-block trees
required by the best previously known algorithm of Kant. Our result provides a
negative answer to Kant's open question about whether is a
worst-case lower bound on the required width. Also, if has no degree-three
(respectively, degree-five) internal node, then our visibility representation
for is no wider than (respectively, ).
Moreover, if is four-connected, then our visibility representation for
is no wider than , matching the best known result of Kant and He. As a
by-product, we obtain a much simpler proof for a corollary of Wagner's Theorem
on realizers, due to Bonichon, Sa\"{e}c, and Mosbah.Comment: 11 pages, 6 figures, the preliminary version of this paper is to
appear in Proceedings of the 20th Annual Symposium on Theoretical Aspects of
Computer Science (STACS), Berlin, Germany, 200
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