scholarly journals Straight-line Drawings of Binary Trees with Linear Area and Arbitrary Aspect Ratio

2004 ◽  
Vol 8 (2) ◽  
pp. 135-160 ◽  
Author(s):  
Ashim Garg ◽  
Adrian Rusu
Keyword(s):  
Aspect Ratio ◽  
Binary Trees ◽  
Line Drawings ◽  
Straight Line

AREA-EFFICIENT ORDER-PRESERVING PLANAR STRAIGHT-LINE DRAWINGS OF ORDERED TREES

2003 ◽  
Vol 13 (06) ◽  
pp. 487-505 ◽  
Author(s):  
ASHIM GARG ◽  
ADRIAN RUSU
Keyword(s):  
Aspect Ratio ◽  
Binary Tree ◽  
Binary Trees ◽  
Line Drawings ◽  
Ordered Trees ◽  
Grid Drawing ◽  
Aspect Ratios ◽  
Straight Line ◽  
Ordered Tree ◽  
Area Efficient

Ordered trees are generally drawn using order-preserving planar straight-line grid drawings. We investigate the area-requirements of such drawings and present several results. Let T be an ordered tree with n nodes. We show that: • T admits an order-preserving planar straight-line grid drawing with O(n log n) area. • If T is a binary tree, then T admits an order-preserving planar straight-line grid drawing with O(n log log n) area. • If T is a binary tree, then T admits an order-preserving upward planar straight-line grid drawing with optimalO(n log n) area. We also study the problem of drawing binary trees with user-specified aspect ratios. We show that an ordered binary tree T with n nodes admits an order-preserving planar straight-line grid drawing with area O(n log n), and any user-specified aspect ratio in the range [1,n/ log n]. All the drawings mentioned above can be constructed in O(n) time.

scholarly journals Analogies between the Crossing Number and the Tangle Crossing Number

10.37236/7581 ◽  
2018 ◽  
Vol 25 (4) ◽  
Author(s):  
Robin Anderson ◽  
Shuliang Bai ◽  
Fidel Barrera-Cruz ◽  
Éva Czabarka ◽  
Giordano Da Lozzo ◽  
...  
Keyword(s):  
Perfect Matching ◽  
Crossing Number ◽  
Binary Trees ◽  
Biologically Relevant ◽  
Line Drawings ◽  
Crossing Numbers ◽  
Parallel Lines ◽  
Straight Line ◽  
Minimum Number ◽  
Number Of Leaves

Tanglegrams are special graphs that consist of a pair of rooted binary trees with the same number of leaves, and a perfect matching between the two leaf-sets. These objects are of use in phylogenetics and are represented with straight-line drawings where the leaves of the two plane binary trees are on two parallel lines and only the matching edges can cross. The tangle crossing number of a tanglegram is the minimum number of crossings over all such drawings and is related to biologically relevant quantities, such as the number of times a parasite switched hosts.Our main results for tanglegrams which parallel known theorems for crossing numbers are as follows. The removal of a single matching edge in a tanglegram with $n$ leaves decreases the tangle crossing number by at most $n-3$, and this is sharp. Additionally, if $\gamma(n)$ is the maximum tangle crossing number of a tanglegram with $n$ leaves, we prove $\frac{1}{2}\binom{n}{2}(1-o(1))\le\gamma(n)<\frac{1}{2}\binom{n}{2}$. For an arbitrary tanglegram $T$, the tangle crossing number, $\mathrm{crt}(T)$, is NP-hard to compute (Fernau et al. 2005). We provide an algorithm which lower bounds $\mathrm{crt}(T)$ and runs in $O(n^4)$ time. To demonstrate the strength of the algorithm, simulations on tanglegrams chosen uniformly at random suggest that the tangle crossing number is at least $0.055n^2$ with high probabilty, which matches the result that the tangle crossing number is $\Theta(n^2)$ with high probability (Czabarka et al. 2017).

Acute Constraints in Straight-Line Drawings of Planar Graphs

2019 ◽  
Vol E102.A (9) ◽  
pp. 994-1001
Author(s):  
Akane SETO ◽  
Aleksandar SHURBEVSKI ◽  
Hiroshi NAGAMOCHI ◽  
Peter EADES
Keyword(s):  
Planar Graphs ◽  
Line Drawings ◽  
Straight Line

scholarly journals Partitioning Graph Drawings and Triangulated Simple Polygons into Greedily Routable Regions

2017 ◽  
Vol 27 (01n02) ◽  
pp. 121-158 ◽  
Author(s):  
Martin Nöllenburg ◽  
Roman Prutkin ◽  
Ignaz Rutter
Keyword(s):  
Routing Protocol ◽  
Graph Drawing ◽  
Geographic Routing ◽  
Np Hard ◽  
Line Drawings ◽  
Simple Polygons ◽  
Straight Line ◽  
Starting Point ◽  
Minimum Decomposition ◽  
Geographic Routing Protocol

A greedily routable region (GRR) is a closed subset of [Formula: see text], in which any destination point can be reached from any starting point by always moving in the direction with maximum reduction of the distance to the destination in each point of the path. Recently, Tan and Kermarrec proposed a geographic routing protocol for dense wireless sensor networks based on decomposing the network area into a small number of interior-disjoint GRRs. They showed that minimum decomposition is NP-hard for polygonal regions with holes. We consider minimum GRR decomposition for plane straight-line drawings of graphs. Here, GRRs coincide with self-approaching drawings of trees, a drawing style which has become a popular research topic in graph drawing. We show that minimum decomposition is still NP-hard for graphs with cycles and even for trees, but can be solved optimally for trees in polynomial time, if we allow only certain types of GRR contacts. Additionally, we give a 2-approximation for simple polygons, if a given triangulation has to be respected.

scholarly journals Distinct Distances in Graph Drawings

10.37236/831 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Paz Carmi ◽  
Vida Dujmović ◽  
Pat Morin ◽  
David R. Wood
Keyword(s):  
Maximum Degree ◽  
Combinatorial Geometry ◽  
Complete Graphs ◽  
Bounded Degree ◽  
Cartesian Products ◽  
Bounded Treewidth ◽  
Line Drawings ◽  
Straight Line ◽  
Minimum Number ◽  
Complete Bipartite Graphs

The distance-number of a graph $G$ is the minimum number of distinct edge-lengths over all straight-line drawings of $G$ in the plane. This definition generalises many well-known concepts in combinatorial geometry. We consider the distance-number of trees, graphs with no $K^-_4$-minor, complete bipartite graphs, complete graphs, and cartesian products. Our main results concern the distance-number of graphs with bounded degree. We prove that $n$-vertex graphs with bounded maximum degree and bounded treewidth have distance-number in ${\cal O}(\log n)$. To conclude such a logarithmic upper bound, both the degree and the treewidth need to be bounded. In particular, we construct graphs with treewidth $2$ and polynomial distance-number. Similarly, we prove that there exist graphs with maximum degree $5$ and arbitrarily large distance-number. Moreover, as $\Delta$ increases the existential lower bound on the distance-number of $\Delta$-regular graphs tends to $\Omega(n^{0.864138})$.

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